Welcome. Thanks for joining us today. If you're joining us live, it is Monday. And if you're joining us recording, who knows what day it is, glad to have you here. We're thrilled to have Timothy Sarantonio with neon one joining us today to talk to us about what time is it, and when do donors donate. So, Tim, we will jump into this conversation. Deep with you but before we do that we of course want to make sure that our audience knows who you're seeing or listening to Julia Patrick is here the American nonprofit Academy CEO. I'm your nonprofit nerd CEO of the Raven group, and we are so thrilled to have our continued supporters. Thank you very much to Bloomerang American nonprofit Academy fundraising Academy nonprofit nerd, your part time controller, the nonprofit Atlas nonprofit thought leader as well as staffing boutique. These companies are invested in you in your community in your mission. They're here to support us and to allow these great conversations, like the one that we are about to share with Tim from neon one. And again if you've missed any of our episodes you can find them on root Roku YouTube fire TV Vimeo, and also in podcast form so make sure that you stream that and you queue up the nonprofit show. And today again thrilled to have Tim joining us and really excited to get super nerdy with you Tim because you just said in the green room chatter that this chapter of the report is your favorite so yeah welcome back our friend. Well thank you for having me and so soon after the last time I know that that's an honor. You know that that not many folks have so I really appreciate being here, but you got me on a topic that I can go on and on about so. We only have 30 minutes I know that so we're really just scratching the surface here. But let's talk about this, this topic for those of you taking a look at the video. There's a nice graphic of this very beautifully done report. We can dive deep into donors and understanding the future of the individual giver and again, you're right this is something I'm sure we could all talk hours about and when we talk about the time continuum. That's certainly something to consider. So, so get us started here what are some things that you'd like to say. Well so for folks that that might not know what this report is this is a new neon one report that came out in March. It's very fresh. Yeah, and we basically wanted to answer. Since March 2020. What has changed and what is maintained about how individual givers behave, and we wanted to answer six key questions who what when where why, how. So, by starting with that each is a chapter. It helps hyper focus the types of research and analysis that we did. And the, when chapter is nump chapter three and that's when things really kicked in the high gear in my opinion I think that the whole report is is very insightful talks about demographics and and kind of the psychology of things but the, the when chapter. We really, I think it just blossoms in terms of of really kind of sinking in and unpacking some of the deeper questions that the data presents us, because ultimately, we want to act on this a lot of reports just end up being like, you know I'm going to sit this on the table. I've been blown away since the last time that we talked. One, how many people have downloaded this thing. Good, good. If you haven't get it. Yeah, and, and to how many people printed it out. Actually, like I got, I got tagged and we got tagged in a bunch of, of like Instagram posts and things like that where people were like showing, they printed it out so. And that's I think it's because people learn in different ways to and some folks like they want to be able to highlight and spend a little bit more time that's one thing that I've had people say is, I want to make sure I have the time to focus on this so ultimately, my final point here is time is so precious. And there's only so much in the day.  And so we have 365 or whatever, depending on a leap your days in a year. 24 hours in a day so what are we going to do with it. What are we going to do with our time and so we're so appreciative that people found this a useful expenditure of their time. Before we get get into more specifics. If you can paint the picture of your research, because you took a lot of information from a lot of different sources. And I think sometimes we look at data and we're like, we're making a lot of decisions but when basically the sample set has been 22 people and half of them were employees and the other halfers family. You're looking at something that I feel in the statistical world is really has a lot of gravitas and really is can support these theories that you've uncovered. Absolutely, absolutely. Thank you for that flag to Julia because we we looked at two primary data sets when it came to time in particular. The first is with our partners at blue meringue donor perfect in Kila, the fundraising effectiveness project, because that is going to show over near 10,000 organizations in 2021 alone, we drew from, and you know $10 billion right like this is a lot of money. And so what was fascinating was to be able to validate, you know, what time of the like what time of the month somebody's giving what time of the year somebody's giving so that was the big data set that we drew from there. But then, what it doesn't tell you is, for instance, what time of the day somebody is giving. We have that those we looked at our, you know, 2.2 billion and. Oh no. Well, it's Julia. Oh, we'll be good. Okay, we'll go times a flat circle she'll come back. So, um, so we, we also looked at our online giving behavior in particular because that's really reliable to know what the donor is doing. Right. Because, and I know for the nonprofit folks were listening out there, you might not always enter the checks the day you receive them. So, so dates are important. But we know if somebody donates on their phone or through the computer, or something like that. There's a timestamp. We know exactly when they hit the submit button. So we can say this is what the donor did not what the nonprofit professional did when it came to something like a check. Still interesting. But with online giving we actually could tell exactly when somebody preferred to give. So, so we looked at a few different data sets for this type of thing that Julia's point but it was really fascinating to kind of see calendar year as well as the day of the week. I had a theory we can get into when appropriate. Well, hey, I'm all for that and we're just going to, you know, do and prop to here thankfully we know the topic and that's something that we can certainly talk about. I'd love to hear your theory and I'd also love for you to demonstrate to us as much as, as well as you can really what the system you've provided on the report it's interactive we can go in we can toggle, you know the time calendar. So this is something that we can all take a look at and really play around with because you're right, guilty as charge we don't always, you know, input the data right when we receive the check, but we can certainly talk in track exactly what our donors are doing and that's really important. So we created a data hub that that lived that lives on our website you don't have to enter an email or anything anybody can go to this it doesn't matter if you're using our product or not. And there's a few interactive elements that play actually pretty heavily with time right now so. So if you just want to kind of get a snapshot. For instance, Jared you can just go and look and there's a clock and the clock will actually recognize what time zone you're in and show you pretty much the the from the report, the the number of gifts coming in. So yeah. So basically let's see oh yeah so so here's the interface itself perfect timing. I'm going to stop using the word time if I can. Yes, like, like, at least in terms of bad puns dad puns so so this is the interface folks it'll it'll recognize your time zone and you can play around and you can say like okay. We did, we did actually think about, should we add toggles for like day of week and things like that. And, and we found for simplicity of understanding that will do this is a blended analysis but the report actually has a graph. And that is much more detailed around the details. And so, but we found that around 1130 in the morning on a Thursday is when people prefer to give online. So, my theory, Jared that I was reference referencing was I when I originally asked to have this data point pulled my theory was that people would get blitzed on the weekend, and want to give at like, you know, after having a few beers. Right. So would we see a spike on a Saturday, and a sun like a Sunday like during a football game or something like that where it's like, right. Code is bouncing right. Yeah the bar code's bouncing and it's like let's go act on it that type of thing. That theory was completely disproven. It is does not bear out other than a small blip at 830 on a Sunday which might correlate to something but, but in general, no people are not bothered to give on the weekend they're stopping to think about it so I have further things that we need to analyze that I think you're going to be really excited about that we're we're starting to think about now from an analysis standpoint, but yeah 1130 on a Thursday, 1130 on a Thursday. So that's fascinating time because we, when we look at, you know, marketing, when do we send emails, you're not necessarily saying this is when you send the communication, you're telling us this is when that communications receiving reaction. Yes. And that to me, you know, I would love to compare this with the marketing to say, when is the open rate. You know how often what what are the times that people are. What do you what do you think I'm going to look at next so we have we have and there's has been so so giving Tuesday or partners at giving Tuesday did work on some benchmarking work with with Mailchimp. Relatively recently but the problem with that data