So, we have now come to the end of the course. This is going to be the my last lecture trying to wind up whatever we had started about the electric field and magnetic field. In our last lecture, we had taken one example and discussed the electric field and magnetic field transformation. We had said that what is what appears to be a purely a magnetic field or an electric field in a frame may appear to be combination of electric field and magnetic field in a different frame. So, whether you term a field as electric or magnetic or partly electric or partly magnetic could depend on the frame of reference from which you are observing. Today, what we will try to do is try to analyze this situation little bit more and try to see how we can think about the presence of these fields or change these fields by taking one simple example of a current carrying wire or current carrying conductor. Then eventually we will show I mean I will not really be able to show because you know this is actually the beyond the scope of this particular lecture series that Maxwell's equations which are supposed to be the basic equations in electrodynamics they remain unaltered even after special theory of relativity. So, there those equations do not change once the special theory of relativity is also introduced. So, they are invariant under special theory of relativity. So, as I said this particular aspect we will not be able to prove as it is beyond the scope of this thing in this particular course, but we will just take one equation and try to sort of convince I would say try to convince that for one equation we will sort of say that you know you can sort of see that Maxwell's equation is expected to be obeyed. So, this is what we are going to do in our last lecture today. So, this is what we have said we recapitulate we discussed an example using electric and magnetic field transformation in our last lecture. To get a physical feeling how the fields could change in different frames we discuss a new concept today which we call as a current density 4 vector. As I said we want to discuss little bit more physically by taking current carrying conductor, but before we come to that situation we will define what we call as a current density 4 vector. Before we do that let us first take traditionally what we mean by a current density. In physics generally current density is considered little more fundamental than the current and current density is always a vector quantity. Because this current concept of current density emerges more from the fundamental aspects of how the motion of charge carriers takes place in a particular wire or by a particular conductor. So, let us assume that we have n charge carriers per unit volume in a given conductor or given wire which is carrying current. Now of course we know that it is actually the electrons which carry the current which are negatively charged particle. But somehow traditionally the direction of the current has been always defined to be the opposite direction of the flow of the electron. So, if the electrons sort of flow in the negative x direction we define the standard traditional current in the positive x direction. So, as we said let us assume that there are n charge carriers per unit volume and each one of them is carrying a charge of plus e. So, I am just assuming that the charge carriers are positive though as I have said that actually the charge carriers in a real conductor is supposed to be electrons which are negative charge carriers. But just for defining the concept of current density we I take this as a positive charge carriers. Now we evolve the concept of what we call as a drift velocity. Actually once we apply electric field these charge carriers start accelerating under the influence of the force which is created by this particular electric field. But because of this scattering that these electrons or these charge carriers whatever they are face within a metal or within a conductor they are not able to move too much ahead they go little bit ahead and then they get scattered then they again accelerate and then they get scattered. And if you remember what we can define that under the influence of electric field these electrons will eventually develop what we call as a drift velocity. It is something like the motion of air breeze. The molecules in air may be moving with very large speeds but we do not feel it. Only when the overall air drifts which may be drifting with a very small speed in comparison to the actual speed of the molecules that we feel that there is a breeze. So, that is what is the drift of this whole air. So, similarly is the concept of electrons. The electrons could be moving with very large speed inside a conductor. But if there is one electron going this way another electron is going this way. So, eventually you do not see any drift. Only when there is a net flow of electron in a particular direction then only we see the current and that happens when we apply an electric field and then we define what we call as a drift velocity. Now current density vector J is defined in terms of this drift velocity as follows J is equal to n e times drift velocity whereas as I have said n is the total number of charge carriers per unit volume e is the charge of each of the carrier which I am assuming at the moment to be positive. Actually it makes no difference if you take per electron we have to take care of science properly. Otherwise this equation is the equation which defines the current density inside a metal or inside a conductor. Now if this situation is viewed from a different frame then obviously you expect that the velocity of the electrons would not be same because we have discussed number of times that the velocity is actually frame dependent quantity. So, if I go to a different frame the drift velocity that you are seeing in a given frame may turn out to be different. It is very well expected. Also I would like to emphasize that even the current density or charge density not sorry current density the charge density would be different and that is mainly because of the length contraction because once we have come to the case of relativity theory you will find that there is a contraction of length and that itself would change the charge density. And if charge density change and the velocity drift velocity change so when I look into a different frame of reference the current density is also expected to change. Therefore I must look on in relativity how this current density will change if I change my frame of reference. That is what is the question that we are going to answer now. This is what I said let us examine this concept from the point of view of relativity. For that particular thing let us assume comparatively simpler picture. Let us not bother about the drift velocities and other things. Let us just take comparatively very very simple picture and try to see how these current densities will transform if I go from one particular frame of reference to another frame of reference. So, let us assume that we have a volume which contains only positive charges each one of which has a value of q which moves with a velocity u. So, let us assume that all of them move in the same velocity because there is no question of drift here. I do not want to confuse. We will talk about drift little later. So, let us assume that they all of them are moving with the same velocity u as seen in a given frame S. Obviously, when these charges move in a particular direction this will constitute current or eventually current density. Let N be the number of charge carriers per unit volume as seen in this frame. We have just now said that this N is also likely to be frame dependent. I will discuss this point little more in detail little later. So, we have a frame as we have been always telling we have to be consistent in our frame. So, there is an observer sitting in S frame which observes the velocity of the charge carriers to be u and once he calculates the total charge number density or charge density. If you let us talk of number density then that is the number of charge carriers per unit volume. He observes that number to be equal to N. Now, let us define a 4 scalar. We have introduced the concept of 4 scalar earlier. This is the quantity which does not change when we change our frame of reference. See like we have defined when we from velocity to momentum. At that time we multiplied by a 4 scalar which we called as the rest mass of the particle. Similarly, here we define a 4 scalar N naught which does not change in your frame of reference. This of course is going to be different from N and this is the number density that is the number of charge carriers per unit volume in a frame of reference in which these charges are at rest. See like rest mass we define that this is the mass what we have called in a frame of reference when the particular mass is at rest. Take analogy from that particular concept. Similarly, we define a number density, total number of charge carriers per unit volume as evaluated in a frame of reference in which these charges are at rest. This N that a person is going to find out in S is going to be different from N naught as we will see just now. Now, the charge density in the frame S is higher from the proper one. This number N I can call as a proper number density and not sorry this N naught we can call as a proper number density because this is evaluated in a frame of reference in which these charges are at rest. Now, what I insist that the N that I am evaluating in S, S is a frame in which the charges are actually moving. In this frame the number density would be different from its proper density N naught mainly because of the length contraction and this length contraction always occurs along the relative velocity direction. So, let us just look into this particular picture. Let us suppose we have a wire or a conductor like this and let us assume that this is x direction. Now, there are charge carriers here. Let us say negative charge carriers whatever are here we have assumed positive. So, let us assume positive. Let us assume that they are moving in a particular direction x. Now, if I view this particular thing from a different frame of reference then in that particular case and let us assume that this particular frame of reference let us say S prime or whatever is that particular frame of reference views a person sitting in that frame views these charge carriers then that particular person would find that this length is contracted. I am assuming that this particular frame is actually moving in the same direction as x. So, we know the standard length contraction formula. So, according to this person the length will turn out to be smaller, other dimensions will not change and this length will become smaller by a factor of gamma u. New length or the length as seen in different frame of reference will be gamma u times the original or the proper length. Therefore, the volume of this particular material will also go down by gamma u because no other dimensions change. So, if you have let us say you know the dimensions are a, b, c then only one of them becomes gamma a and b and c remain same. So, when I take the volume, volume goes down by a factor of gamma u and therefore charge density the total number of charge carriers per unit volume go up by a factor of gamma u. So, that is what I say that if I have come to this particular frame of reference s in which I find that these charges are moving then this number density that will be evaluated in frame will be different from its proper number density because of the length contraction and therefore this particular factor will get modified or get multiplied by a factor of gamma u. Here is the current density if I have to write in s frame. I have to write the total number of charge carriers per unit volume which I know I have earlier written as n will actually be gamma u times n naught where n naught is actually a full scalar. This factor is n and this n is gamma u times n naught using the definition of charge density that we have used earlier. This equation we can write that current density is total number of charge carriers per unit volume which is gamma u times n naught