For this problem, we need to calculate HDD for one full week. The data that is given is each day what is the outside temperature, average outside temperature. For example, Sunday, the average outside temperature is 49 degrees Fahrenheit. Monday, 47 degrees Fahrenheit. Tuesday, it's 51. And Wednesday, it is 60 degrees Fahrenheit. And on Thursday, it is 65. Friday, 67. And on Saturday, it is 58. So we need to calculate heat and degree days for each day. One day, that is Sunday, the outside temperature is 65 minus 49, the outside temperature will give you the degree days. Similarly, 65 minus 47 degrees Fahrenheit, one day times 65 minus 51, in this case, one day times 65 minus 60, and one day 65 minus 65. This will be 0. One day, 65 minus 67. Remember, when it exceeds 65, the heating degree days would be 0. In this case, also it is 0. One day on Saturday, it is 65 minus 58. In this case, it is 7. These two are 0s. This is 5. And this is 14. And this happens to be 18. And here, it is 16. So the total sum is, for one full week, it is 60 degree days. This is equal to 60 degree days.


My last question and I feel this is like the most important question that's why I saved it for last you would ask them so what's up you guys sit down and welcome back to my channel thank you guys so much for joining me today I am so grateful to each and every one of you you guys have been following around this journey from PA school to now PAC and I wouldn't have it any other way you guys have been amazing so interview season is in swing people are also getting their applications in for PA school and it's pretty crazy because school is actually about to start in just a few weeks and that's why I wanted to make this video for you guys I want to give you guys at minimum five questions that I feel you should ask on your PA school interview it is important that you guys have at least five questions at least you know you can have more but at minimum five questions that you can put in your back pocket for when you are asked on your PA school interview do you have any questions for us because the one thing you do not want to do you guys is be like uh um uh no I don't have any questions thanks for having me because in their mind they're thinking yes thank you for coming we are not going to see you in the next few short months because you want to actually show them that hey you are like thinking about stuff you know you've thought about what the school is about and maybe even the fact that hey you've done your research on this school that's why you should always do your research as well you guys so this is going to be just like my five questions that I kind of asked and I thought about when I was on the interview process and I think that it's beneficial for you as well so you can take them and use them as you see fit but you guys don't all use them at the same interview because they're gonna hear the same question be like hmm where did that come from okay so try to be creative now the first question that I think you should ask um which was important to me was do you do graduate assistance so do you have like a graduate job assistance program so my particular program actually had a fellowship where a lot of the students that went to my school went into this fellowship which meant that they had jobs directly out of PA school yes it was paying them a little bit more but they would also get their doctorate degree and then be able to go on and do whatever they wanted to do after that year of service in the fellowship so technically it was like a graduate assistance in that the school kind of shuttled you into a job but I mean there are schools that actually like have prospective jobs come to you and you know you kind of get into like the same type of program thing where you give them back like a year or two of your time and they help like pay off your loan or whatever the case may be so that is something that you should probably look into when you're trying to find the right school for you and then when you're on the interview process just ask and see because you have not because you ask not right so that I think is one thing that you should definitely think about when you're applying to PA school and getting these interviews so the next question that I think you should ask is what do you feel is the strength of your program now I know I know some of you may be a little leery because you're like oh my little oh me just a pre-PA trying to get in wherever I can to ask them what is the strength of their program but you are a prospective student that's going to be paying tens of thousands of dollars to this program so it is actually important and speaking of strengths go ahead and hit that like button and subscribe right now because it will strengthen my youtube algorithm which will help my channel out a lot a third question which was important to me was do you all allow outside rotations now this is not something that like every school does and it might not necessarily be important to you like for me you know I had a family that I picked up and I moved all the way to the school that I went to and I was like you know I'm not really trying to stay here for two and a half years because I wanted to go back to the state that I was from I love the DMV area so I wanted to go back there and work and build relationships there so that's why I was like it's important for me if I'm able to do my rotations back home because that is where I'm trying to go when I finish school and that's where the relationships that I need to build are so that I can get an in when it's time for me to get a job so that was an important question for me it may not be for you and maybe you can insert another question in there but I think that is important for you to source out what are the like main things that you're looking for in a program I also think a fourth question that you should keep in mind is what is the culture of your program now culture meaning like how does your program run is it very like small groups based do you guys do a lot of activities together are you really all about like the mission trips and are you all outdoorsy because that will make you and help you understand like hey am I going to fit into this culture because you want to actually be able to fit into where you're going into school right you want to be able to be comfortable and have like that be one less thing that you need to worry about which is fitting in so definitely ask that question it's really important because you want to be comfortable while you're going through the screwling slog of didactic year and then again when you're in clinical year having friends and people that you can