LET THERE BE... Voltage? | Maxwell's Equation #2 Explained for Beginners

what's up you lot park here and today I'm going to be talking about one of Maxwell's four equations of electromagnetism now I've already made a video about one of them in the past and you guys seem to enjoy it so check it out up here if you haven't seen it already although don't worry you don't need to have seen it to understand what I'm going to be talking about in this one but still go and check it out if you haven't already it's actually quite a good video if I do say so myself anyway let's get into it well no hang on I can't do that yet I have to say first of all thank you so much for 7000 subscribers that is ridiculous we are well on our way to 10k which is you know pretty cool so thank you so much if you subscribed already and if you have it then please do consider it depending on whether or not you like this video so without further ado let's get into the video so you can figure out whether or not you like it so if you don't know Maxwell's equations are a series of four equations compiled by James Clerk Maxwell that describe everything to do with electricity and magnetism collectively known as electromagnetism now in the previous video I made about Maxwell's equations which I linked up there earlier I discussed one of the four Maxwell equations and that one dealt with the fact that there is no such thing as a magnetic monopole and as a consequence of this we discussed the idea that magnetic field lines must not have a start or an end point they must be closed loops now that sounds relatively complicated especially if you don't know what any of that means but I tried to make it as easy to understand for somebody who's never seen this kind of stuff before so that's what I'm going to be trying to do in this video as well now the reason that I brought up the previous video is because in that one I use differential notation I discussed what these little upside-down triangles mean however some of you in the comments said that you would like to see the next Maxwell equation discussed in the integral form which looks a little bit something like this for the previous one that I discussed and that actually really helps me out because the differential notation and the integral notation are just two different ways of writing exactly the same thing they're just two ways of looking at the same thing from different angles and the reason that your request helps me out is because while the equation that were going to be looking at today is actually much easier to discuss in integral notation so thank you for that ok so the equation that we're looking at today is this one this is what it looks like and hopefully by the end of the video we'll be able to understand exactly what all of the symbols in this equation mean so let's start with this s-shaped symbol which we can see on both sides of the equation it's known as the integral symbol some of people have seen this symbol already from your studies of calculus but if you haven't or ready seen the symbol then let's quickly go over what it means to do this let's first imagine that we're dealing with a mathematical function let's say we're dealing with y is equal to x1 of the one of the simplest functions that we can think of this function is such that whatever the x value at a particular point along that line the Y value is exactly the same as the x value and this is true along the entire length of the line so that's why the function is known as y is equal to X because on that line at every point the Y value is exactly the same as the x value now let's imagine that for some reason we want to try and find the area underneath this line and above the horizontal or x-axis and we want to do this between X is equal to 0 and a specific value of x let's say X is equal to 10 so now what we've drawn is a shaded triangle so we can work out the shaded triangles area by recalling how to calculate the area of a triangle we can remember that the area of a triangle is found by multiplying 1/2 by the base by the height now we know that the base of the triangle is 10 units long because we're calculating the area of the triangle from X is equal to 0 all the way to X is equal to 10 however we don't immediately know the height of the triangle but we kind of do because we know that the function we were dealing with is y is equal to X therefore if we pick any point on the line and in this case we'll pick the most convenient one that we know that the x value is equal to the Y value this means that if we pick the point where X is equal to 10 then Y must also be equal to 10 therefore we can see that the height of the triangle is 10 units and in doing so we can calculate the area of the triangle as 1/2 multiplied by the base which is 10 multiplied by the height which is also 10 and that ends up being 50 units squared now this is easy enough right but this kind of stuff only works with relatively simple functions such as y is equal to X what do we do in a situation where we've got a more complicated function let's say Y is equal to x squared now this curve y is equal to x squared is defined in such a way that whatever the x value is at a certain point on the curve the Y value is that x value squared that's why it's known as Y is equal to x squared for example if we take the point x is equal to 2 then on the curve y is equal to 4 and if we take the value x is equal to 5 then on the curve Y is equal to 25 so coming back to what we were trying to do earlier let's now try and find the area underneath the curve above the horizontal axis and between X is equal to 0 and X is equal to 10 now we don't have a triangle anymore so it's not quite as simple as it was before in fact we've got this whole curvy bit right that's what's preventing us from calculating just the area of a triangle