multiplied by h q multiplied by the velocity I am assuming all of them are moving in the same velocity. So, this will be the current density as seen in this particular frame of reference. So, the only way of writing only reason why I am writing this equation in this particular form because I am using a full scalar here. If I would have written just n this n would have changed if I want to different frame of reference while this n naught is going to be same in all the frames. Now, instead of number density sometimes we define charge density. This charge density is very simple just multiply whatever is n naught multiplied by the charge which is q. So, this is a very simple definition most of the time we talk in terms of charge density rather number density. So, we define the charge density rho and like we have defined a proper number density we can define a proper charge density rho naught as follows. So, rho naught whatever was the number density multiplied by charge is very simple just charge density the total number multiplied by charge. Similarly, the charge density in a particular frame is the total number density as seen in that frame multiplied by q. So, this rho is equal to n q and rho naught which is the proper charge density is n naught q and because as we have discussed n is equal to gamma u n naught. So, this particular factor rho naught this factor this equation rho can be written as gamma u rho naught because this is n naught q this n I can write as gamma u n naught n naught if I absorb here this becomes rho naught. So, this becomes gamma u rho naught. So, rho charge density in a particular frame of reference can be written as gamma u rho naught and only thing which I insist that this rho naught is also a 4 scalar because n naught is a 4 scalar and q the charge in relativity we do not expect this to change once we change the frame of reference. At this point we have not specifically mentioned earlier, but let us emphasize it now the charges do not change once we change the frame of reference. Therefore, this quantity rho naught is also a 4 scalar. So, we can write the current density now in terms of this equation gamma j is gamma u rho naught u this is what we have written here earlier except this gamma u n naught this n naught q's I am sorry have been written in terms of rho naught that is all I have done in this equation. This is j is equal to gamma u rho naught u now I am in a position to define current density 4 vector. If you remember in one of our earlier lectures we have talked about the velocity 4 vector we have said that u x, u y, u z they are not the components of the 4 vector or the velocity 4 vector. So, remember p x, p y, p z are the first 3 components of the momentum 4 vector, but u x, u y, u z are not the components or not the first 3 components of velocity 4 vector. In fact, what are actually the first 3 components are gamma u u x, gamma u u y, gamma u u z this aspect we had discussed much more in detail in when we are discussing the velocity 4 vector. So, this velocity 4 vector the components are given as gamma u times u x this is the first component, second component is u y, third component is u z this is the I c all have to be multiplied by gamma u.  This is what we have defined earlier. Now, I can use this particular thing to define charge density 4 vector all I have to do is to multiply by a 4 scalar which I can do by multiplying by rho naught because we have earlier said that rho naught is actually 4 scalar. So, we define the components of current density 4 vector by multiplying this u 4 vector by a 4 scalar which is rho naught. So, this has been multiplied by rho naught all other components are exactly same only thing this has been multiplied by rho naught. Now, we realize if you look at back at the equation of current density this I can write component wise if I write component wise it will become j x whatever x component will become j x gamma u rho naught u x the y component of this equation becomes j y is equal to gamma u rho naught u y things like that.  So, what we see that the first three components of current density 4 vector are actually the real current density as seen in that particular frame it is like momentum, momentum 4 vector the first three components are the momentum as observed in a particular frame of reference. Similarly, current density components are the first three components of the current density 4 vector and the fourth component like in the case of momentum 4 vector dependent on the energy of the system here the fourth component of the current density 4 vector depends on the charge density. So, the fourth component is rho see like in that particular case of momentum it was I e upon c the fourth component dependent on the energy here the fourth component depends on the charge density. Now, once we have found this particular 4 vector we know how to transform it if I go from this particular frame of s to any other frame of s and if I know the relative velocity between s and s to be v I can always transform this particular 4 vector components into any other frame by using the standard matrix equation which we have been doing earlier a number of times. So, the current density will obey the following transformation rule if we go to another frame s and this is the current density transformation. So, we have in a s frame these are the components the current density is J x J y J z these are the x y and z components of the current density the fourth component is I rho upon c where rho is the charge density as seen in this particular frame of reference I want to find out the current density and the charge density in a different frame s frame of reference and let those current densities begin be given by or the components of the current densities in s be given by J x j y j z and let the charge density be given by rho then these J x prime J y prime J z prime rho prime would depend on J x J y J z and rho by this particular matrix equation transformation equation where of course this gamma beta all these depends on the v the relative velocity between the frame between s and s prime the standard way we have been dealing in relativity so far. So, now I can expand this particular thing we have done it a number of times so we will not spend too much of time in expanding this particular thing I will just write the expansion of this particular equation immediately in the next transparency this is what it becomes J x prime becomes equal to gamma times J x minus v rho J y prime remains equal to J y J z prime remains equal to J z and the charge density now will be different in a different frame s prime as given by this particular expression rho prime will be gamma times rho minus v J x upon c square this is somewhat like your x y z if you want to remember somewhat like x y z t transformation x prime turned out to be gamma x minus v t so it is a similar type of equations if you go back in that particular case J x prime is gamma times J x minus v rho and rho prime is rho gamma minus v upon c square J x I can always write the inverse transformation equation for that particular thing as we have said change prime to unprimed quantities and vice versa make v minus v so this what happens as the inverse transformation it means if I know the current densities and the charge densities in s prime frame of reference I can find out the current density and the charge density in s frame of reference by using this these inverse transformation equations so this is what has happened to my transformation of current density now let us come back to the situation of current carrying conductor my idea of using this particular example of giving current carrying conductor is only to look little more little more deeply into the origin of these electric fields and magnetic fields as we change from one frame to another frame of reference and this particular example generally reasonably illustrative because many times we have a question that suppose there is a particular frame of reference in which we see only magnetic field and you say that you go to a different frame of reference now in addition to this magnetic field you also have an electric field so from where this electric field has come because once we say that equations the Maxwell's equations which are the basic equations do not change in relativity so what is the origin of these electric field so let us what we are trying to do by giving this particular example to go little deeper into this particular aspect try to somewhat understand this thing so now let us come to a realistic conductor and let us assume that it has it is carrying a current in plus x direction as seen in frame s so there is a frame s in which you see a current flowing in a particular wire or conductor along the x direction let us assume that charge carriers are actually the electrons it means actually the electrons must be drifting in minus x direction and that is the reason that the current is actually in the plus x direction as we have said the direction of the motion or direction of the drift of the electron is opposite to the direction of the conventional current. Now I want to I mean when we talk about the current we do not talk about this so clearly but let us be very very specific I am looking for this particular thing from a frame s and in this frame s I assume that the positive charge carriers remember the conductor overall is electrically neutral it is a slightly different situation from what we have discussed just now I am talking of now a realistic conductor if there was no electric field this particular conductor was electrically neutral and their negative charges and their positive charges both are there now these electrons tend to be somewhat free or approximately free and when we apply these electric fields these electrons are the one which actually give rise to the current but those positive charges which are sort of which are generally believed to be immobile which are ions which have been left behind they remain at rest that is what is the normal picture of current that is always told to us in high speed so I am looking at the same frame in which this picture is correct it means I am assuming that the positive charges are at rest and it is the negative charges which are drifting in negative x direction to give rise to a current in plus x direction that is what I have said here in this frame the positive charges are at rest and electrons drift in negative x direction giving a current in positive x direction alright now let me assume we all of all of us know that if there is a current carrying conductor let us assume that is a very long conductor this conductor will generate magnetic field around it so because of this particular current there is a magnetic field which is created around it we know how to find out the direction of the magnetic field if the current is going like this now you have a you can find out there will be all the magnetic field lines of forces will be circular with you know the wire passing through the center of the circles and these are well known actually let us not discuss them in detail what I am also insisting that is all generating only the magnetic field without any electric field so there is no electric field okay and the reason there is no electric field because there are positive charge carriers as well as negative charge carriers and in a given volume whatever small element that we take we assume that they are same number of positive charge carriers and same number of negative charge carriers so the net let me just write it here if we take a small element in a wire in this element negative charge carriers and positive charge carriers both are same therefore if you take a small section of the wire you do not find a net charge density here and therefore there is no electric field which is generated outside there is only a flow of electron at all the times this charge neutrality is maintained in a small section of this particular wire therefore there is no electric field but there is a magnetic field. So this is picture with which we are quite familiar in our high school then we have been discussing various laws related to magnetism at that time we have been talking about this type