actually get along with and do things with and fit into this culture is important because it's just going to be great for your mental health my last question and I feel this is like the most important question that's why I saved it for last I heard it somewhere and like you know like interview tips or something like that and I thought that it was like a really good thing that you should use when really applying to any program and like any job or anything like that but honestly I feel like it's a really good option for PA school as well and the question is to whoever is asking you the question well do you have any questions for us what was the characteristic traits of the most successful students at your program now you it kind of sounds like oh like what is this girl talking about but honestly depending on the way that the professor or the director of the program like answers that question to you will help you decide like is this the program for me as well for instance like let's say they were like oh you know the students that have been most successful at our program studied their behinds off they were like in this place 24 7 some of them even slept here and you know they never really went home I really don't know if they socialized as much but they got the work done you could throw anything at them and they were ready and willing to study and get that work done now I mean obviously you want people that are studying hard or hard workers and things like that but is that the like life that you want where you don't have a life but PA school and all you have to do is study and sleep in your PA programs like building no if that's not what you want then that might not necessarily be the school for you so these are things that you have to take into consideration because it's like all right I want to know what the culture is but I also want to know like hey am I going to have a life and if I will fit into that culture because you are seeing that successful students at your program really need to do extremely extremely hard work so that I think is something that you should really take into consideration remember that question put it in your back pocket and know that I told you that one I got it from somebody else but you know I told it to you here first so definitely ask that and keep that in mind when you're going through these interviews because you have to ask them something when they say that question you're not trying to be the person that looks like when they ask you do you have any questions for us okay all right thank you guys so much for watching if you have any questions for me please leave them in the comment section below and also please leave me like any suggestions that you may have for videos as well because I do take those into consideration when I am planning out these videos for the various different weeks if you have not already done so go ahead on over to my inner instagram at adana the pa and please follow me on instagram thank you guys so much for watching I will talk to you guys next time bye


So, what I showed you before were really quite regular wave ripples, and you can see them in this image as well. Some of the crests are nicely preserved, and you can see this nice wave form actually in the sediment here. So, one of the interesting things about this image, though, is that above the waves, you have some planar laminations and planar surfaces, and there are actually some wave ripple laminations in here that show dip in both directions, but the actual ripple crests were eroded away before the sediment up here was being deposited. If you look carefully down here, just above the pen or pencil, you can also see that some lamina are going this way, and some lamina are going the other way. So, we have the really nice wave ripples preserved here, unless regular wave ripples preserved in both of these other layers. The image on the right here also has wave ripples, so we have a boundary here between coarser sand and fine sand. And then if you look within this fine sand zone, there are lamina that dip in a lot of different directions. So, there are some that are coming down here, there are some that are coming up. You can see some of the crests here, there's a trough in here, crests coming up here, probably some oblique layers in here. In this particular case, there are a lot of lamina going in different directions, and they tend to be quite rounded, and the lamina are dipping in both directions. So where we see some of these more rounded lamina, they meet the symmetry criteria, and then we also have lamina dipping in two directions. So the ripple cross lamination can really look quite different depending on the characteristics of the waves. So in this particular example we just discussed, it's really quite irregular, and that suggests that the waves probably moved in irregular directions as well. We're on the left side here where we have these really nicely regular symmetric ripple marks, probably the waves were coming in in one direction and very consistently through time producing a much more regular lamination. So cross ripple lamination from waves can really tell you a lot about the environment and the dynamics of the waves. Thanks for watching.


Hello it's Missy! Thank you for watching! Today's video is a side braid. The French braid version of this style is really popular so I thought it would be fun to share an elastic version. This is a little bit easier in my opinion but let me know what you think down below in the comments. If you haven't subscribed to my channel yet and you're new, click that subscribe button and click that little bell then you'll be notified every time I post a new video and let's get started! For this hairstyle I'm placing my braid on the left side of my head so I'm going to have my hair parted on the right side so I have most of the hair to work with on the left side. I'm going to first divide off a section of hair that I'll leave out to frame my face then I'm going to pick up a section of hair next to the hairline brush through it to use for the first section of my braid and for each section I'm going to have two separate pieces so once this piece is divided out I'm going to pick up another section a few inches back from the hairline from along the part and I'm going to combine these two sections together. Using a clear elastic band I'm going to tie the two sections together a few inches down from the part line. I want the top of the braid to be softer so I'm not placing the first section right on top of my head but instead a little bit lower. Once it's tied off I can gently pull on small pieces of hair above the elastic. These will kind of slip out of