is is it isn't necessarily really drilled down an indexed against when people might be clicking through and activating on a form. Right. The Mailchimp callback data is is kind of labyrinthine and if I'm going to be honest, but, and now they're actually more geared towards small businesses because they got acquired by the people make quick books right so for us, we have our own email engine. That thing sent out 250 emails last year I think that's a pretty good data set that you can draw on. So that I hinted at it that's actually where I want to go next Jared is to do that exactly is to see when are people sending the emails and when are people opening the emails and if there anything that lines up. I'm so glad that my crystal ball is working. I'm going to go buy my lottery ticket numbers now. What about the calendar impact how does the calendar you mentioned 365 days a year unless it's leap year. So how does that impact our culture of philanthropy in relation to that donor engagement. One of the facet this actually, they're all fascinating obviously but I think that that one of the most interesting questions that fundraising consultant T clay buck brought up when he started thinking about this was, Have we conditioned donors to give in something like December. Is that we, is that when we're asking them, or do they prefer that. And I think there's some interesting things that don't paint a clear one way or the other that I can unpack right now. So one of the funny parts is that like one of our best performing blogs for instance, is end of the year statistics that every nonprofit fundraiser should know it's just like everybody like Google's it and it pops up. And, and one of the ones at the top is like 31% of gifts come in December and I was like, that's true anymore. So, we're going to have to update the blog spoiler and, and, and, and that's fine that's fine and it actually turns out that that was some old data from the network for good, from about, I don't know like five years ago, and, and that's part of the things that like I encourage everybody. When you're doing your research when you're trying to understand what's happening in your own world. Just check the attribution and the methodology this goes back to Julia what you're talking about like this survey come from 22 people and three dogs and then that's it right like, like, what is the demographics that's why in our center. In our report, extensive methodology is outline. Same with fundraising effectiveness project we lean very hard on transparency there so you can trust it. And so, for us, I actually for for calendar year giving, I didn't want to use ours I actually wanted to see fundraising effectiveness project data so they gave that to me. And it found that 20% or so give or take a few percentage points which is still significant but like, it's not no 31% I'll say that. And so it was about 20% 1720% came in December. Good. That's a lot in one month. But it actually some of the more recent research that FEP had come out with shows that November December, especially in the past few years are starting to trigger some really interesting donor behavior. So, I'll talk about FEP but I do want to mention something we saw internally in our in our research to for something like online giving. So, some research that we found is that I'm going to read it some read doing it correct. We also found that donors whose first gift first gift to an organization ever in November or December has retention rate that is 53% higher than those making their first gift in the period of January for and through October we also find that they're typically making their second donation which we both are we all all of us now is way important for any times more important than the first gift. And that's coming around 11 to 13 months. So, this begs the question. Is it because we're slamming people with end of the year updates and communications, and then they're giving a year later, typically, because there's a big spike that happens at that 11 to 13 months. Is it because we're slamming them with communications, or is that when they prefer to give the final interesting point that leads me to believe that I actually think that there's a fair amount of people who do like to give in December. As their preference was that FEPSA that donors who gave in the last day of the year have a lifetime value 15% higher than the average donor. And that if you play around with our interface there. We actually have day by day analysis of December because it's so significant. And we show that, and it spikes way more than giving Tuesday, even online donations, because if it's if it's offline, going back to the thing. Yikes, I forgot to enter all the checks to get into all the checks right like, like offline is not as sure fire as online. And so with online I said, Okay, what happens there, bunch of people really love to give on the 31st way more than giving Tuesday online. And that makes me feel like either they're getting slammed with so many communications, which yes that might be happening but if they didn't care they wouldn't be given, or they would give another time right. That is also geographically so I live in Arizona we have our qualified charitable organization tax credit state right we can give to several different buckets for for state credit, if you will. And so for me my cycle is not December 31. It's tax day.  It's coming up. This is fairly very relevant. How many, how much money do I need to put and then what buckets buckets. Well, guess what, my second favorite chapter is the where chapter, and, and, and we are working on geography as as part of that interface on the crystal ball I knew it, I knew it. But now I had to index it now now you have me intrigued to actually go back and look at state giving by month to not just as an aggregate that's that's probably another round of things but probably mid March, mid April, you know, kind of that November December trigger it's also that March, April. Okay. And I would also see to it really depends on what part of the country let's say for example, you're in the south where or the Gulf States where you are prone to weather disasters that happen in a specific time of the year that then drives community response, the Plains Middle America, another time with year where you have major environmental things that can, can, storms happening on the east coast the northeasters that we're typically not going to see during those summer months, like we're going to be, unless it's a sandy situation, we're typically not even worrying about stuff in a major way. Maybe it's kind of weird flooding season that's starting to happen but yeah it's. And there is some some really interesting research around disaster timing. There's, I remember a graphic that I saw very early on. During the pandemic, and then there was also some follow up analysis by fundraising effectiveness project on kind of what happened in 2008.  And then a major drop off. And then you're going to see actually a second. I don't want to use wave but you understand another bump of giving that that occurs, especially from those major donors the more committed people who are starting to come back. And we're seeing this the fundraising effectiveness project data for 2021 just dropped. That November December analysis was only a snapshot the larger report just came out AFP FEP dot org. And there's a nice visualization that that giving Tuesdays done there too. And we're finding that those first time. Those first time donors have stuck around, especially if they're on the larger side, but there's a lot of concern on the smaller gifts that are just being left behind and ignored kind of those those midsize donors and things like that.  Let's get even nerdier acronym soup. Let's move into these NTE codes right and let's just start off with what the heck is this. Yep. I know it has something to do with the red tape and the IRS and you know, always naming something differently. So what very basic Tim NTE. This is the many ways the self reported designation that you put for what do you do it you're not what what is your nonprofit do. That's it this is what the IRS. That's the IRS has this is the IRS naming of it. IRS designated category. Yeah, it's the category so they have two levels they have the major category which is what we filter by here. Is it healthcare is it human services is it animal and environmental is it religion. And then there's the sub designations that like number in the hundreds that overwhelm we just kind of draw it up at the top level but we could get down here and we have done that before where it's like. Wow, it really looks like food pantries did this or interesting that type of, you know, labor union did that like you can get to that level of detail but but most analysis doesn't, you know, start with that. There's a Canadian version to of this that they're starting to do mapping between all these different government designations for for international analysis. We don't have much time you mentioned early in the episode which was just about 20 minutes ago that you could talk about this for hours so graph that we have up on the screen it does show pretty pretty pretty much the same pretty equal except for there's a huge spike. And if my plesural eyes are reading me correctly, it's December 2020. And that's where it really jumps not completely off just the chart, but doubles the majority of the other dates. And that maintained through 2021 to so when we get a we just again there's so from a nerdy standpoint. The reason you don't see a lot of reports coming out or pretty much you shouldn't see reports on 2020, you know the previous years giving in January February is lagging lagging data entry. And we see those checks, and we see that the kind of the, the, the, the lagging data that comes in the drift is the proper term for it so the data drift is people who are like, Oh right I forgot to put like the date on this and then they go back and then suddenly the data