of behavior. So this is what I said let the negative and positive charge densities in this frame as b rho e and rho p this e symbol I have said for negative charge carriers and this p symbol I have said for positive charge carriers. So let us assume that the charge density for negative charge carriers is rho e and for positive charge carriers is rho p of course because this is charge density so this is negative because the charge is negative though number density is not negative number density has to be always positive while positive charge density is positive because the charge is positive and because we are saying that there is no electric field in this particular frame of reference I expect that if I take any small unit volume by small unit necessarily small infinitesimally small volume you are always find that this rho p plus rho e is equal to 0. So there is no charge density in a smallest volume a small volume which you take along the wire I am not writing in the differential form etc because that is not the idea we want to make things simple. So just want to say that I expect that this rho p plus rho e must be 0 in this particular frame because I do not see an electric field I see only the magnetic field there is a current there is a current density but there is no electric field this current and current density causes the magnetic field. Now we realize that because in this particular frame it is only the negative charges which are moving so the current density the net current density is being caused only by the motion of negative charge carriers positive charge carriers would not contribute to any current density which is obvious because they are not moving they are not moving in this particular frame of the first. So this is what I have said in this frame S the current density is solely due to the electrons therefore net current density I can write as equal to J x e I am writing only x component because current density current I have been told that is only in the x direction current density is also in the x direction J x e I am using to mention the x component of the current density for negative charge carriers. And as we have said and J x is the overall net the total current density as we have said because positive charges are immobile so this J x is solely because of J x e and that will be given by rho e multiplied by the drift velocity of the electrons or negative charge carriers negative charge carriers we mean electron. This is what it will be given as rho e times u d e where rho is the charge density u d e is the drift velocity we have already defined this particular thing is exactly the same thing. Now let us go to a different frame of reference S in which the drift velocity if electron is found to be 0. So I have already found out what is the drift velocity if I know the current density I know what is the drift velocity if I know the drift velocity I can always go to a frame of reference an inertial frame of reference which has the same relative velocity with respect to S as the drift velocity of the electrons. What it means is that if you have a wire in which these electrons are moving in this particular direction this is plus x direction let us consider one electron which has exactly the same speed let us not bother about that you know let us just consider a drift velocity and assume that overall electrons are drifting in this particular way. I go to a particular frame of reference which has exactly the same velocity of course this frame of reference also has to move in this particular direction. So that they find that overall there is no drift of electrons in this particular frame of reference. So in this particular S frame of reference I am defining this particular frame of reference S as one frame of reference which of course is the inertial frame of reference because the drift velocity is supposed to be constant for a given electric field so long current density is constant the drift velocity is also constant. So I am looking at this particular aspect from a frame of reference S in which electrons are not found to be drifting so that is what I said let us observe this wire from a frame S in which the drift velocity of the electron is found to be 0. Let us see what an observer in S frame would notice would it notice that the current has become 0 and therefore there should not be any magnetic field that is not a current picture. Actually in this frame the positive charges will be observed to be moving in plus x direction see remember it was in S frame that your electrons were drifting and the positive charges were stationary. But once I have changed my frame of reference to make the electron stationary in that particular frame of reference in that particular frame of reference the positive charges will move and because as we have said here this velocity is in the negative direction. So positive charges choose a mobile earlier which would appear to a person sitting here to be moving in positive x direction. Therefore an observer in S frame would feel that all these positive charge carriers are moving in plus x direction. So there is a current in that particular frame of reference but that current is being caused by the positive charge carriers and this positive charge carriers will actually produce a magnetic field. Therefore a magnetic field is expected to be present also in S frame of reference. What about electric field? If you look at these equations these are the transformation equations which I have written from S to S frame of reference. Of course in this case v happens to be negative let us forget about it I am just writing in the number form. So if you look at this particular equation of course ex will turn out to be equal to ex but ey will depend on vbz, ez will depend on vby depending upon which point you are looking or the field at which point of course the current is in the x direction you will always find you may always find at least a component of by or vz or both. Even though ey and ez are 0 because we have said in S frame there is no electric field but there is a magnetic field and this magnetic field has to be in some direction of course it cannot be in x direction because the current flows in the x direction and the magnetic field is being caused only by that current flow and I am assuming this to be infinite wire so sort of infinite wire. So therefore at least one of these will be non-zero it means in this particular frame in addition to the magnetic field the observer would also find electric field this is going to happen if my transformation equations are correct. So these electric fields are present in this particular frame of reference from where are they arising earlier we have said that their charge densities were just neutralizing at each small volume of the wire and therefore there is no net charge in any small section of the wire and therefore there was no electric field earlier from where the electric field is coming. So now we will realize that is what we will do just now to show that that was too earlier in S frame in S frame of reference if you take a small section of the wire now you will find it to be charged and if you find it to be charged it is this charge which will generate this electric field. So electric field will be present in this particular frame of reference because of that particular section I mean sections of wire being charged how do I do that for that I must do a charge density transformation that is what I am going to do next. For that let us evaluate the charge densities in S frame of reference I have just now evaluated I just now found out the transformation equation relating to the charge densities let us do the same thing here. Find out what will be rho prime if I know rho in S what will be rho prime of course relative velocity of between the frames I have taken specifically equal to UDE of course I have reserved symbol U instead of V though when we originally defined charge densities we had used symbol V but I am using U because V we have as I said always reserved for the relative velocity between the frames of course in this specific example I am taking V is equal to UDE because only in that particular case the electrons or the negative charge carriers will be at rest this is my transformation equation for the negative charge carriers rho E prime is equal to gamma rho E minus UDE because V is equal to UDE J XC whatever was the current density divided by C square this is gamma rho J XC I can write as rho E into UDE. So, this rho E I have written here this becomes UDE square divided by C square this gamma this rho E I can take it out this will be the gamma rho E multiplied by 1 minus UDE divided by C square but this is also squared UDE is the drift velocity of electrons as was seen in S frame. So, this will be the charge density of electron which will be seen in S prime frame of reference what will happen to the charge density of positive charge carriers I will write exactly the same equation gamma rho P minus V J XP divided by C square where J XP is the current density of positive charge carriers in S frame but I know that this J XP 0 because the charges were not moving in S frame of reference therefore current density in that particular frame has to be 0. Therefore, this quantity will be 0 and this I can write as gamma rho P as we can see that the charge density is as seen in S frame of reference are not same because of this transformation. Therefore, this is not equal to this they were earlier same their magnitudes were earlier same good signs were different their magnitudes were same in S frame but in S frame of reference even their magnitudes have become different. So, let us evaluate the net charge density in S frame of reference which is just the sum of these two which earlier was 0. So, I have just used this particular equation which is from this particular transparency this plus this this plus this this what I have written here this gamma rho E would cancel out here remember this rho E was equal to minus rho P in S frame of reference. So, this rho E has been changed as minus rho P here. So, once I do that this gamma rho P and this will cancel out and this will become gamma rho P UDE square upon C square. So, we find that there is a net charge density in this particular wire. So, in S frame of reference it will find that the wire is charged if you take one particular section of the wire will be charged and if it is charged of course, we will generate electric field. Now, what happens to charge conservation some of you may ask that does it mean to say that this wire overall has become electrically non-neutral originally we have said that the conductor was electrically neutral. Now, we are saying that there is a net charge density does it mean to say that it is no longer electrically neutral. What happened from where we got the additional charges is the charge conservation obeyed or not the question is that see normally you will not have strictly speaking infinite wire you will always have a loop. So, once you are supplying the current there will be current flowing and then eventually the current has to flow back into a particular direction. So, whatever it is a situation you will always have situation somewhere where you have wire and this wire eventually has to close here there has to be some current source. Now, as you can see the current direction in this section of the wire is going to become different from whatever is here the if I am looking at the positive x direction the charges are going let us say the current is going in this particular direction this way while here it is going this way. So, in one section of the wire if I am positively finding it positively charged the other section of the wire I will find it negatively charged. So, overall the wire will not be will still remain to be electrically neutral. So, that nothing happens to that particular thing must be realized that thing, but if you take a specific section the way we have been defining in this particular thing that we feel only because of this particular wire that comes because of this positive charges which are being in that particular frame which you know which views this particular charge carrier this