the elastic and this will add some volume and create a fuller section. Now I'm going to create a new section of the braid so I'm dividing off two new pieces of hair on either side of the previous section. The forward section will be from along the hairline and the back section will be from the top of my head behind the first section that I already tied off. I'm going to combine the two sections together on top of the previously tied off section. Then using another clear elastic band I'm going to tie the two sections together about an inch or so down from the previous elastic. After it's tied off I'm going to gently pull on the two sections just above the elastic band so the hair slips out a tiny bit and this will make it a little puffier and fuller. Now I'm going to divide off two new sections of hair for the next part of the braid so I'm separating off a small section from along the hairline underneath the previously divided off section. Then I'm taking a section from the top of my head behind the previous section and I'm going to combine the two pieces together over the top of the previously tied off section. I'm going to use another clear elastic band to tie them together and then go back over the hair and carefully puff them up just by gently sliding small sections up and out of the elastic band. Now I'm going to pick up two new pieces for the next section of the braid so I'm dividing off a section from above my ear and another section from the top of my head. I'm combining them together over the previous section and then tying them off using a clear elastic band. Then I'm going back over the two sides and gently pinching and pulling on small sections of hair carefully sliding them up and out of the elastic so the two sections are puffier and fuller. Then I'm going back over the two sides and carefully pinching and pulling small sections of hair so they slide out of the elastic so the two side sections are puffier and fuller. I'm going to repeat this process working my way down the left side of my head dividing off two sections of hair one from the front side of the braid and one from the back side of the braid from along the hairline and from the top or back of my head and I'm combining them together over the previous section each time and then tying them off using a clear elastic band. Then I'm going to puff them up by gently pulling small sections out of the elastic band and when I'm bringing in hair from the opposite side of my head so from the right side on the other side I like to work my way along the part line first bringing in sections from the very top of my head and then once it reaches the crown I'm going to start bringing in sections from the right side and kind of work my way from the top of my head down towards my ear and then the nape of my neck and I'm going to drape these sections around the back of my head so the back side of the hairstyle looks soft and it creates a side swept effect and as the braid gets lower on my head and curves over my shoulder the hair from the right side of my head will sweep around my head in the direction of my left shoulder and then be combined into the braid using elastic bands for each section will help ensure the style stays nice and secured throughout the day there's less chance the style will slip out or become undone as long as the elastic bands hold I think this is a really fun alternative to the french braid I've seen this done with different braids but using elastic bands is just a little bit easier it still looks just as cool but there's no need to know how to braid or braiding involved once I've brought in all of the hair from my head I found that I had three separate sections or ponytails of the braid that's left hanging down so using the very bottom or lowest section I'm going to split the hair into two equal sections and then wrap them over the top of the rest of the hair so I'm continuing the same pattern or style of wrapping the hair on top of the previous section and then once it's at the top or the front side of the braid I'm going to tie them together using a new elastic band and continue creating the elastic sections of the braid I also continued to puff out each section by gently pulling on the hair to loosen them out of the elastic you can pull the sections a lot and make them really big and full or keep them smaller depending on the look that you're trying to achieve I kept mine relatively uniform not too big not too small I did find that the bigger I made them the kind of messier the style got so if you want your style to be more casual and soft you can pull them out a lot if you want it to be more tight and concise then you can pull them out less or not pull them out at all and I created about five or six sections of the braid draping over my left shoulder I wanted to go down as far as I could as long as each section was maintaining the same size as the rest of the sections so once they got to be too small or thin I stopped splitting the hair and tying new sections of the braid after I tied off the last section of the braid I decided I wanted to hide this last elastic band you could hide each one if you wanted but I decided not to do that so to hide the bottom elastic I took my topsy tail tool and I found this on Amazon and I pushed the end of the tool inside or through the elastic so the loop was at the top and the tail was poking out the bottom then I'm going to divide off a small section of hair from the ponytail and I wrapped it a couple of times around the elastic then I pulled the end of the wrapped piece through the tool through the loop at the top and then I'm going to pull the tail of the tool down so the top loop is squeezed through the elastic band and this will pull the end of that wrapped piece through the elastic and that is what will hold it in place without needing to use another elastic or a bobby pin I think this style is so cool because it looks just like a side braid but there's no braiding involved and because each section is tied off with an elastic it's a lot more secure and should hold throughout the day thank you so much for watching I hope you enjoyed today's video comment down below what you think about this hairstyle if you haven't subscribed to my channel yet and you're new please subscribe click that notification bell then you'll never miss any future uploads and I will see you next time