is popping up so we don't, we kind of wait, because we know historically there's, there's a period of time that's like the drift will be over. And so that's why you don't see data like like 2021 in December until now, but held strong same idea, people just do give a lot in November and December this is this is the big aggregate time. And a lot of people do their appeals as well so they're definitely as correlations there but there's also a lot of, even if we go back historically, when you know, the Christmas giving in, in, in England would occur and it's our Christmas appeal right and, and things like that and so that did carry over that did carry over. And I think it's cultural and I think we see that with Ramadan and, you know, different different times of when our people are called by an action of faith or witness of faith, you know to participate that way and look at the data, sorry Julia but like yet to just kind of show that in terms of even the data, like a lot of the stuff we don't, we serve some religious organizations but a lot of it is, is even not logged necessarily by the IRS, for reasons. And so that makes up a huge pie. And so all the major monotheistic religions have big times and periods in December. So there's there's a correlation. There's a lot of globalization and correlation aren't always related but there's, you know, there's definitely something happening. Well yeah I think there's a call to people and I think also to it becomes part of that, you know, piece to where we, we fulfill that prophecy because to your point you know we're having more events. We're sending out more communications we're doing more walks or appeals, I mean it all works together. I mean that's why I found your report so amazing. Is there any correlation between Easter and Tax Day? Who knows, right? I really don't know but hopefully one of our viewers and listeners will know. Maybe shed the light on that I don't know. So thank you so much for having me as well. It's been great. I want everybody to know where they can get this report. It's absolutely fascinating. We see a lot of data, a lot of reports that come to us at the nonprofit show and a lot of them can be really Mickey Mouse where you're like really wondering what the gravitas of this research is and how was it put together. It's like a master class and how this is all done. So Tim I'm delighted that you would come on and then also that you could actually communicate this to us and we could discuss it because that's the other part is the interpretation and the vocalization of this and then also the visual elements. So how we can get to this information. It's just so impressive. And think about that from the nonprofits perspective they should take inspiration from that that like the best way that you can actually tell your story isn't through data. It's through making the data very simple to understand. And so when it comes to your impact, it's actually a lot more effective to use an individual story to demonstrate that, right. So, so absolutely and I'm so so so thankful and appreciative of that feedback. Well, it's really cool and it's, it's the old econ major side of my brain coming out. I went to school for history. I don't know how I got into this so I love it. Well, here's Tim's information again neon one. Check them out. You can download this report. It has amazing infographics that time slide riveting I played around with that for 30 minutes trying to figure out, you know, keep adding to it to we're going to keep adding to it. Good, good, good to know. Here's Tim's information. Again, I'm Julia Patrick CEO of the American nonprofit Academy been joined by the nonprofit nerd herself Jared ransom CEO of the Raven group. Again, we want to thank all of our sponsors, and to let you know that if you want to share this episode with somebody else you can find us on Roku YouTube, Amazon Fire TV, Vimeo. Again, all of our presenting sponsors that are with us day in and day out. Thank you from Bloomerang American nonprofit to your part time controller the nonprofit nerd fundraising Academy the nonprofit Atlas nonprofit thought leader and staffing boutique. We say thank you thank you for your support. Wow. Okay Tim, we are buying you a beer at AFP icon in Vegas I'm just with I'll see you there I'll get I'll get that stuff to you after. Thank you so much. Thanks again for having me. Yeah that have joined us. Please stay well. See you can do well. We'll see you back here tomorrow.


Let's talk about some other cross stitch items right now. And as I was doing my research, I learned some stuff. So I think you guys are gonna learn something too. Wait until you see this one brand. I've never heard of it. Cross stitch kits definitely can be a big money bolo and I'm gonna show you what I mean in just a second. Hey, bolo buddies. Thanks for watching. It's from my little pony. Amazing, right? All right, let's get started. They do range from bread and butter to big money. So definitely do your research. What's really great is cross stitch kits right up in the corner of every cross stitch kit. If it's not in the top corner, this is dimensions. So you find it up in the corner. You type that into the search bar along with this one is dimensions, the brand. And you're probably going to find your item. Now, this one, I couldn't find it.  This was in a recent video that I did and I had an offer of $125 that I declined. And let me show you this one.  They are advent calendars. Advent calendars in general, you wanna pick them up because a lot of them are big money. And if they're not big money, they're bread and butter. And they're usually a quick flip. So this is that dimensions. It is a cross stitch kit. It is not the gold collection. The gold collection are amazing. This is just a regular dimensions kit. It is from 1984, so it is vintage, but it says Advent ABC's Chris Davenport's colorful design makes the long wait till Christmas fun for the whole family, hang special treats, candy or small trinkets from each star, then remove one by one until December 25th. At the same time, children can practice their ABC's. And it's an 18 by 14. So there's the information right there that I just read to you. And again, I could not find any solds on this. So I priced it high, I think I priced it at like 200 plus, but it was on sale 30% off, I believe right now for 175. I turned down 125, but somebody offered me 150. So I went ahead and took that. Now, if I would have waited until closer to Christmas, I probably could have got in the $200 range. That's just me speculating, because honestly, there's no comps. So $150 for this. I picked this up at a thrift store. There's a video on that. It looks like this. If you wanna go check out what I paid for it. Here's what the back looks like. So it's got the thread and it's a kit. It's a cross stitch kit. So let's talk about some other cross stitch kits. And I'm gonna show you some big money Bolo's to be on the lookout for. Let's get started. Okay. So I just shared with you that awesome Bolo that I sold. Super excited about that one. After I didn't accept the 125, I was kind of like, now you get that? Should I have accepted it? But I just, I knew that I could get more. And I really think that I probably could have got more than what I took, but that's okay. I'm super happy with that sale. Let's talk about some other cross stitch. I'm gonna show you some of my items right now. And as I was doing my research, I learned some stuff. So I think you guys are gonna learn something too. Wait until you see this one brand. I've never heard of it. Like we all know about dimensions. And we're gonna talk about that a little bit. Which ones to look for. But this brand never heard of it. And there's a whole bunch that go for big money. So stay tuned for that. All right. Let's get started here. So first off, here is my sale right here. And I noticed that I didn't put calendar in the title. But if you look here, it's not actually numbered. This is an ABC Advent. Okay. But it works the same way. But it's basically doing the ABCs. Okay. So this one, I went ahead and just typed in Advent Calendar just to see what else would come up on the high solds for cross stitch. And there is one that is the Gold Collection Advent Calendar. And this one went for 143.50. So mine was not the Gold Collection and it went for more. But from what I have found, the one that I found is really, really, I'm gonna use the word rare because no comps, people were messaging me about it. I had multiple offers. It was definitely an item that's hard to find. So this one right here is the Gold Collection. It's the Joyous Advent 143.50 and 129.50. Two of those that sold. Now here's one that it says Dimensions Christmas Plastic Canvas Kit, Teddy Bear Advent Calendar. So this is the plastic canvas and it's probably shaped like a tree. That one went for 49.99 on Best Offer. So not really 100% sure what that sold for. Here's another Dimensions. Here's what I'm gonna say about these two. These were put on a buy it now. I personally, anything Advent and Dimensions, I would price higher. But maybe this is kind of the going rate for this calendar, but I typically price up, especially on things that I know people collect and that are in high demand. And if you guys don't know about the Avon calendar, that's another great one to be looking for. It has nothing to do with cross stitch. It's just an old Avon Advent calendar that is big money. So here's a few more of the Dimensions. Just some different styles. These are bread and butter, you know, $35 or less, but still great pickups. If you're at a garage sale and you see these and they're a buck or two, definitely pick them up. They will sell. So I always pick up pretty much any cross stitch if I can get it for a dollar or less because it's gonna be at least bread and butter, sorry. This one, I don't know if it's, it kind of looks like it's just the cardboard there. So not 100% sure on that. All right, so let's get into some of these cross