particular wire as a charged charge. Now, with this particular thing as I said this probably the last section that we have wanted to cover formally I want to just introduce how the Maxwell's equation we expect them to remain invariant. In fact, if you remember the electric field and magnetic field transformation we had obtained from the force equation the Lorentz force equation and saying that this force must obey a force transformation law. In fact, they can also be derived by maintaining that the Maxwell's equations are unaltered or unaffected when I change my frame of reference. We do not change any magnetic equations unlike we have changed many of the classical mechanics equations the equations pertaining to electromagnetics electromagnetic theory they do not change once we change my frame of reference the equations which are the equations Maxwell's equation for Maxwell's equation they do not change. So, I will not prove in general which is beyond the scope of this course which require if you want a general proof much more many more details. We will just take one simple example and convince you know I am always trying to say that I am trying to convince that Maxwell's equations are expected to be moving nothing like that nothing more than that. So, this is what I have said in relativity Maxwell's equations are unaffected we should not give the general proof we are just trying to convince you. So, these are the four Maxwell's equations which are very very standard equations the first equation is essentially what we call as a Gauss's law which talks of divergence of electric field in terms of the charge density. This is a similar Gauss's law for the magnetostatics which says divergence of B is equal to 0 and this quantity is 0 because it always says that there is no monopole. This is what is the curl of electric field which is called Faraday's laws of electromagnetic induction and this is Ampere's law with the correction of Maxwell pertaining to this displacement current which gives you curl B. So, the two equations are having diverges there are two equations which are having curl and these are the four equations which we call as Maxwell's equation. So, I will take one of these equations this particular equation the third equation and out of that I will take one component and try to convince you that this does not alter once I go from a frame S to S. So, this is what I am trying to do in this particular as the last part of this particular course. So, we said we shall take third equation and try to write this in S frame but before we do that let us expand this curl into individual components. So, remember it has a curl here, there is a curl of electric field and here you have derivative time derivative of magnetic field. So, let us first write this into the component form. Then this components will of course be in the form of x, y and z and on the right hand side you will have time derivative. Now, I will change these x, y, z to x prime, y prime, z prime this t to t prime assuming that x prime, y prime, z prime, t prime are related to x, y, z, t by Lorentz transformation because that is what is the relativistic transformation and try to see that I can write exactly similar equation in S frame of reference where e will be replaced by e prime and b will be replaced by b prime and this curl will be all with respect to x prime, y prime, z prime and this t will be with respect to t prime. Once I am able to write that I have shown that this Maxwell's equation remain invariant. Of course, we can do similar type of things for all other equations. We know how the charge density transforms we have just now discussed. We have also known how current density transforms that also we have discussed. So, we know the transformation of all these equations. So, in principle, you could have taken any equation and try to prove it. But as I said, I am not doing that in general just taking this particular equation trying to convince you. So, let us first expand this curl del cross e. So, left hand side of that equation had just curl of e that is del cross e. Of course, you can use any method of expanding it. You know some people try to write in form of determinant then try to expand it. I have written in a particular vector form which I am finding which I find generally much more easy to remember. But in any way you would prefer to write this particular vector one can expand this curl. So, this del operator as we call can be written as i del del x plus j del del y plus k del del z where i, j, k are the unit vectors. Then cross e I can write in the vector form as i, e, x plus j, e, y plus k, e, z again i, j, k are unit vectors in the x, y and z direction the standard way. So, I have to expand this curl it means I have to take this cross this, this cross this, this cross this, then plus this cross this, plus this cross this, plus this cross this, then this cross this, this cross this, this cross this. If I take first term i del del x when take i cross i I will get 0, i cross j will give me k and i cross k will give me minus j. Similarly, j cross i will get minus k, j cross j will give me 0 and j cross k will give me i. Here also k cross k will give me 0. So, all I wanted to say that normally if you would have expanded this, you would have got 9 terms because one has to operate on 3, second also has to operate on 3, third also has to operate on 3, we will get 9 terms. But out of those 3 terms will be 0, one because of i cross i, another because of j cross j and third will be because of k cross k. So, eventually you will be landing only with the 6 terms. I am not giving the details of these things, I can think you can work out simply and try to convince yourself that whatever I have written is probably correct. So, these are my cross products and all I have done is expanded into this particular form. As I have said that this expansion you could have worked out in any other way which is comfortable to you. So, all I have written is 6 terms and contained, kept those terms along the x direction here, y direction here and the z direction here. So, this del cross E I can write like this that this has to be equated to the right hand side which had del B del t, the x component and y component and z component of this equation in fact can be written as i del B x del t plus j del B y del t plus k del B z del t. Then this x component can be equated to the x component of the curl that we have obtained, y component can be equated to the y component of the curl that I have found out, z component can be equated to the z component. So, I will get 3 equations which I am writing in the next transparency. So, these are collecting them, we have collected all the components and equated to the right hand side. So, I get these 3 equations. Now, I will use only one of this equation, this particular equation, the second equation, try to transform it into S frame of reference by using the standard equation corresponding to the partial derivatives. I will not go into the details because this requires lot of mathematics as I say my idea is only to convince you and not to give you too much of details of the mathematics which I say as the beyond of the scope of this particular course. So, we shall use the standard partial differential formula and use it along with the Lorentz transformation. What is this particular formula? This del del x can be written as del x prime del x del del x prime plus del y prime del x del del y prime del z prime del x del t prime del x del del t prime. This is the standard partial derivative formula when you have there are 4 variables on which it is dependent. I know from Lorentz transformation x prime is equal to gamma x minus vt y prime is equal to y z prime is equal to z t prime is equal to gamma t minus vx upon c square. So, let us examine this particular term. If I take del x prime del x, this is x prime, this is partial derivative. It means all other variables I have to take them as a constant. So, t will turn out to be constant. Once I take the partial derivative of x with respect to x prime with respect to x, so I will just get gamma. This term will give me 0. So, this becomes gamma into del del x prime. Similarly, del y prime del x, if I take partial derivative with respect to x, it means y has to be treated as constant. So, this will give me 0. Del z prime del x will give me 0. Here del t prime del x, t prime does depend on x. Once I take partial derivative, t has to be taken as constant. I take derivatives, I will get minus gamma t v upon c square. So, this is gamma, not t minus gamma v upon c square. So, minus gamma v upon c square into del del del t prime. So, this operator del del x can be replaced by this operator once I change my frame from s to s prime. I can write similar equations for y which will give me very clearly del y is equal to del y prime. This you can try to work it out yourself and exactly similarly I will get del del z is equal to del z del z prime. Again del del t will be different from del t prime. I use exactly the same thing. Again you can work it out yourself. Del del t will turn out to be equal to gamma del del t prime minus v del del x prime. So, now I know how to replace this partial derivative from s frame to s frame of reference. So, this things I will substitute in this particular equation. So, this del del z, del del z I will replace by whatever we have evaluated earlier. Of course, as far as del del z is concerned, it is just z becomes z prime. Here there is a partial derivative with respect to x. So, this once I change to del x prime, I will get another term. Similarly, when this partial derivative with respect to time, I will get another term which I am doing in the next transparency. I am rather going fast, but idea has to say is to convince you. So, this equation as I said remains identical. Del del x, I am replacing by this equation. So, this once I go to prime frame of reference, this becomes del e z del x prime minus v upon c square del e z del t prime. This del t also gets changed and this becomes del v y del t prime minus v del v y del x prime. So, this equation now becomes in this particular form. What I will do? I will collect terms corresponding to partial derivative with respect to x and with respect to partial derivative with respect to t. So, if I do that, this equation becomes in this form. This is interesting because normally if a similar equation has to be obeyed in s prime frame of reference, I expect del e x prime is equal to del z prime minus del. This is an equation which I expect to be true because this is exactly the same equation. I am sorry, there should be a prime here because this is what I expect. If the same equation I am able to write in s prime frame of reference, I will be able to say that things are consistent in s prime frame of reference. This is the equation which I have written here. So, if I write this equal to, if this happens to be equal to e z prime and if this happens to be equal to b y prime, I know that these equations will be identical in s prime frame of reference. This is precisely what I know is true from electric field and magnetic field transformation. Of course, e x must be equal to x prime. So, from this, I can write these equations. So, these equations will be consistent in s prime provided we have these three equations valid. I know from electric field and magnetic field transformation that these equations are correct. So, I expect this equation to be valid also in s prime frame of reference. So, what I have done is shown only for one equation. You can try to show for other equations. You have to do somewhat more mathematics and try to convince yourself that Maxwell's equations are actually consistent in all the both the frames. They remain unaltered. We do not change Maxwell's equations once I go from s frame to s prime frame of reference. So, this is my summary. We discussed current density four vector. We try to analyze the current carrying conductor in two frames, especially discuss how the electric field is generated in that particular frame of reference and given an hint how Maxwell's equation remain unaltered. So, this happens to be the end of the course. Best of luck, all the best.