So, welcome to 7th lecture. So, till now we have covered vector calculus and electrostatic fields. So, from now onwards we will see for next 2, 2 and half lectures magnetic fields and time varying fields. So, which are you know very important for this course because most of the time we are interested in calculation of magnetic fields in electromagnetic devices that we deal with. So, when we talk of magnetostatics we start with Biot-Severt law. So, for you know line current carrying current i the value of magnetic field intensity dH at point P at a distance r capital R is given by this formula right with alpha as the angle as shown there. So, in the vector notation it becomes this formula right. So, this is well known Biot-Severt law. Now, thing to note here is that either you have line current, you can have surface current or you can have volume current. All 3 cases that overall unit. So, is ampere meter ideal is ampere meter k ds k is the surface current. When, when does surface current come into picture? For example, when we will see later on when a high frequency excitation is there and skin depth is very small current tries to remain at the surface and then you have what is known as surface current density. We will see talk little bit more about later on this. So, k ds, so k since it is surface current density as shown here this is the surface here and the corresponding k is the unit is ampere per meter because it is only surface. So, one dimension is not there because it is just a surface very thin surface. So, ampere per meter into meter square is again unit is ampere meter and the third well known thing which we commonly use in field calculations is J. J is volume current density ampere per meter square into volume meter cube again the overall unit is ampere meter. So, all these 3 the unit is overall unit of current multiplied by corresponding dimension is ampere meter. So, always remember I in ideal I is the scalar dL is the vector. So, dL is the one which gives the direction to the current element, current whereas here k and J they are the vectors. So, coming next to current element which is like a line current infinite line current as shown here. So, you have again at point P the magnitude of the vector H is given by this formula where alpha 2 and alpha 1 are the angles as shown in this figure. And if it is a infinite line current for finite line current of course, this is applicable then alpha 2 and alpha 1 will come infinite line current alpha 2 will because when this goes to infinity alpha 2 will tend to 0 and alpha 1 will tend to 180 degrees. And then this formula reduces to well known formula H is equal to I upon 2 pi rho. Always remember in case of electrostatic fields and this magnetostatic fields because E and H are equivalent. E in electrostatics or electricity is equivalent to H in magnetic fields or magnetics. So, if E is in we saw in electrostatics E was proportional to 1 over r square is it not? Here H is also 1 over r square variation. This well known formula I upon 2 pi r or 2 pi rho and there 1 over r dependence 1 upon distance dependence is for only the infinite line current case. But for a finite current element it is always 1 over r square variation for H this point to be noted. The question is if it is a infinite what do you mean by infinite charge distribution? No, if it is a infinite line charge it is q upon 2 pi epsilon 0 r l that is you know it is a 1 over again distance dependence for a line charge. So, here another thing to notice which I have also explained earlier del cross H is equal to J this is the point form of Ampere's law and integral H dot dl is equal to I is the integral form of the same equation. And since del cross H in general is non-zero so this will be a non-conservative field. We have earlier seen what is the difference between conservative and non-conservative field. So, this is a this is an example of non-conservative field. Now coming to magnetic flux density we know B is equal to mu naught times H in case of free space. In materials we will see later how this gets modified and how mu r relative permeability comes into picture. Now you know there is often confusion between this what H represents what B represents. So, one can say that H represents a source which is current and its unit of course unit of H is Ampere per meter. So, you can say it is a macroscopic you know representation of magnetic field whereas B you can visualize at a point is it not because H you cannot really visualize at a point. So, you can say that H is representing the current source which is you know producing it. So, B definitely can be visualized at point we say flux density at this point is so much Tesla or Weber per meter square. So, that is why it is you know B can be called as macroscopic you know representation of magnetic field. So, again it is a very you know subtle difference between the two and you know only one of them is good enough to you know represent magnetic fields if it is free space both B and H are not required only one of them is sufficient to you know describe the magnetic fields if one is known other is automatically known. So, then this is a Gauss's law of magnetics closed surface integral B dot ds is equal to 0 if you apply divergence theorem then you will get divergence B dB is equal to 0 that leads to the second Maxwell's equation in point form divergence B equal to 0 which tells that there are no magnetic sources or sinks and flux lines they are always closed. But we should always note that if this surface integral if you make it open surface integral integral B dot ds open surface integral gives you the flux crossing that surface moment you make it closed it becomes 0. This point I already explained free space only one of them is sufficient to describe magnetic fields. Now, divergence B is equal to 0 and we know that divergence of curl of a vector is always 0. So, then we can define B vector as curl of some vector. So, curl of say A vector what is this A we will see in the next slide. So, B we can define as curl of A right. But we know B from you know Bayard-Severt law which we have seen in the previous slide H H was this quantity that quantity into mu 0 gives you B. Actually by using these two formulae del cross B is equal to del cross A and B is equal to this you can then derive expression for A as given here. So, A is integral mu 0 i dL upon 4 pi r and integrated over the you know length L right. Again as I mentioned to you whenever you see in most of the equation whenever i appears involving vectors i will be you know accompanied by dL vector that only gives you the direction of current right. Whereas the same equation if you want to represent in terms of if there is a volume current distribution then this i dL will get replaced by j dV where j is the vector dV is just a scalar. Now we come to a very important quantity in electromagnetic which is called as magnetic vector potential you know because you have potential in electricity which