stitch kits. If you search, I do wanna show you this, let's go to a cross stitch kit and we are gonna search ended recently, okay? All listings, not just auctions, all listings. And we're just gonna kind of go through here and look at some of the solds. And what I want you to see is prices, sold prices are all over the place. We've got 22 on this one. This is a Dimensions. This is a gold collection. If you find the gold collection, always start them high if you're gonna auction them or comp them out and price, this is my recommendation, price on the high end of comps and put a best offer on it. That is what I would do if there's a few listed. I also recommend getting these items on other platforms because sometimes these items are saturated, let's say on eBay and Poshmark and Mercari may only have one or two listed or maybe none. You may sell it for more on those other platforms. I start my items on eBay and then I use list perfectly to cross post my items to the other platforms. List perfectly, I use it to cross post. Now they do have a catalog where you can list into the catalog and I'm gonna have more information on that soon. But what I do is I just get my items in eBay and then get my items over to the other platforms. If you wanna try list perfectly, there's a demo video down in the description that shows you how it works. And if you're interested, you can use coupon referral code BOLOBUDDIES, all one word to get 30% off your first month. I've been using it since 2019, you guys, and I love it. So here's probably something you shouldn't do is start your auction super low. This one went for $5, but we are gonna go into this. This is a do it yourself DIY diamond painting. They're calling this cross stitch. And again, we're gonna go more into this soon. So stay tuned for that, because this is something that I found from doing this research. So here's a JLIN or JANLIN. These typically do not go for as much. That doesn't mean there's not some out there in the wild that are worth more, but typically bread and butter on that brand. Here's a Bucilla. That's another good brand. This one went for 69, another gold collection. They had it at 49.99. So like I said earlier in the video, it's really easy to comp these because they have an item or a cross stitch kit item number. So it's right up here in the corner, depending on which one you have. This one went for 9.99. I think this looks like maybe a smaller kit. So these are just some of the recent solds. Here's one, a gold collection that sold for 399.99. So those gold collection are definitely big money if I find those. And I missed one at a garage sale. It was actually in my church sale video. Somebody's like, you had a gold collection in your hand and I think it was one of the smaller ones. I have no idea why I sat it down, but I did. So we all make mistakes. All right, let's move on here. So here's one. Another good thing aside from gold collection cross stitch kits to look for are any stocking cross stitch kits. Now, if you can find the gold collection, that's a double bonus, okay? But even if it's just a stocking kit, those make great bread and butter, but some of them can go for 100 bucks even if they're not the gold collection. So look for the stockings. This one is a gingerbread haunted house. So this is Halloween themed, but I also noticed the word sampler. It's right here in the title, the Victoria sampler. Not sure if there's another one that I'm gonna show you that says sampler also that went for big money. This one sold for 153.61. So this is Halloween, it's vintage and it's a sampler. This one here is another gold collection stocking, 162.50. So you've got the stocking. Here is another stocking gold collection, 165.50. Look how cute these are. I mean, they're amazing. Now, this one is just a gold collection dimensions and this one has, the picture could definitely, it could be better, for sure the photos could be better and collections not in the title. I would definitely put gold collection in the title, but it still sold for 182.50 even though it was not a great title and not great pictures. Gold collection right here, 202.50, tender loving care. This one's called and another gold collection. So as you can see, gold collection is a Bolo. And as I was searching, I kept seeing these do-it-yourself diamond art kits. And if you look up here, people are putting them under cross stitch kits. So even though it's not in the title, when I search cross stitch kits, these come up.  So they're calling it a cross stitch. I don't know why because it's not cross stitch, but maybe because it kind of looks like a cross stitch when it's done, I'm not 100% sure. But apparently some of these are big money Bolo's. This one sold for 225.50 on auction. And again, the photos are not fantastic. So if you look here, I went into this seller's feedback and look, diamond art. Let's see, diamond art club, diamond art club, diamond art club, 140, 250, 251, 130, 122, 162. Let's look at this one right here. 251 on this one. So definitely if you see these out, look them up. Looks like they have a barcode, but this is not a scannable barcode. So you are gonna have to put love at its darkest, do it yourself, diamond art kit and search it that way. But those were all solds and over $100 for these. Here is the other one that I was talking about that says sampler. Morning has broken sampler. And this one sold for $264 in auction style. And it looks like somebody has written on here. So this one is it, it says new. I was gifted these from an estate and I will be listing many items over the next week. Most items on are open and have original price stickers. I don't know. Looks like someone wrote on it, right? Okay, the next one here is Nan's boutique, just Nan, okay? Pay attention here. This is, because I didn't know this, Nan's boutique. Delightful decorative details. This is just a detail. How many of you would have walked right past this? Just Nan, you see it right here? Just Nan. And it's a tassel. This sold for $74.11 plus shipping. Now are you ready for the cross stitch kits? $98 on this one. This one's called Just Nan. Right there, Just Nan Harmony. Let me know in the comments, did you know about this brand? Because I did not. This one sold for 98. Just wait. To be continued, Lady Scarlet's Journey by Just Nan. This is another kit, okay? It looks like this one comes with some extra pieces. This one sold for $109.50. Scream House by Just Nan. So this one kind of got your Halloween theme. 127.50. Christmas Workshop by Just Nan, okay? And these are a little bit fancy.  Here's another one. This one is called Merry 15 by Just Nan. Again, a Christmas themed item. 202.50 on this one. And it, no pins. It's missing the pins it looks like. Here's another one. Cross Stitch Silver Needle, 2009. Just Nan, Merry 15, $368 plus shipping. And you can see right here, wonderful kit. Thank you so much, $368.00. This item was paid for. So if you are in the listing and you wanna know, did the person actually pay for the item? This is one way to check. You go into the feedback. And it looks like this person sells a lot of Cross Stitch. You put in, like, let's just put in Just Nan. And you can see here, $368.00. Here's one that sold for $57.42, $47.00 and $113.00. So that's their sales just on the Just Nan. Let's go in here. And let's go to Sellers Other Items. Let's go over to Sold. And let's search highest first and just see what else they're selling. It looks like this is, Needle, Needle Smith is the name of the store. So, oh, look at this one right here. Here's another one. Tea with Honey, Just Nan, 250. Fancy that Santa's Workshop. Cross Stitch Chart Cardstock, 191.26. Fancy that Tom the Toymaker. Fancy that. So here's another one. Fancy that must be a good one. Let's go in here and see if we can look at what the packaging looks like because they have it zoomed in. And this is one that I don't know about. Okay, lookie there. Fancy that. There is another one that I have never seen before and probably would not have, I probably would have picked it up if it was cheap, but I wouldn't have known that it would be a big money bolo. So, and I don't know if they all are, you definitely got to do your research, but here's another one. Fancy that, 152.50. The Plantation Sampler. Again, sampler seems to be like a word that's important that goes, when sampler's in the title, some of those go higher. Here's some more fancy that. So, wow. Definitely glad we came in here and looked at this. So this is another great way to research. Here's the JustNan. This is, looks like just the one piece there with my needle. 50 bucks on that one. JustNan. All right, you guys, I don't know about you, but I definitely, definitely learned a lot from researching the cross stitch kit.  I wanna show everybody that this sold. And then I was thinking, you know what? Let's just kind of do a little research and I'll record it and kind of share with everybody what I find. And I actually found a lot of stuff I didn't know. So I'm super excited about that. I have done other, I think I did one other cross stitch video, but it's been a while ago. So this is an updated version. If you would like to see more videos like this Bolo category videos, I have over a hundred. And I have a whole playlist and it says category videos. If you wanna see more videos just like this, check that out. And my channel is all about Bolo items. If it's your first time here, I would love for you to subscribe. Bolo means be on the lookout, be on the lookout. Items to be on the lookout for, items you can hopefully buy low and sell high. That's what I talk about. I do also have a new series called, do not sell these items. And these are items that have been removed from eBay or people have gotten suspended. They are Vero's items, do not list. So check out that series. I think I probably have five, six, maybe seven up already and more to come. And thank you for being here. Leave me a hat emoji if you made it this far. And thanks for watching.