Statistics and Excel correlation large data sets focused on z-score relationship part number one. Get ready, taking a deep breath, holding it in for 10 seconds, looking forward to a smooth soothing Excel. Here we are in Excel. If you don't have access to this workbook, first a word from our sponsor. Yeah, actually, we're sponsoring ourselves on this one because apparently the merchandisers, they don't want to be seen with us.  Like our accounting rocks product line, if you're not crunching chords using Excel, you're doing it wrong. A must-have product because the fact as everyone knows of accounting being one of the highest forms of artistic expression means accountants have a requirement, the obligation, a duty to share the tools necessary to properly channel the creative muse. And the muse, she rarely speaks more clearly than through the beautiful symmetry of spreadsheets. So get the shirt because the creative muse, she could use a new pair of shoes. If you would like a commercial free experience, consider subscribing to our website at accounting instruction.com or accounting instruction.thinkific.com. It's okay because we'll basically build this from a blank worksheet. But if you do have access, there's three tabs down below example, practice blank example, in essence, answer key practice tab having pre-formatted sales. So you can get to the heart of the practice problem, the blank tab blank worksheet with just our starting data in it so you can practice formatting these sales within Excel as we work through the practice problem. If you don't have access to the data set, you can find sample data sets by going to resources such as Kaggle.com. Let's go to the example tab to get an idea of what we will be doing, where we will be going. We're looking at correlations once again, different data sets to see if they have a mathematical relationship or correlation between them. Are the data points moving in the same direction? For example, if there is a mathematical relationship or correlation, the next logical question would be, is there a cause and effect relationship? And if there is a cause and effect relationship, the next logical question would be, what's the causal factor in that relationship? We're going to be looking at two data sets of heights and weights. This will be a little bit different than what we've seen in the past in that we're going to have a lot more data. So these are more extensive data sets that we will have, and we're going to do our standard calculations to the mean, the standard deviation. We'll plot this out, noting that both the height and the weight have a similar look to them and that they have that bell-shaped curve. That doesn't mean necessarily that they're correlated, just like we saw in prior presentations when we had the uniform distributions, when we had random number sets, didn't necessarily mean that there's a correlation even though there's a similar kind of shape to the curves. But these, of course, both do conform to more of a bell-shaped type of curves because we're talking about things in nature. And in this case, you would think that there might be a relationship, of course, between heights and weights possibly. So then we'll do our calculations for them with the z-scores to calculate the correlation. According to our formula, we'll also do the formula calculation with Excel's data tools. We'll do the other data analysis with our tools. And then we'll also take a look at the relationship between the z-scores to get a better sense of what's happening with regards to these z-scores because, of course, the z-scores is kind of the heart of what we are doing with this correlation analysis. So having a better understanding of it would be a good idea. Let's go to the blank tab to then get started. I'm going to delete this Kaggle thing. And we're going to first, I'm going to format the entire data set by going to the triangle up top, right-clicking on the selected area, formatting the cells. I like to go to currency, negative numbers, bracketed and red, no dollar sum. I'm keeping the decimals this time because there's decimals in our data sets. We'll keep that as the default. Let's go ahead and say, okay, I'm going to make it bold as well. Home tab, font group, make it bold. And let's center these top ones and make them into our header. Home tab, font group, dropping down bucket black and the white for the lettering, centering it and the alignment, wrapping the text and centering. So we have our height and inches, our weight in pounds. Now obviously both of these are measuring, and also note that the data set is quite long if I say control shift down goes down to 25,000 rows down here. So if I go back up top, now just intuitively we would think, oh, these both sound like they would kind of correlate to a bell curve possibly because we're looking at something in nature. We can kind of test that out. And we'd say, well, there's probably, there might be a relationship between the two of them because you would think of someone's taller that they might weigh more. And then you might have the question, well, what's the causal factor? If I was going to make a hypothesis about it, I would think, well, if someone's taller, I would think that would have, that would be maybe a causal factor of that would be causing someone's weight to be heavier. So that's kind of like my thought process going into something like this. Let's do some let's let's check it out. So I'm going to make a skinny C, a skinny C, and let's do the same height and weight headers here. Let's just name it high. I don't need the inches. Let's just say height, h e i g h t, weight, weight. That's weird spelling for both of those height and weight, home tab, font group. I would not win the spelling contest when they gave me those words. I give up. I give up. I'll sit down. That's what I usually did when I was in grade school. Oh my goodness, height. I just give up. I'll sit down. I know I'm out. I'm out. So we have the mean and the standard deviation. So on the height, let's go ahead. This is going to be equal to the average tab. And we're going to go to the heights control shift down and enter. And there it is. Let's do the weight. I can just copy this across to the right now. Let's just copy it across. And let's make this a little wider so we can see the weight 127 on the mean. And then we're going to say the standard deviation standard deviation. And for the sample standard div s is going to be this in this stuff control shift down 127 is the is the mean. I'm heavier than that. But I think it's because I'm like totally yoked up with muscles and everything and muscles like as you know, muscles heavier than fat. So that's why I'm heavy, heavier than anyway, I'm just guessing here. Let's go ahead and add our bell curve. So I'm going to select the this data control shift down and control backspace to get back up top. And let's see what this data set looks like on its own by going to the insert tab charts. Let's put a histogram in their histogram. And let's say that there's our histogram. So obviously it looks somewhat bell shaped. Let's say that this is going to be H H E I G H. Alright, there we go. And so there is that. And then I could do the same for this data set and say, well, do they look similar? Maybe control shift down, control backspace, insert charts histogram. And so there's this one also looks kind of similar, looks kind of similar. I don't know looks kind of similar to me. Let's say this is going to be weights. And we'll put this down here. So heights and weights now they both look of course bell shaped doesn't necessarily mean that there's a correlation between them, the fact that they both kind of conform to a bell shaped kind of curve. But it might we might, you know, might give us some indication that we could say, maybe that's more likely that there's a correlation or something like that. But again, it's not necessarily the case that there would be. Alright, let's make column G a little bit larger here. And we'll do our calculations for it. I'm going to copy the formula to do so. And we'll do our calculations of like so let's make this a little smaller. Bring this up top, up top is where you need to be right there. And we'll make this one a little bit larger. This needs to fit in that space, fit in the space here, everybody needs their own space. Okay, so then let's copy this data, I'm going to put my cursor on column A, and select to column B, control C, going from A to B, and then control C on the keyboard, and then we'll paste it down. And let's calculate our Z scores, I'm going to put my cursor on column I, right click and insert. And this is going to be the Z of H. So we'll do the Z of H. And this is kind of a pain because we're going to do this entire data set, which we saw was very long data set, but we're using Excel. So not a problem really. So we're going to say this is going to be equal to brackets, the Z score is each individual x minus the mean divided by the standard D of the sample. So we'll each individual x divided by the mean, which is this 6799 F4 on the keyboard dollar sign before the E and the two closing it up and then dividing it by the standard D standard deviation right here at four on the keyboard dollar sign before the E and the three and enter. And so there we have it. Let's double click on the fill handle to drop it down. So there it is. And let's do the same thing over here for the Z scores of the weights, Z of weights, something like that. I'm going to say let's make this same formatting home tab format paintbrush it right there. Z score is going to be equal to brackets each individual y minus we're on y now this one minus the y's mean F4 in the keyboard dollar sign before the F and the two closing it up and then dividing by the standard deviation F4 on the keyboard dollar sign before the F and the three and then enter double click it on the fill handle to drop it down. Now we'll multiply them together. So we're going to multiply these two together and then we'll sum it up. We'll have then the numerator. So this is going to be then the Z of H times the Z of W. I'm going to format paint it again, home tab, clipboard format paint right here, multiplying this out. This is going to be equal to the Z of H times the Z of W and then I'll double click that down. Boom. Let's make these a little bit thinner. This one could be thinner. These three can be thinner for sure. These three can be thinner and so then we can say okay. So now let's do our the rest of the