is V and then E we calculate in terms of V as E is equal to minus gradient of V. But in magnetics you know you have only B and H fields and then there is you know no other there was earlier no other quantity in terms of which you can calculate B and H fields. So, then you know using all these basic equations and the properties and the Maxwell's equations and whatever we have seen the researchers coined this magnetic vector potential A and later on I will explain you this potential that word actually is not really a correct word because it is not really a potential because potential means it is like a scalar right. But this A is actually vector. So, Maxwell originally called is magnetic vector potential as electrokinetic momentum vector. So, it basically was actually it should have been called as vector but this magnetic vector potential this you know these words have been very commonly used in now literature and most of us are used to it. But we should remember that it is vector and as Maxwell correctly termed it is electrokinetic momentum vector. How that is momentum vector we will see later when we see time varying fields. So, del cross H is equal to J and then you know B is equal to mu naught H. So, this equation can be recast like this and then B you replace by del cross A and then we know del cross del cross of a vector is given by del of divergence minus del square of that vector. So, this is what these are common vector identity and is equal to mu 0 times J. Remember that we are still in you know we are not describing materials yet. So, we are in free space that is where mu 0 is appearing.  So, now actually we you know divergence A. So, till now what we have done is B is expressed as curl of A. So, curl you know see in vectors by definition of vector a vector is completely defined if it is both curl and divergence are defined. So, till now we have defined only curl of a vector. So, B B is equal to curl of A. So, divergence of A we have not yet defined. So, we will now define divergence of A as this mu 0 epsilon 0 daba B by daba T. Again this is free space that is why here mu 0 and epsilon 0 is appearing and this is called as Lorentz gauge. Now this is not an arbitrary definition. This Lorentz gauge is again it can be proved that it is consistent with continuity equation. This you can see in all textbooks it is you know that derivation is given straight forward derivation. So, this Lorentz gauge is consistent with continuity equation. And now hence this A gets completely defined because we have defined curl of A which is B and divergence of A as given by this expression. So, A as a vector is completely defined. Now at low frequencies this right hand side term will be almost negligible because you know frequency is small. This again daba D by dt or daba V by daba T is given by j omega V in frequency domain. And since these mu 0 and epsilon 0 are generally very small numbers is it not 4.2 10 to the power minus 7 8.85 into 10 to the power minus 12. So, these are the product is small and if the frequency is small this is going to be a very small number is it not. So, that is why this is 0 at low frequencies. So, divergence A becomes 0 at low frequencies and then this is called as coulomb gauge. So, that is why you know in low frequency electromagnetic divergence A is always taken at 0. When you are dealing with high frequency electromagnetics involving wave propagation, antennas and whatnot, divergence A is taken to be by this expression Lorentz gauge. So, this is the difference which clearly separates low frequency and high frequency electromagnetic field computations. So, now if we all agree that this is consistent theory then we will take divergence A equal to 0 and then del square A will be then equal to minus mu 0 times j. Now compare this equation with Poisson's equation in electrostatics which is del square v is equal to minus rho v upon epsilon naught again this is free space in free space and the corresponding expression for v integral rho v dv upon 4 pi epsilon 0 r. So, now this expression for v and the expression that we saw in the previous slide for A you can see the similarity if mu 0 and epsilon 0 appear individually if one is in numerator other will be in denominator we will always find that here mu 0 ideal here you know rho v dv upon 4 pi epsilon 0 r. So, here mu 0 is numerator here epsilon 0 is in denominator. So, you can 4 pi 4 pi is common here it is you know line source that is why ideal is there here it is a volume charge. So, it is rho v dv. So, there is a similarity between the expressions. So, now this is a you can say this is a vector Poisson's equation del square A equal to minus mu naught j is a vector Poisson's equation that can be split into three scalar equations del square A x is equal to minus mu 0 j x the j the x component of j then similarly y component of j and z component of j will be correspondingly be equal to del square A y and del square A z. So, these are basically three scalar Poisson's equations right now little bit you know more understanding of magnetic vector potential we will do. So, of course, we have understood its you know expression and its you know genesis on the point of your definition of divergence and curl, but let us understand in space how it you know works. So, again here i he i is the line current here. So, if the i is directed like this through this del vector see again I have shown here purposely del as a vector here I have not just shown i because this del is the one which will give the current the corresponding direction right. So, we should be consistent whenever we show something we should always remember this small small things that will make our understanding clear. So, for this current along this direction you have the magnetic vector potentials also directed in the same direction because you know in the previous that expression you have on right hand side there is only one vector D L. So, the direction of that vector will be the direction of A is it not. So, D L is in the vertical direction through which the current is flowing. So, A is also vertically directed and the magnitude of A is reducing as you go away from the source because it is inversely proportional to the distance. Now here del cross A is not 0 obviously because it del cross A is B, but how do we you know sort of understand it here suppose we take this as the z direction these are z direction then A z you can see the A z component is changing the direction is fixed, but the magnitude is changing is it not. So, that means A z magnitude is changing with respect to x right and in the curl expressions you have now for example here I have written that we will see