Hello everyone! I'd like to talk to you a little bit about radical isogenes, which is a topic in isogeny-based cryptography that has applications to a lot of different schemes that use group actions. So one of these schemes that uses group actions is called seaside, which is a certain post-quantum key exchange protocol. So we have a starting public elliptic curve E0 and Alice and Pop both have a secret element of a certain group that acts on elliptic curves. So what they do is they both act on the starting curve and then they send the result to each other and then they act on what they receive from the other person. And as long as the group action is commutative, which it is in this case because the sea and seaside stands for commutative, the result they get in the end is going to be the same. Now what's really cool about seaside is that it has very small keys. So compared to all the other post-quantum key exchange protocols, it has the smallest keys, but it's relatively slow. And the reason it's slow is because computing the group action is slow. So the secrets of Alice and Pop correspond to maps between elliptic curves. There we call isogenes and these maps typically have a very large degree. But they decompose into lots of smaller maps of smaller degrees. So essentially computing the group action boils down to computing chains of isogenes of small degree. So how do you do that? So for example, how do you compute a chain of 5 isogenes? One way to do it is to sample on each of the curves in your sequence a point of order 5 and then to set the next curve to be the quotient by this point. And that works. But it kind of sucks because first of all generating this random point, it can fail with a certain probability and then you have to try again. And this is not super good for for example constant time implementations. And also, as mentioned, it's quite slow. So the dream would be that instead of having to sample a random point every time for there to just be maybe some kind of formula depending on the coefficients of the curve that just always gives me the next curve in the sequence. And that's exactly what radical isogenes does. So for example, this is radical 5 isogenes. So let's say that we have an elliptic curve that is of this special form for some value of the parameter B. Then the radical 5 isogenic formula tells me what the next curve in the sequence of 5 isogenes is going to be in terms of this parameter B. So it's this very simple expression that depends on the fifth root of B. And this is much faster. So compared to sampling random points, this is much better. And this is currently the fastest way to compute long chains of 5 isogenes. And you can infect truth so you can show that for every degree n there exists a radical isogenes formula. But the question is how do you find them? Like how do you find this magic expression? So that's not super obvious, but that's basically what we did. So we found a new method to obtain the formulas which is based on the theory of modular curves that extends the range for the n for which we know the formulas. And then for the formulas that we have, we also rewrote them a little bit so that they are more efficient to evaluate. And we show that this leads to significant speed-ups for particles such as seaside.


In the last lecture, we had introduced the concept of electric potential. What we said is that electric field being a conservative field, by definition what it means is that the curl of the electric field is 0. Now, in that case I can express the electric field itself as gradient of a quantity by convention we take negative gradient of a quantity which you call as the potential. Now, the word potential reminds us of potential energy, but I emphasize again that though the two things are related, potential is not the potential energy. What is the relationship we have talked about last time and we would amplify on the concept of potential as we go along in this course. Loosely speaking as I mentioned towards the end of last lecture, the potential is very similar to pressure in a fluid. For example, if you have water flowing in a tube and at one end of the tube you have a higher pressure than at the other end, then water tends to flow from a region of higher pressure to that of a lower pressure. Similarly, the potential in an electric field essentially gives the measure of a level. So, therefore, if a charge, a positive charge is at a higher potential and there are regions of lower potentials around it, then it would tend to move to the region of lower potential. Today we will be talking about potential, its connection to the energy, the potential energy and in general we will see some applications of this. So, that is what is the content of today's lecture. Suppose I want to bring a charge Q, I want to bring a charge Q from a point A to a point B. Now, I know that there is electric force acting on it and since the charge is in an electric field, I have to do work to bring this charge from wherever I am bringing it to this point. So, I have to do work which is given by minus because it is not the work done by the charge, but work done by the external agency. So, supposing my reference point is A having a position vector r A to r B and it is f dot d l, this is the work done. Now, since the force on the charge Q is Q times the electric field, this can be written as the integral from r A to r B of E dot d l. We have seen that in terms of the potential, the electric field is negative gradient. So, it is minus grad phi dot d l and when we discussed the meaning of the gradient, we had seen that this quantity is nothing but the differential d phi. So, therefore, what I have there is a Q there. So, what I have is Q times phi at the point B minus phi at the point A. So, with respect to some reference point, the potential of the point B is phi B minus phi A. Now, I could choose the reference of point A to be the point where the potential is 0, in which case I find the potential, the work done is simply equal to Q times the potential of the point B provided phi A is taken to be 0. So, notice that this amount of work that has been done must now become the potential energy of the system, but this is the potential energy in supposing I have a large number of charges and it is the charge Q which has been brought into the field of these large number of charges. Then it is the energy of the entire system, the potential of the energy of the entire system has changed by this amount namely Q times phi B, but in the total expression for the potential energy, this term has explicit reference to the charge that is being or considered by us which is Q times phi at the point B. So, in some sense the potential energy of the system associated with my test charge is Q times phi B and the potential of that charge at the when it is at the point B is nothing but the potential energy associated with a unit charge when it is brought from wherever the 0 reference of the potential energy is to the point B. So, as I said that potential is not potential energy, but there is a deep connection between the potential is potential energy is measured in joules, but potential the unit of potential is joule per coulomb which has the name a bolt. Now, often we deal with surfaces or curves on which the potential remains constant. In other words, a charge placed at that point experiences no force, the gradient of the potential is 0. Such surfaces or curves are known as equipotential surfaces or equipotential curves whatever the appropriate situation is by definition by its the way we have defined equipotential surface, they are perpendicular to the electric field lines because there is no work done when you move a charge in an equipotential region. So, here for example, I am showing you the field and the equipotential correspond to a single positive charge which is marked by the blue in the picture. As we know that the field lines are symmetrical and are going out if it is a positive charge. Now, the equipotential surfaces would be surfaces which are perpendicular to it. In other words, they are systems of spheres of various radii around which are perpendicular to these field lines. Let me now illustrate the concept of potential by calculating potential for couple of special cases. The first problem that I am going to take up is the potential of a charged ring on its axis, the potential of a charged ring on its axis. So, let me redraw this picture. So, I take the charged ring in the x y plane, take the charged ring in the x y plane and this is my z axis and this is a