calculation. Now we're just going to sum up this outer column that'll give us the numerator divided by N minus one. Let's make a little worksheet on the right to do that. Making a skinny M to do so. We're going to say this is R or the correlation and I'll make these three black and white home tab font group black white and then I'm going to say this is the sum of Z H times Z W putting this in the outer column because it's going to be our numerator equals the sum brackets of this outer column Z of H times Z of W control shift down and enter. There we have it. Now I'm going to calculate the denominator with a sub-calculation. I'll pull into the inside and indicate it is a sub-calculation by saying N minus one colon and that means I'm going to do a little calculation below it which is going to be N space space because I just want an N and then less one which is going to give me the answer or the denominator N minus one without the colon and if I indent this it's going to look like this I'm going to select these three home tab alignment indent and I'm going to indent this one here as well. So now we've got the count which is going to be the count of the columns. There's a bunch of them equals the count and I can select any of these columns because I just want to count the number of rows control shift down enter there's 25,000 rows minus one and then I'm going to put an underlying home tab font group underlying subtracting this out as our formula says to do 25,000 minus one is 24,999 and then we can do our division problem here which is going to give us our R or correlation because now we've got the numerator we've got the denominator we're going to divide out the numerator divided by the denominator and we get the 0.01 let's make this a little bit larger home tab home tab number and make it a little bit bigger now something doesn't look right to me right here because I would think that the correlation would be higher than that and if I analyze what happened what happened here if I go on over here and by the way I can double check this by saying let me see the computer do it and see what the computer comes out to and then we'll fix it so I'm going to say over here if I go to the the data and use the analysis data analysis if you don't have that you can find it in the file tab on the left you can go down to the options and then down to the add-ins and we want the excel add-ins go the data analysis tool pack and so once we have that we're going to open up the tool pack and then I want to do a correlation okay and then the range is going to be this range I'll select the headers including the headers control shift down and control backspace so I have that whole thing I'm going to select this item I want to pick up the labels because I included the labels and then I want to put it somewhere on the range I want it on my range over here otherwise excel puts it on someone else's range and I want it on the home on the range this is my home I want it next to my home on the range so I'm going to say okay let's say okay and so it should come out to according to excel 0.5 so if I go through this and say okay what did I do wrong here I think my mean calculation is right my average the standard deviation is correct it looks like the z score over here is what is wrong and what happened I divided instead of subtracting so I'm going to fix that and say this is going to be this minus this and then copy it down and see if that fixes everything else should populate properly and you can see it fixed my z score over here so notice it's useful excel's spitting out the answer like this is useful sometimes it's used it's also useful to do it ourselves because then we get a better sense of what is happening and it can give us more understanding about the data and then we can also simply use as we did here which we totally did on purpose of course for demonstrational purposes that you can use this just kidding but we could use this as a a double check for what we what we put in place all right so let's do some underline here we're going to say font group and underline I'll make this black and white home tab font group we can make this black and white let's center this home tab font let's go to the alignment let's wrap it and center it okay it's already centered okay let's make a skinny queue now I'll clean this up a little bit now and then next time what we'll do is we'll go in and look a little bit more at the calculation of the z scores and because you would and see the relationship between the z scores for these two data sets because that's a key component of one of the calculations we're making here let's let's do one more thing with our toolkit though before we do that let's go to the insert or data tab up top let's go to the analysis group the data analysis and let's go to our descriptive statistics and say okay and then input a range so the range is going to be this again so I'm going to say control shift down and control back so there is our range and I'm going to add the labels the output is going to be where do we want the output I want it over here we want this range on the v and I'm going to say give me the summary statistics and the con and the confidence at 95 okay and then okay so that gives us our nice summary data takes a little while to analyze because we had a whole lot of data this time so it's spitting it out give it a little give it a little time here and we'll say so that gives us then our calculation of the means uh standard error median mode standard deviation sample variance but once again there's static numbers so this can give us kind of a double check of this of the mean standard error and so on I still think when you're building a model that's useful to do your calculations yourself and then use this kind of as a double check so that you have a dynamic worksheet one that changes as you make adjustments to it but let's make this one home tab font group black white let's make all of this blue and bordered make it look nice so I'm going to make this bordered and blue if you don't have that blue right there it's in the bucket drop down standard blue and let's just do that to the rest of it and then we'll continue on with the z score stuff next time control shift down I'm going to make this blue and bordered or border blue the other way around and border blue selecting this stuff and control shift down border blue selecting this stuff border blue selecting this stuff border blue let's make a uniform skinny column control I'm holding down control and selecting all the skinnies so we have a uniform skinny width and we'll then make it like that skinny that's the skinniness that we want it to be so that looks much better mucho mejor and next time we will continue on with some analysis of the z scores


In the last fifty years, ninety-two percent of the freshwater aquatic life on the planet is gone. Eighty percent of the biomass insects are gone. What's our plan? Suicide, is it? There's one more danger to Bulgaria, our Honorable Minister was pointing out today. If the temperature in the world goes up by two degrees, then the wind speeds in the world will pick up. If that happens, suddenly one day you may find millions of tons of sand from northern Africa may just land in Bulgaria overnight you can turn into a desert. It's foolish not to foresee things and to mitigate those situations. If we are a smart generation, we will mitigate the disaster before it comes.


Hey Virgo, welcome back to the channel. This is Kelly from House of Virgo. Hope you're all doing well. If you're new, welcome. Please do smash that like and subscribe button if you are new. And if you enjoy the reading, do keep in mind this is a general reading. Some messages may or may not be for you. This is a channel for Virgo, Sun, Moon, Rising, or Venus, but all signs are welcome. So Virgo, if you want to get a personal reading with me, there's a link in the description box. My readings for July are heavily reduced or heavily discounted, quite inexpensive. So definitely take advantage of the opportunity. Yeah. All right. So let's see what's going on here, Virgo. What's going on for you, Virgo? Sun, Moon, Rising, or Venus? What is going on for Virgo? Okay. Not enough. Wow. Frustrated in a relationship, lack of confidence, self-sabotage, fear, ego issues, and jealousy. Okay. And you also have Cupid's arrows again. Have faith, love is coming, surprise, invitation, or meeting hesitation. Yeah, that's still happening, Virgo. But I kind of feel like you, gosh, you got the grim reaper. What the heck? What the heck? Let's see. The relationship is over. No second chances. Grow and transform your life. Okay. So somebody's still very much in love with you, although they don't feel like they're ever going to be enough for you. And they know that this relationship is over and you're not taking them back. Okay. You're done. Point blank, period. Virgo is one and done here. Okay. Let's see. You've got the magician. Let's see what's going on here. What's going on here with you? Virgo's like, look, I don't want the love. I don't want the offers. I'm not trying to make any decisions about this. I'm manifesting new starts, new beginnings, have other options, and I'm not so really into something here anymore. Okay. That's you saying this. So yeah, you're very focused on what you're doing in your life at this time. And you're just even not sure to some degree. Maybe a little bit apathetic towards love at this time.  Excuse me. Eff it. It's just like, no, I think I'm just going to forget about it. Forget about love. I'm just going to do me. Yeah. So I'm kind of picking that energy up right now. Yep. Trying to maintain a positive mindset, be successful, do the things you need to do for yourself. It just kind of feels like, you know, I've with the Cupid's arrow, so I do feel like someone is really, they really want to like connect with you. But I don't know, Virgo, I kind of feel like you're just indifferent to someone. It could be a Gemini. It could be a Leo. You just seem indifferent. Or there's not enough there for the relationship to keep moving.  You're done. Second chances. Let's see. It's interesting. Yeah. Too much stress, Virgo. Somebody lies. Somebody cheats. You're done with it. Too much stress here. Okay. Too much. Causes you too much anxiety. You're trying to be with the sun right there in the center and the sun being the center of the universe. Or at least our area here. You're trying to be in like the center. You're trying to be the center of your own life, right, at this time. I just feel like it's, you know, too much, I don't know, something just too sneaking around or you're just saying no. Okay. You're just saying no. You want love with somebody else maybe, or you just want to have a new beginning in your life. Let me get a clarifying card for the magician. Show Virgo. Oh, they gave you two. Okay. Yep.  