that in the next slide also B is given by this expression daba a z by daba y A x hat minus daba a z by daba x A y hat. Now here we are talking of two-dimensional approximation we are not actually that means current is in z direction and we are we are basically you know the B field will be that is why in x y plane is it not right. So, here moment A z is changing with x that means at least one of this term is going to be non-zero that means B exists right now what we have considered here is this. So, this is z and suppose this we are talking of x so what we have shown is here this is you know current and A is we have shown like this and only the we are considering the x component, but if you rotate this by 90 degrees here basically daba A z will change with respect to y are you getting what I am saying right. So, here it is changing in this plane which is z x plane right it is basically the variation with respect to x after 90 degrees rotation in phi you will have A z varying with y and at any intermediate angle A z will vary with both x and y. So, that clearly you know helps us understand this magnetic vector potential and its relations relation with B right and how the A vector varies with you know distance from the source. Now, this slide is particularly very important for 2D FEM calculations that we are going to see throughout this course our course will be dominated by 2D magnetic field computations and that is why this side is quite important. So, psi this denotes flux is open surface integral B dot ds remember again I am repeating close surface integral B dot ds is identically 0 open surface integral B dot ds is equal to the flux causing the that surface right. So, now B you replace by del cross A right and then you apply stroke theorem you will get psi flux is equal to close counter integral C stands for counter close counter integral A dot dl. Now, if the flux is in webers this is meter. So, the unit of A becomes Weber per meter right. So, unit of A magnetic vector potential is Weber per meter. So, line integral of A around any close path is equal to flux passing through the area enclosed by the path right this is what is describing this equation. Now, again let us understand in space what is happening. Now, here you have current again in z direction. Now, here again I have marked the coordinate axis z is vertically up x is in the horizontal direction. So, y is going into the paper because it is like x is horizontal y goes inside z is up right. So, it is basically the fingers of the right hand you turn around from the x to y you will get the z direction. So, x y into the paper and z up right. So, now again you know you have this current source directed along the z direction. So, you have this A 1 and A 2 at two points which again will be in z direction.  If you take the top view of this in the x y plane you will see this current source adjust by dot is it not. So, this will be a dot current is coming out and the field lines are as per the right hand rule again thumb pointing in the direction of current field will be given by the fingers right. So, the field will be as shown there right. So, this is x y now again here x y so z comes out. So, remember here y was going in but here moment you show this as x y z comes out. So, it is basically by the right always go by right hand rule okay. So, now this and if you see here the same thing in 2D I have shown it just by a dot here. So, here of course I have shown the field but if you just have to understand by you know points. So, A 1 and A 2 will be at 2 there will be the such two points in this x y plane right. You should remember the flux is always associated with the corresponding area through which it flows. Now, again you know what is the area that is corresponding to this flux is flowing through such an area where one dimension is along z axis. So, here actually when we say the flux is really flowing flux really does not flow to this paper surface. Flux actually you can always have to visualize that flux is crossing some surface. So, what is that surface either it will be z x z. So, this will be you know either this surface. So, that means suppose these are current source the flux is crossing like this is it not. So, either it will be suppose this is z it will be either z x plane and the corresponding surface or if you turn by 90 degrees in phi it will be z y right. So, always it is crossing some surface which is you know and at interval it will be you know some surface which is either in z x plane or not in z y plane is it not some intermediate angle. But whereas, if this is there it will be z x this will be z y because this is y as shown here y is into the paper right. So, you should be very clear about this. So, flux crossing this surface and it is going in. So, flux goes into this surface right and that is given by integral b dot d s a n hat where a n is the unit normal to the surface. Now, again I mentioned in one of the previous lectures the direction of this a n will be decided by now this counter integral is it not because this is open surface. So, open surface has two possibilities of n either you know it is you know it is either it could be this is open surface it could be this this unit normal or it could be this unit normal. But now we are taking some counter integral. So, that counter integral decides the direction of unit normal for that open surface right. So, I am again and again repeating some of these points. So, that these are very important for you know in general vector calculus point of view and electromagnetic field distribution in space and visualizing those. So, now in this case it will be a y hat for the x z plane surface shown in the figure right it will be a y hat for the x z plane shown. So, now this psi is equal to a close counter integral a dot dl now that actually is you know if you evaluate that it will be simply a 1 l l is the length of this you know segment.  