charged ring having a charged density lambda. So, what I do is this, I take an element along this ring supposing this is taken at an azimuthal angle phi and I take d phi as this angle. So, that the length of this arc is r d phi and what I am going to do is to calculate the potential at a point here, point p which is has the coordinate 0 0 z. Since the length of the arc here is r d phi, I will put a prime here phi prime because these are the integration variable. So, let me say that the charge that is there is given by lambda r d phi prime. So, that is the d q which is the charge on that length element. Now, this if this radius of the ring is capital R and this of course is the height is z then it follows that this distance is r square plus z square. So, therefore the expression for potential at the point z is given by 1 over 4 pi epsilon 0. Now, I have to calculate the potential at this point. The potential at this point is I take the charge element here find out what is the potential due to this which is 1 over 4 pi epsilon 0 lambda r d phi prime divided by this distance and I have to integrate this from I have to integrate across the angle which is 0 to 2 pi. So, I got lambda r d phi prime divided by root of r square plus z square. Now, notice that everything else here is constant other than the integration over d phi prime which gives me 2 pi. So, therefore I get lambda over 2 pi 2 epsilon 0 times r by square root of r square plus z square. So, this is the potential and the corresponding electric field is minus gradient of phi by symmetry the electric field obviously is along the z direction. If you look at the field due to this the field due to this is directed along this extension of this line, but as I go along by symmetry the component perpendicular to the z axis will cancel and I will be left with only a electric field component along the z direction. So, as a result my gradient phi is nothing but unit vector k times d by d z of phi with a minus sign. So, let me put a minus lambda over 2 epsilon 0 r and d by d z of 1 over square root of r square plus z square this of course, gives me minus will go away because of the differentiation k lambda r by 2 epsilon 0 into z divided by r square plus z square to the power 3 by 2. So, this is this is the final expression as we have obtained for the potential by due to a charge ring along its axis. Now, let me continue with some more examples here I am trying to calculate the potential of a uniformly charged spherical shell. So, I had a line charge now I have a surface charge uniformly charged spherical shell with a charge density with a charge density sigma. So, what I do is this that this is my sphere and it is a shell. So, the charges are all on the surface. So, I take a an element area element on the surface. Now, you recall that in the spherical polar coordinate the area element is given by r square sin theta d theta d phi here we have used for running variables primes. So, it is r square this r is the radius of the sphere because everything is on the surface. So, therefore that does not change sin theta prime d theta prime d phi prime. So, as a result the potential at the point r let me go back a little bit. So, notice this that whichever point the sphere is perfectly symmetrical whichever point you want to calculate the potential let me join that to the origin and call that as my z axis call that as my z axis. So, the distance along the z axis is z, but I could jolly well later on replace it with r because it is sphere is perfectly symmetric with respect to the distances around it. .. Therefore, my phi of r or phi of z is 1 over 4 pi epsilon 0 and my charge that is there which is sigma d s prime divided by r minus r prime.   So, what is wrongly labeled here as r is actually the vector r minus r prime.  So, this is equal to let us write it down r square plus z square minus 2 r z cos theta prime. So, this is same as phi of z because I have defined my z axis that way. So, notice this that the integral has no dependence on phi phi prime. So, as a result the azimuthal angle integration gives me a factor of 2 pi. So, let us pull this out. So, I have got 1 over 4 pi epsilon 0 2 pi sigma r square these are all constant. I have left with sin theta prime d theta prime can be divided by this and let me let me introduce a change of variable by taking mu is equal to cos theta prime which means d mu is equal to minus sin theta prime d theta prime. The limits the recall that we are talking about theta. So, theta limits are from 0 to pi and since I am now doing cos theta prime. So, 0 to pi means 1 to minus 1, but I can make it from minus 1 to plus 1 by accommodating this minus sign there. So, therefore, I have my integral from minus 1 to plus 1 sin theta prime d theta prime is d mu over square root of r square plus z square minus 2 r z mu. Well, this is an integral which is easily done. So, I have got sigma r square by 2 epsilon 0 and if I integrate this I get minus 1 over actually I get 2 r z because there is a 2 r z there. So, but there is a square root everywhere. So, when I integrate it I get to the power 3 by 2 the 1 over. So, this is raise to power minus half. So, I get minus raise to power half divided by half. So, that 2 and this 2 will take care of.  There is a reason I have written it like this because we have to take care while taking the square root that the positive signs square roots are taken. So, let us look at what it implies. So, this means that if I am talking about points inside the shell the for points inside the shell radius r is greater than z. If the radius r is greater than z then phi of z is given by now you recall what did I have I had sigma r square divided by 2 epsilon 0 into minus 1 over r z I have to be a careful little careful that I have a square root of r minus z square minus r plus z square. So, I have to take the square root properly. So, in this case I should take r minus z because z is less than r. So, therefore, I get here minus 2 z and that cancels giving me sigma r square by 2 epsilon 0 times r recall that area of the sphere is 4 pi r square. So, as a result so this 2 and this 2 actually cancels out. So, I do not have a 2 there. So, I add I multiply and divide it by 4 pi I get 4 pi epsilon 0 r and 4 pi r square tie sigma is the total charge in the shell. This is the field inside. So, the field inside is given by q by 4 pi epsilon 0 r you notice that this is independent of the distance. This is independent of the distance. On the other hand if we are talking about points outside the shell z is greater than r then I have to write down phi of z by taking appropriately the square root and I would get instead of this factor being minus 2 z I will get minus 2 r and as a result I get this as q divided by 4 pi epsilon 0 z. So, it goes as 1 over z. Now, both these expressions I can combine into a single expression by writing this in terms of water known as theta function theta of r minus r times q by 4 pi epsilon 0 r notice what I have done is I have made now z going to small r because I told you that I could do that because of symmetry. So, in principle since distances are usually written as r let me write it as r and then plus theta of r minus r q divided by 4 pi epsilon 0 r. Now, this theta function is known as a step function whenever the argument r is greater than r if r is small r is greater than r then this function is equal to 1 and if capital R is greater than r then this function will be equal to 1. So, this is the way to write it and the electric field is obtained as a negative gradient of this potential this is a constant field. So, it does not really give me any contribution. So, a theta function is something like this. So, this is a theta. So, this point is r this is theta r minus r. So, this is given by this is obviously along the radial direction times theta of r minus r q divided by 4 pi epsilon 0 r square which implies that for points outside this sphere it would seem like all the charges are concentrated at the center of the sphere. So, let us summarize what we have learnt from this example. This is a uniformly charged spherical shell first thing to notice as I expect is that the potential is spherically symmetric it depends only on the distance from the center. Second thing is the potential is constant within the shell potential is constant within the shell this also follows from the fact that since all the charges are on the surface of the sphere using Gauss's law you can easily show that the field inside is equal to 0 because q enclosed is 0 e dot d s is 0. Now, since the field