I'm protecting what's mine. Yeah, Virgo. You're really Virgo. You're not playing whatever a situation is here.   I feel like you're just in this magician mode, like manifestation, totally focused creativity. You want success. You want good things to happen in your life for you going forward. So you're making some strong chess moves here. Let me see the sun. What's going on? Oh. Oh yeah. You're tired of being left out in the cold. Get some of that heat and melt all that snow. Virgo, you know what?  Somebody was just leaving you broke busted and disgusted, feeling kind of out in the cold and you're done. You're done with that. So it's too much stress. There's too much anxiety attached to this grim reaper energy. In July, it looks like you're doing something here. You are adios everybody. You're doing you. Okay. You're doing you. So that's good. Let's see what this other person is going on with this other person. How they're going to react to you. Just not giving them any time of day. Well, of course, they're going to put you on a pedestal with the Empress.  I don't feel like you're trying to do anything here. It's just whoever this grim reaper person was, they were just sneaking around and either sneaking around or they were just lying or being deceptive, that kind of thing. And maybe betrayed you or maybe they ran away from the relationship caused you too much anxiety or in and out kind of energy. So, you know, what are they going to do? Hold you in high regard. Because you know, you're creating value. Valuing yourself is creating value. Okay. What did they say? Hold on. Let me try to remember. Access is contempt. People have access to you all the time. Hold you in contempt. I don't know what that means, but somebody needs that. No longer an access. Oh my gosh, look at them here. Now they're gonna, now they're gonna, I feel like this person just kind of felt like they didn't have to choose and you'd always be there. Okay. Looks like you're not being there. Looks like you're done. And now they're standing here like, uh-oh. I need to make some plans here. Three of wands. I need to figure out what's going on. I'm reaching out to Virgo and I'm not getting anything in return. Virgo's not responding. Virgo's not messaging me back. I'm gonna have to go get Virgo. Oh geez. I'm gonna have to go get Virgo. I am so passionate about Virgo. I want Virgo back. I need to keep Virgo where Virgo belongs. Locked up in the dungeon somewhere. All right, let me see. In my pocket. So yeah, you just cut it off Virgo and you just push back on that grim reaper energy. And I'm not talking about like your person that didn't treat you badly. I'm talking about someone who is lying, who is dishonest, who caused a lot of stress and anxiety and relationship. That's the person I'm talking about. Okay, but you're, you're like listening to this and you're like, yeah, this person is going to come out of whatever hermit mode or silence they've been in in a reaction to you. Okay, they're definitely going to come in with some lusty, flirty energy for sure. A little bit of romance or kind of energy as well. Someone who uses like the lover boy, lover, lover girl technique to try to like really in a little bit here. I feel like this person, you know, it could be another Virgo as well. I feel like this person, you know, with the hermit has been kind of like in hermit mode, just not really talking, no communication in the reverse. I feel like they're going to start opening up here and communicating with you and trying to, yeah, just flirty messages or something like that. Yeah.   But maybe Virgo, maybe some of you weren't showing your value, right? In the sense of like, maybe you were too available or you were always letting somebody come back in. Who's lesson is it? Is it yours? Is it theirs? I don't really know, but I just kind of get a sense where I feel like this person is going to abandon ship and swim to your shores here, wherever they've been out there on that rock amongst the waves, just all by themselves. Yeah. So here they come. Here they come, Virgo. The rest, at least from now until the end of July, they're coming to get you, Virgo. They're coming after you. I mean, they're going to try, at least, they're going to try. What you do with that's on you. You want to talk to them, talk to them. If you don't, you don't know, you got unrequited love. So for some of you, it's basically like, you know, there's not enough attraction or chemistry to keep this relationship going. And that's what I was saying. Not enough right here. Not enough, not enough just to keep it going. There's frustration in this relationship, losing confidence, this person's self-sabotages due to fear and ego and jealousy. Okay. So I feel like for you, Virgo, it's like, nah, we're just not right for each other. I don't necessarily feel like the love is not there. I just get a sense where the love is not enough.  You know, like, I don't know. But if you don't want them to anymore, that's fine. Because you have love options coming in. Let go of control issues, Virgo. Allow situations to just unfold naturally in your life.  That's what I'm hearing. Just leave it be. Leave it alone. Very soon, you will be able to decide what you want so that it comes to you now. If you have a decision to make in regards to this relationship, some of you might be kind of like, hmm, they might come this person might come back in and, you know, try to reconcile with you and the decision you might have to actually, if you haven't fully made it, I mean, some of you might have just stopped talking to another person and nothing got confirmed as far as like breakup or separation might just like ghost might have been a ghost situation. So, but if you ghosted a really long time or you were ghosted and you didn't chase somebody, I feel like right now, um, this energy is some, there's going to be some kind of contact and I feel like you're going to have to like actually say something. You might have to actually address it and say, you know, whatever you need to say about this relationship, whether that's, you know, I don't want it anymore. I'm sorry. Wish all the best, that kind of thing or don't call me anymore or I'm changing my number or something along those lines. Like a decision to say something is what I feel. Okay. All right. And this person's going to come in and apologize for sure to you. I'm going to try to, I mean, you can expect to hear that. But I mean, is it sincere? It could be. But it feels like based on this reading, it might just be because, you know, you're done. You don't feel like there's anything here. It's not enough for you.  It may just have been a situation where it's like, no, it just doesn't work for you and you change your mind. This could be somebody that you've just been dating for like a couple of weeks and just not going the way you want it to go. Um, it's just not, you guys aren't jiving. I don't know where I got that word. You're not jealous. You're not vibing or you're not jiving like some 1970s pimp cat. I don't know. You're not jiving with each other. And yeah, I feel like this person wants to jive with you though, Virgo. They want to jive with you.  So maybe this is a twin flame connection for you. That's what all this struggle is all about, Virgo. For sure, lessons to be learned here. Oh my goodness. Oh my goodness.  If you want to work it out with your twin flame, have at it. I can't tell you, but I know that there's some kind of, there's love and then there's struggle at the same time. You know, deep intense passionate love and then this severe struggle and anxiety where you're almost at the breaking point, like completely stressed and mentally exhausted.  Some of you want to have children in this situation that never felt like you could have that with this person. Maybe you had children with someone else or maybe not, but it's kind of like that's your, for some of you, that is a goal for some of you, for others. And also I will say with the Empress here, this person may be like, yes, I want to have, I want to have that baby too. Oh boy.  You know, if you want kids, I know not all of you do. Okay. So this is for those of you who are still at the age where you can have children or to adopt children. There may be an issue also with children connected to this relationship. There might be some conflict with grown adult children as well. I'm feeling that has caused a lot of problem. Maybe somebody's son or daughter is a grim reaper in this relationship. Okay. There's rejection here.  This is an apology from a twin flame, a rejection issue, children being important to this topic, Virgo. Now I feel like this, I mean, I don't know who's feeling rejection. There's some kind of rejection, which is actually, it says divine protection. So you know, I don't, for some of you, there might have been rejection where it's like, oh man, I didn't realize what a blessing that was. And now I'm going to keep it moving here. All right. So you've got passion. Try something you've never done before. And marriage, the relationship will, is moving towards a secret union. You know, just because a train is heading down the track doesn't mean you need to board. Okay. Just keep that in mind. You know, you could always say no about things, you know, or it might have an issue to do with marriage. Maybe you're already married to someone and this is some kind of a relationship that was interfering or maybe you were in a relationship with someone that had, was married to another person who had children for some of you. Okay. For others, yeah, I mean, marriage with the twin flame. Yeah, but look, look at all the, look at all the ups and downs in this twin flame relationship. We got apologies and then there's passion, which keeps you guys connected. And then there's all this rejection.  