So, a dot dl will be 0 on this and on this and on this side a and dl they are oppositely directed because a is vertically up dl because we are taking counter integral. So, dl is down is it not. So, that is why you have got a minus sign. So, finally and then this again is 0 this does not contribute to the line integral. So, then you have got psi is equal to a 1 minus a 2 into l and for 2D approximation again in one of the previous lectures we discussed that whenever we do 2D approximation we take 1 meter depth in z direction and that is why if we do that then l this l will be 1 in z direction and then this will just reduce to a 1 minus a 2. Now, coming to this point again. So, now here if you take two points on one counter these points 1 and 2 are at same distance from the current source. So, then a 1 z minus a 2 z will be equal to 0 is it not because psi is 0 because no flux is crossing the surface. So, that is why they are called as see this concentric counters are called as equi A or equi potential counters or flux counters because on this counter since a a 1 and a 2 magnitude is same. So, no you know flux crosses in fact flux is going along suppose these are the two points suppose these are current and these two points are there. So, the flux is going along the points it is not crossing any surface which is you know subtended by those two points are you getting what I am saying the flux when flux has to cross the surface is it not. So, here those two points the flux is just going along those points. So, it is like a parallel it is going parallel to those points whereas when the flux has to cross it has to go perpendicular for example here the flux crosses that surface is it not. So, it is basically here though that is why the flux is crossing the surface here and that is why there is a difference between a 1 and a 2. Since here flux is going along the you know counter joining points 1 and 2 right a 1 minus a 2 will be 0. So, this is another important point that flux can be set up by using only boundary conditions in terms of A. Now, for example here these one rectangular you know geometry this could be a magnetic material right and you want to set up the flux flowing like this that you can easily do by imposing the boundary conditions like on this side there is a a 1 now remember a 1 means a 1 is z directed. So, this is the x y plane and a all along this will be vertically directed as shown here as I am showing here it is a 2 again you know vertically directed, but the magnitude everywhere is same a 2 right and on these two horizontal you know lines you have homogeneous Neumann condition. I have explained in one of the previous lectures what is homogeneous Neumann condition that is derivative normal derivative of that field variable is 0 that means the value here. So, now for this horizontal you know line then unit normal will be this is it not see why it is normal because normal we always also set with surface is it not. So, then you may be wondering where is the surface here always remember practical things are always 3D. So, here also it is 3D figure approximated as 2D. So, what is the third dimension into the paper and it is 1 meter depth. So, that means this length into 1 meter depth and a corresponding surface form for that surface this will be the unit normal is it clear. So, unit normal is always to a surface right. So, always you know this you have to imagine this is a 2D approximation. So, what in the background there is a 3D actual figure and corresponding surface here is formed by this line and a 1 meter depth you know segment and a corresponding you know rectangle is it clear and the unit normal then this will be the become the unit normal depending upon you know whether it is how the counter integral is taken and all that either it could be this or it could be you know the other way right as I explained earlier. So, so difference in a values matters that means you know you just a 1 minus a 2 is we have seen flux is it not. So, you take the difference accordingly to set a given flux and flux density. That means you know it does not mean that there is no current source anywhere current source is there that current source could be you know distributed current source here somewhere and it could be a distributed current source. That current source I have shown purposely a dot here because then that dot only will produce the flux in this direction. So, there is a distributed current source which will give this condition that a is directed vertically up because this current is coming out. So, similarly a will be coming out here and the magnetic vector potential is same all along. Why I am saying here distributed current source if it is a point source you will not get constant a values on this is it not then it becomes a circular the values will be constant only on the circle. So, that is why I am saying some distributed current source only will give you constant a values on this vertical line right. Now, so vertical boundaries we have diraculate boundary condition right and horizontal boundaries we have homogeneous Neumann boundary condition. Now, this diraculate boundary condition these are not homogeneous because there is some finite value. If suppose this a2 you define it as 0 right because only the difference matters. So, a2 you define as 0 and a1 correspondingly some number which sets up a given flux then this a2z will become homogeneous diraculate boundary condition. Whenever the potential specified is having some finite value it is non-homogeneous if the potential specified is 0 it is called as homogeneous diraculate boundary condition right. So, now this v is equal to del cross a and then you know this is a normal our curl formula and this expression in fact we saw earlier here this expression comes because there is only a z a x and a y are 0 right because a current is directed in z direction. So, a is directed only in z direction. So, a x and a y components are 0. So, that is why you get a simplified expression for v in two dimensional case remember this two dimensional case. So, now this is a very this last point is very very important not described in many you know textbooks of electromagnetic or even in on numerical techniques on electromagnetics. But this actually explains the magnetic field lines and electric field lines very very clearly and corresponding difference between them. So, now on the vertical boundaries here we have defined a z is it not we have defined this a z values two a z values that means daba a z by daba y is 0 because a z is constant on those vertical boundaries right. So, daba a z by daba a y is 0. So, that means this this term becomes 0 right that means only this term exist b y exist is it not only the y component of b exist right. So, in this case the flux line or the equi a line and a b line they flow together because both are in the same direction y direction clear. So, that is why magnetic flux flows along equipotential lines in magnetics right. Now, let us see the corresponding thing in electric field lines case electrostatics. So, in contrast e is equal to minus del v and then you have got this expression for del v and now let us take again a simple parallel plate capacitor case. Now, here I am taking the capacitors plates which are you know vertically oriented. So, this is one plate and is another plate. So, you have v 1 here specified v 2 here okay. Now, the we know this will be the