is 0 the potential has to be constant and which we take to be the 0 potential is constant inside. The other thing is that if you look at the expression for the potential. So, what I meant is that the field inside field inside is 0 the potential is constant. Now, if you look at the expression for the potential then you notice that as small r becomes capital R this expression is the same as that expression. In other words the potential is constant across the surface. So, potential is constant inside spherically symmetric outside going as 1 over r as if all the charge was concentrated at the center. The potential is constant across the spherical surface, but the electric field is 0 inside the shell, but electric field outside goes as 1 over r square q by 4 pi epsilon 0 into 1 over r square. So, on the surface very close to the surface when r is equal to capital R, but just outside the electric field magnitude is 1 over 4 pi epsilon 0 q divided by capital R square where r is the radius, but the moment you come infinitesimally inside the electric field is 0 the potential is constant. What it tells me is that across the charge spherical surface the electric field has a discontinuity the electric field is a discontinuity. Now, we will see that this is true in many other cases as well. Let us go to another problem which is calculating potential for a charged disc. So, I have shown here a charged disc that is a in principle a much simpler problem. So, I have a disc and I am interested in calculating the electric field along its axis. Now, this is a problem which obviously has cylindrical symmetry. So, if this distance here now what I do is I take a concentric or I take a shell of radius r prime and width d r prime. So, that the area of that shell. So, let me redraw it on the paper. So, this is a disc and I am looking at the axis on the center of this disc. So, what I have done is to take a concentric annulus here of radius r and width d r radius r prime width d r prime. So, that if the charge density is sigma the total charge contained in that annulus is sigma 2 pi r prime d r prime. Now, all these points on this annulus is symmetric with respect to this. And so, therefore the potential at the point z what I need is just to take any of these points. And you notice if this distance is r prime this is z this is simply square root of this is actually the this is a perpendicular. So, this is r prime square plus z square. So, phi of z becomes 1 over 4 pi epsilon 0 integral this charge which is there sigma 2 pi r prime d r prime divided by square root of r prime square plus z square. Now, this integral is trivially done 2 pi 4 pi goes I am left with sigma by 2 epsilon 0 this is already root of r prime square z square plus z square r prime d r prime is differentiation of r prime square. So, this is nothing but square root of r prime square plus z square only. So, once again for the same reason the factor of 2 cancels out because there is a 1 over there is a 1 over square root there and r prime square gives a factor of 2. So, this is from r prime equal to 0 to capital R. So, let us look at what does it give me. So, this is equal to sigma by 2 epsilon 0 r prime. So, this is r square plus z square minus I have to put r prime equal to 0. So, you notice what I get is square root of z square which is nothing but modulus of z. So, this is my potential at the point potential at the point z. Now, let us take some specific limits supposing this point z this point let us say p is far away. So, that z is much greater than r if z is much greater than r I can write this sigma by 2 epsilon 0. What we do is this that in this case since z is much greater than r I have to do a binomial expansion of this one. So, that what I get is I take modulus of z out square root of 1 plus r square by z square minus modulus of z. So, this is equal to sigma by 2 epsilon 0 r by modulus of z. Now, look at what it actually means it the potential the expression for the potential that we are finding at very large distances at very large distances it is the it is given by now notice that the total area if I put in because this is just pi r square this is pi r square. So, I can take out sigma times pi r square is equal to q and write it in terms of charge q and I will get then the expression that is the at large distances it looks like a single point charge, but more interesting point is what happens if this is actually there is a factor of 2 there because there is a binomial expansion. So, it is 1 plus 2 times r square sorry 1 plus half r square by z square and. So, therefore, it is there and accordingly we could do that this is r square by z because there is a z with z square cancelling out and I am saying that pi times sigma r square is the charge q divided by 4 pi epsilon 0 z which means that at large distances the at the point p the potential is same as that due to a single point charge. Now, all that is physically meaningful because when the distances are very large then obviously the disc which is of small size it appears like a point charge. On the other hand the more interesting limit is when the radius r is much greater than z what does it mean? It means that I am very close to the surface now when we are close to the surface the expression is sigma by 2 epsilon 0 square root of r square plus z square minus modulus of z f z is very small then this is simply equal to sigma r divided by 2 epsilon 0 this is sigma r divided by 2 epsilon 0. Now, notice that when you are very close when you are very close to a surface so far as that point the observation point is concerned the disc looks essentially of infinite extent. So, the expression for the potential that you would get would be identical to that due to the infinite charge plane. Now, the thing becomes obvious if using this you are to calculate the electric field which is minus gradient of fire and we use the cylindrical symmetry. So, that this is given by sigma by 2 epsilon 0 d by d z of this quantity here square root of r square plus z square minus modulus of z and that is equal to minus k sigma by 2 epsilon 0 z by root of r square plus z square which is the differentiation of r square plus z square now this is a modulus of z. So, its differentiation with respect to z should give me 1 if z is just it is z minus 1. So, if z is greater than 0 it is minus z. So, I just get 1 and if z is less than 0 then this z is negative. So, I must write plus z and differentiate it. So, I get sin of z sin is S g n namely the signature the positive or the negative sin of z. Now, notice that near the disc where z is equal to 0 the electric field this term vanishes because there is a z in the numerator and I have left with k sigma by 2 epsilon 0 sin of z. So, if z is positive it is k by sigma by 2 epsilon 0 and if z is negative it is minus k sigma by 2 epsilon 0. Once again you notice that while the potential was continuous across the surface the electric field is not the electric field as a discontinuity the electric field as a discontinuity because it is plus k sigma by 2 epsilon 0 plus sigma by 2 epsilon 0 above the plane and below the plane it is minus sigma by 2 epsilon 0. Before I go to this other example let me make one observation there are two cases that we have talked about. One was the charged spherical shell another was a charged disc in both cases we have charged surfaces and what we noticed is that whenever there is a charged surface I am generalizing it because I will prove it in general. Whenever we have a charged surface the potential across it is continuous, but the electric field suffers a discontinuity as I go from above the charged surface to below the charged surface. We will see that this can be proved as a general property. Let me continue example of a or calculation of potential for a three dimensional case. So, let in this case I am talking about a uniformly charged sphere containing a charge cube. So, my charge cube is given by the volume 4 pi by 3 r cube times the charged density rho which is being taken as uniform. Now, this is a problem which we have seen in the past as an application of Gauss's law. Remember that Gauss's law tells me that the flux through any surface a real or imaginary is given by the charge enclosed divided by the flux through any surface is given by charge enclosed divided by epsilon 0. So, as a result if the point at which you are calculating the field. So, let us suppose this is my sphere this is of radius r. Now, if I am calculating the field at a distance r smaller then I enclose it by a an imaginary Gaussian surface of radius