And then there's an apology and then the passion heats up. And then there's rejection again. And then there's another apology. It almost feels like a cycle where nobody's really, like the, the sole contract is not officially been broken yet. It has to be broken. So I don't know if that's like, I feel that they don't just get broken. I feel like that's why people go for years and years and years and years and years in these cyclical toxic relationships. I feel like someone has to break the cycle. That's when the sole contract, I feel like ends when someone has had enough and has learned the lesson and then they end it. You know what I mean? Is a struggle, you know, go back to Monday's reading or Sunday, Sunday's reading positive energy for the rest of this week. But I do feel like you're going through some major shifts right now in your life. Your focus is on your better health and your better mental health and making your relationships better. Really, really important to you at this time as it should be. Okay. So let's see what other messages are for you here. Oh, okay. Yeah. Look, transition. There you go.  No one said this would be easy or that it has been the season in your life is transforming, bringing you into a new hemisphere. Allow it to happen and go with the flow. It's now time for change. So I kind of feel like no matter what happens, someone trying to get you back or not trying to get you back, you're still transitioning.  Having this new start. Okay. And taking time to reflect. So when it comes to this relationship, for a lot of you, I feel like getting distance, going into your cave, taking time to regroup and recollect. A lot is going on. You need some alone time. You need to create distance from a toxic kind of relationship, staying the stillness there and in the quiet, but don't get lost there. Don't stay down there too long. Just take all the time you need. You can tell other people you're taking a vacation or you're just taking some time to yourself. Just whatever if you feel obligated to say something to other people or to this person. But the thing is put yourself first. You need a recharge. So put yourself first. You can say, I'll get back to you or yeah, I'm going on vacation. Talk to you next week or whatever, whatever you got to do, whatever you got to do to kind of protect yourself and figure out what it is you want. Like be, I just feel like it's the most important thing is to get distance from this. The most important thing.  So we're going to shorten it a little bit tonight.  Dreams come true. Your hope, faith and wishes are being restored. Make a wish right now, Virgo. Pause the video. Make a wish right now. Believe it is yours. Okay. Yeah. So where are my, let's see what the Chinese signs are between you and this person. Let's get a little more clarity. If the video, you guys, if the video stops when I'm in the middle of the reading, I may just leave it as it is. Okay. But I might not. We'll have to wait and see. So if it just happens to stop my apologies, you can expect the apology on here. I'm sorry. And we'll keep it going. Okay. Year of the goat, year of the dog, year of the horse, tiger, pig, snake, tiger again, a monkey and dog again. One more and a rat. Okay. So that's you or your person. All right, Virgo. I can't believe how fast the reading went. It's only like 20 minutes. All right. Let's see. Huh. Okay. Yeah. Somebody's sitting on the beach drinking a Mai Tai. That's for sure or will be in July vacation. It could be a family vacation or going somewhere with family, doing some kind of family vacation thing. Oh, geez. A lot of beach energy happening right now. Sitting thinking about decisions taken maybe a little time out sitting by the water could be a beach could be a pool could be wherever a lake sitting by some water thinking about the family thinking about marriage thinking about happily ever after taking some time to figure out maybe just taking some time out because you're unable or unwilling to make a decision at this time following your heart rather than trying to make others happy pretending maybe there is no problem or just kind of taking the time out to just ignore it so you can get a little peace of mind. Some of you might be over analyzing the situation as well. Okay. Something to do with like I said somebody who yeah just was a real cheat caused a lot of stress caused stress to a mother figure or a mom as well. Now I just want to say also I see someone cutting a steak or eating a steak and also see like a jewelry box going through a jewelry box and someone playing some kind of outdoor game. I know for some reason it's really hot maybe it's an indoor outdoor kind of game with some kids or some children also maybe it's like indoor mini golf or something. Okay. Oh right. Also connecting with some old friends as well or childhood friends or two kids meeting for the first time two little kids playing together meeting for the first time. Okay. And I'm also picking up 1972 1967 and I'm also picking up September 10th and Leo Capricorn that's double Capricorn and Libra. Okay. September 4th. September 9th August 26 August 29 September 21st someone is 60 someone is 46 someone is 19 area code 6 1 0 they're giving me or 6 0 1. Okay someone is 54 I have number 5 1 0 5 3 1 5 8 1 0 9 so they might be phone numbers something along the lines there and also Tars again Leo Aquarius let's get one more Aries Sagittarius and Gemini. Hello Gemini you're always here in my readings. What's going on? All right let's get some charms for my camera dies. Oh it's interesting. Okay. So just one more. All right three interesting little messages has a lot to do with animals. I will tell you that. So we have a rhinoceros you have a feather and you have a little I don't know what those hot dog dogs are called. I always forget what they're called. So maybe somebody has one of those dogs. What are they called? Shweiner weiner dog or something. I don't know you have a pet rhinoceros. I don't know what's going on with you. I don't even know where that would come from. That's just related to something very old school like not old school but age old something to do with age that rhinoceros someone who's like real narky but maybe even horny someone who's old and horny what and also a feather here. Okay so obviously the feather does represent as always an angel wing. If you find a feather see a feather somewhere that's kind of like just like that it might be like a quill feather that just fell off a bourbon any kind of feather significant maybe somebody's going to the zoo. Do they have rhinoceros at the zoo? I don't think they do. Do they? Are they extinct? Are they even alive anymore rhinoceros? I guess they are. I've never actually seen one I don't think. I mean I've seen hippos.  It's not a triceratops but I don't know there's something weird happening here with these terms. I just that's what I'm picking up here so I'm going to leave that there and let you figure that out.  What is going on? Dogs, birds and triceratops rhinoceros and feathers you guys something's happening here with with animals. I will tell you that maybe there's this animalistic feeling of something going on or this you know that it is the venus retrograde too so a lot of time there is a resurfacing of past people or issues that have to do with past people whether it's energetically something you need to close on your mind or something of a reconnection that does happen. I think July 22nd is when it actually starts or the 21st but we're pretty close you're already in the pre-shadow period so yeah keep that in mind Virgo. All right I'm going to leave that there hope you enjoy that battery's about to die don't forget if you want to donate to my fundraiser below please do thank you everyone who's been donating or donate to my website with the missing kids all the links are in the description box below helping you find missing kids or traffic children and yeah don't forget if you want to get a personal reading also there's a link in the description box below. Do go and grab that as the discount is pretty awesome. All right you guys I will see you tomorrow.


Welcome to Patriot Bootcamp. We're here in Lehigh, Utah. Mx is hosting a fantastic event and part of the reason why it's so fantastic is we're joined by a very special celebrity, Chef Robert Irvine. Sir, welcome aboard. Thank you. A lot of people don't realize this chef. Well, a lot of people do know that chef is a veteran himself and an entrepreneur and he's here sharing his expertise with our participants. Chef, thank you for being here first off. Well, it's very inspirational. It's very important. DAV is doing such an amazing job, but it's bootcamp, taking people from zero to hero and everything in between. And something that I never had, I wish I had you 20 years ago in DAV to help me go through the stages of growth. It's exciting. I mean, 30 veterans in there with 31 mentors, which is really special. One-on-one, lots of different excitement about different companies, different developments, stages, strategies, etc. So it's really fun. We mentioned this earlier, but Chef Irvine, his TV shows are the only thing that he doesn't completely on. Everything you see with Chef's brand comes from your family of companies. Tell me, why is it so important that you maintain your branding, maintain your reputation? Well, unlike other celebrity chefs out there, I own everything that we do. I don't license my name. We own the manufacturing for all our families or friends. And for me, it's important that people know that we control that. It's the quality of the food, the liquor, the nutrition, the clothing. I want people to trust our brand. And when you become a trusted brand, it's easier when somebody walks in and says, oh, DAV, love what they do. I'm going to buy into that. And that's what the Irvine company of brands are. They're a trusted brand across the board in retail, in the military, in military retail, and so on and so forth. And a huge part of that is give back. And the chef has done tremendous things for DAV, supported us over the years, and giving back here. Tell me about your foundation. So the Irvine Foundation is all about mental and physical health. How do we do that through programming, through dogs, through wheelchairs, through fitness programs, through unions, from breaking bread for heroes, and obviously all the stuff we do with DAV, which is really important to me. I mean, DAV has been around for 100 years. You do some amazing things. And it's so interesting because you do things that I don't do in our foundation. And our foundation does things that you don't. We play in separate lanes yet together. We're going to make amazing changes. That's great. Thank you for everything you're doing, chef. Everyone tune in. Follow us at www.patriotbootcamp.org, DAV.org. Follow the chef. And thank you. We'll see you next time.