equipotential lines vertical and the e lines electric field lines will be orthogonal. Let us verify that they indeed are orthogonal by through this equation. So, now here on vertical these bondage we are specifying v 1 and v 2. So, here daba v by daba y is 0 is it not daba v by daba y is 0. So, only e x exist right. So, that that is clear that only e s exist and that is what we have got e exist in x direction only. So, now here e and v lines they are orthogonal to each other. So, you can say electric flux which in case of electrostatics it is e or d is the flux remember it is e or d that flows perpendicular to equipotential lines right because see v you are specifying v right on the vertical. So, v is constant on as a function of y. So, as y is varied v is constant. So, daba v by daba y is 0 let us select previous in the previous case of magnetics right. Now, here this shows clearly the same difference between electric flux and magnetic flux. So, here in this slide you have these two plots one for electric field one for magnetic field in case of electric field you have a high voltage conductor in the vicinity of some ground plane or ground conductor which also could be concentric in this case because the field lines are symmetrical about the circle. So, you have here you have equipotential lines circular and the e fields are crossing orthogonally to the equipotential lines and as you see here the there is no electric field inside this conductor because there are no field lines because the conductor is assumed to have very high conductivity tending to infinity and that is the main difference between the electric and magnetic fields. In magnetic field case you have a current carrying conductor here and there is internal as well as external field. So, the field is there inside as well as outside and you can see there are no orthogonal lines crossing each other here. So, that confirms the statement that we made that in case of magnetics equipotential flux flows along the equipotential lines in case of electric field the flux which is no e field flows orthogonally to the equipotential lines. So, that clearly brings out the difference between the electric and magnetic fields. So, with this we end 7th lecture from tomorrow we will further tomorrow's class we will further see the magnetics and later on we will see time varying fields. Thank you.


Let's look at the magnetic force on charged particles and do some example calculations. So first of all, this is the equation, which I've introduced in another video, that explains that the amount of force is equal to the amount of charge times the amount of velocity, also called the speed, the amount of the magnetic field, sometimes called the magnetic field strength, and the angle theta, which is the angle between the velocity and the magnetic field. Now just to give you a quick picture of that, and we'll be referring back to it, what we've got here is that both velocity and magnetic field have directions. So maybe the red is my velocity and the B is my magnetic field. Then the angle between them is the angle theta that I would use. We're going to start with just one calculation here, where I'm going to plug in some values. And I want to make sure that you can plug these in and get the same value in your calculator. So for the charge, you're going to have something in coulombs, and it's generally a small amount. This case, this is 4 times 10 to the minus 6 is actually for microcoulombs. 45 meters per second, 5.4 Tesla, and you want to have coulombs meters per second and Tesla so that the way the units work out is going to be Newtons. And again, I worked on those units in the original video that introduced this equation. Now for the angle, you could have it in degrees like I have here, or you could have an angle in radians. Just make sure that if you've got the angle in degrees, your calculator is in degree mode. Or if you have the angle in radians, you have your calculator in radian mode. And in this case, when I plug it all in, you should get 0.000687 Newtons, or 6.87 times 10 to the minus 4th Newtons. So it's a fairly small amount. Now one of the things that we could do here is we could change the direction, getting smaller angles and larger angles. And I'm just going to show some calculations that I've already worked out. So for example, if instead of 45 degrees, I had a smaller angle like 30 degrees, I get a smaller value for the force. And if I had a larger angle like 60 degrees, I get a larger angle for the force. And that's because of the way the sign calculation works. I'll remind you that the limits for the sign calculation have to do with the 0 and the 90. Sign of 0 degrees is 0, and so you'd have no force. The sign of 90 degrees is 1, and that's going to give you the largest force. Just to show you that it really does give you the largest force, if I go ahead and grab some larger values for the angle, we'll see that our force starts to drop back down. And the sign of 120 is symmetric with the sign of 60, and so that gives us the same force. The same thing that 150 is symmetric with the 30. And then when we get out to 180, we're back to 0 newtons. So that tells us how our values are going to change here depending on our angle. In most problems, you might be given the angle as an actual angle measurement. In other cases, you might have to figure out the angle because they'll start describing things, like if the velocity goes towards the right and the magnetic field goes up. And once you sketch that out, you'd see, oh, that's a 90 degree angle. Or if the velocity went to the right and the magnetic field went to the left, that's 180 degrees. So you can start to see how you can use those words and then draw a quick diagram just to confirm what kind of angle you're going to have.


You know, there's a huge difference between setting a boundary from your soul and setting a boundary from your ego. When the soul sets a boundary, it feels like it's divinely guided. It feels somehow intuitive and healthy, and coming from the best part of us. When the ego sets a boundary, it's often coming from reaction towards someone, something else. It's often coming from fear and anger and the desire to control an outcome that we don't want. In other words, when the ego sets a boundary, it doesn't come from our best and highest self. It comes from a poorer version of us. So when you set boundaries, ask yourself, is this my ego trying to control outcomes? Or is it my soul divinely guiding me into something that's coming from the best version of me? I think you find there's a huge difference in how those boundaries feel to you and to others. I hope that helps you guys. Love you.