smaller and I find that the flux then is 4 pi r square which is the area times magnitude of e which is equal to the amount of charge contained which is this cube full cube divided by epsilon 0 which gives me the electric field this time I will put its radial direction cube by 4 pi epsilon 0 1 over r square unit vector r which is like Coulomb's law as if the entire charge is concentrated at the origin. However, if radius r. So, this is r greater than r if the radius r is less than r the electric field is given by 4 pi r square e equal to q enclosed by epsilon 0 and the q enclosed is not the entire cube, but q times small r cube by capital r cube because that is the fraction of charge that is included within an imaginary surface of radius small r which is less than capital r. So, this will give me if r is less than r the electric field will be given by 1 over 4 pi epsilon 0 q by r cube times small r and of course, the direction is still along the radial direction. Now, what I will do is this I will integrate to find out I will integrate to find out the potential in the two cases. So, in the first case for r greater than r the potential v of r remember negative gradient of the potential is electric field. So, this is simply given by 1 over 4 pi epsilon 0 1 over r for r less than r. Now, this already assumes this already assumes that the 0 of the potential is at infinity. Now, having chosen the reference once I cannot change the reference for the second part of the problem. So, one has to be careful when you calculate or you determine the potential at the point small r which is less than the radius. So, I need the potential of that point with respect to the surface of the sphere. So, in order to find out what is the potential on the surface of the sphere I go from infinity to the radius of the sphere which is of course, a result that I already know. So, that is one part and the second part is to go from the surface of the sphere to the point r and take this expression and integrate it. These are both of them are trivial integration. If you calculate it properly you will find this is given by 8 pi epsilon 0 1 over r 3 minus r square by r square. We have taken a few examples of how to calculate the potential in different cases. .. Let me return back now to this point that we have been making that whenever we found that there was a surface charge the electric field had a discontinuity the potential did not. Now, in all the words I am looking at the behavior of the field and the potential at the boundary what happens at the boundary. So, let us look at a situation like this I have an arbitrary surface here. Now, I already know supposing these are infinite charge surfaces I already know how to calculate the electric field above and electric field below. Now, notice we can obtain the same thing like this let me enclose this by a Gaussian surface. So, I take a rectangular parallel pipe of height epsilon half of it is above the surface and half of it is below the surface. Now, I know that the flux through this is equal to q enclosed by epsilon 0 and how much is q enclosed remember the charge is only on the surface. So, if the surface area of the top happens to be equal to a then I get sigma a is the charge divided by epsilon 0 this is flux. So, therefore, this is e dot d s and as I make this epsilon becomes smaller and smaller I find that the contributions are only from the top and the bottom phase which means that normal component of e above the charge surface minus normal component of e below this should have been below is sigma by epsilon 0. So, this reemphasizes the fact that across a charge surface we have discontinuity of the electric field I will elaborate on it in our next lecture. Summarizing we have looked at potential it is meaning and calculated it in a few cases we have talked about a charged ring a spherical shell and a uniformly charged sphere three different types of problems. Important point to find out was that whenever there is a charge surface we have found electric field has a discontinuity. Now, this is a general problem these are known as electrostatic boundary condition which I will take care of in the next lecture.


So there's a recipe called Delicious Cake, which is in a cookery book called Simple Recipes and Menus for Camp, Home and Nursery, which was written by a woman called Lucy Kahn, who was actually living out in Calcutta at that time. Shall we have a go? Yes, let's have a go. I thought it would be fun if you make it a recipe with a KitchenAid and I'm going to try and do it by hand, like Lucy Kahn would have had to. Yes, she wouldn't have a KitchenAid, would she, back then? She would not. She might have had a servant, but she wouldn't have had a KitchenAid, that's kind of what they ended up replacing really. So Polly, I've got my mixer on slow, so I'm giving you a little bit of, making it a bit more of a fair contest. It's not really fair, is it? Well, wooden spoon, mixer, it's a little bit one-sided, isn't it? So we're sticking to the original recipe. It's a bit lacking in instructions for method. If you didn't have any idea about cooking, it's sort of... You'd have to know your way around the kitchen, wouldn't you? You'd have a servant that moves the way around the kitchen. Exactly. And of course, this is the moment where the servant crisis starts because so many women left domestic service to work during a wartime and then didn't go back. And then, of course, the story of the 20th century is in part how domestic appliances sort of help and supplement labour in the kitchen. Certainly labour-saving, isn't it? Yes, you're gloating now. Lucy Kahn didn't specify the number of eggs, does she? She specifies the weight of eggs. Ounces, yes. So it's quite weirdly. You weigh them in the shells. You know, you would think that you'd weigh them out of the shells. So you weigh an egg in a shell? Yes. So it's four ounces of eggs. Which equates to two eggs, fair of ours. Yes, yes. There are a lot of cookery books to choose from in those days. You know, the British Library's got hundreds of thousands of cookery books. But 1919, hardly any were published. So this is quite unusual, in a sense.  How are you going with yours? Do you need a bit of extra elbow grease? I need a different arm. Polly, here's my delicious cake. How are you going with yours? I'm nearly there. That looks terrific. Lucy Kahn would have been proud of you, I think. She was, sprinkling of success here. I'm coming up on the inside, though. Good luck with it.


So with the WTA Finals coming to an end, a day later than it was supposed to, finishing on Monday, we have a change in the rankings. Big change to the top of the WTA rankings, but let's go have a look at what happened in the last week. The WTA Finals we had in the final. Shafiontek taken on Pagula, both hadn't lost a set all week, as Shafiontek dominating in that final 6-1-6 love in under an hour to take her first WTA Finals trophy. And look, Pagula had a good week, but Shafiontek was just on another level. Let's go over to the rankings now and we have a change to the top. Egas Shafiontek ends the year as World No. 1. She had to win the title, had she had lost to Pagula, she wouldn't have got. Back to the top. She is No. 1. Sabalenka down at No. 2. Going into the new season, Sabalenka was No. 1 for 8 weeks since the U.S. Open, but Shafiontek ending the season with a win and she had to win. We have Goff coming in at No. 3 to end the season, Rabakron at 4 and Pagula at 5. Gondrussever is No. 6. We've got Jabaret at 7. Mukher at 8. Sakria at 9 and Krijikovar rounds at the top 10 for the year. And that is the WTA done and done. And again, familiar faces at the top, especially Shafiontek, second year in a row that she is the year in No. 1. So there it is, the WTA Finals ending the season and what a way to end the season with the best player on the planet back to her best and dominating in that final series. She beats Sabalenka, she beat Goff, she beat Pagula along the way. All in straight sets and she regains her top spot. Well, what's going to happen next year? Let me know in the comments below. Do you think that Shafiontek can be No. 1 another year, third year in a row, something that Bardi did in previous years? Or do you think Sabalenka can regain her No. 1 spot very quickly? Of course, a lot of points to save at the start of the year. She's got to defend those Australian Open points to Sabalenka. So maybe not early next year, but Shafiontek, she's back on top. She's back 200 points ahead. Can she stay there this time for another year?