however we could calculate the area of this triangle that I've drawn in right now and just say that this is an estimate of the actual area admittedly it's a pretty poor estimate and it's an over estimate of the actual area because we're also saying that the area under the function includes this shaded bit here when in reality it doesn't so we've overestimated the area now a way to improve our estimate is to not say that the area is a triangle but rather divide up the area under the curve into trip easier then what we can do is to find out the area of each trapezium and add it up all together to find the area underneath the curve or at least an estimate of the area underneath the curve now the reason that this is still an estimate of the area will be a better one than before is because we have straight lines at the top of each trapezium rather than the curve which is what we'd have if we had exactly the area that we're trying to find but anyway so the way to find the area of each trapezium is to multiply the base length of the trapezium by the height exactly down the middle of the trapezium and we can do this for each different trapezium and then add up all the areas to find a better estimate of the area underneath the curve now another thing that we're going to do is to call the base of each trapezium DX this is just a notation thing but it'll become clear why we do this in a minute so don't worry about why it's called DX it's just that's what we're gonna call that length and so essentially what we're doing is dividing the area that we're trying to find into trapezius such that the base length of each trapezium is the same as the base length of every other trapezium each trapezium has a base length of DX then we can work out the middle height of let's say the first trapezium so we can say that the center of the base of the trapezium has an x coordinate x1 and then we can follow up to the curve and we can work out what the y-value is because we know that Y on the curve is equal to x squared so the area of the first roupies IAM is the base DX multiplied by the middle height which is x1 squared the area of the second trapezium is DX multiplied by x2 squared the area of the third for pzm is DX multiplied by x3 squared and so on and so forth and to find an estimate for the total area underneath the curve we add up all of these areas so we get DX multiplied by x1 squared plus DX multiplied by x 2 squared plus so on and so forth and then we can factorize factories up DX and what we've got in brackets is X 1 square plus X 2 square plus X 3 square and so on and so forth then what we can do is improve our estimate of the area even further what we can do is to divide up the area into much thinner trapezium we'll still say that each trapezium has a base of DX but this time we've got a lot more of them yes there are still problems because we're still using straight lines to approximate curves but these problems are much smaller than before so increasing the number of trapezium is reducing the error in our estimate our estimate is getting better and better than more trapeze here we have and we could keep doing this until we have large numbers of trapezium representing the area underneath the curve then the total area under the curve becomes the sum of the base DX multiplied by x1 square plus x2 squared plus x3 squared plus however many midpoint X's that we have squared and a short way to write this is as the sum of xn squared where n takes the numbers 1 2 3 4 5 however many that we have and this weird-lookin symbol which is actually the Greek letter Sigma represents a sum over all of these terms now don't worry too much about this bit the next bit is what's important basically what we can do is to then say that our area is now divided up into infinitely many trapezium and each trapezium is infinitesimally thin in other words it's as thin as can be we can make it as thin as we want it and if we were to zoom into the curve the base of the trapezium would still be thinner than we can ever imagine so to reiterate we've divided the area of the curve into infinitely many trapezium and if we divide it up into infinitely many trapezium then now what we have is no longer an estimate for the area of the curve we have the exact area of the curve because remember as we increase the number of trapezium representing the area of the curve our estimates got better and better and better and if we make DX so small that we have infinitely many trapezium then we now have the exact area under the curve in this situation the squiggly sign the Sigma becomes an integral sign and that's where the integral sign comes from it represents an adding up of lots of little bits and pieces to give us the overall whole thing and the reason I said it so abstractly was because it doesn't necessarily have to represent the area underneath the curve the idea is that you add up lots of tiny elements to give us the whole thing now of course when we're doing this properly and rigorously the symbols are slightly different we use Delta X to represent the non infinitesimally thick for PZ abases but that's not really relevant to the video so let's move on like I said the integral sign is used to add up lots of little tiny elements to give us the total whatever it is that we're trying to find and I described it very visually using the area underneath the curve but this can be used for many different things including the equation that we were looking at earlier and when we apply this train of thought to our equation we see integral signs on both sides of the equation this means that on both sides we're adding up lots of little things to find something bigger but what are these little things on each side well to understand what we've got inside these integral signs on each side of the equation we need to remember the meanings of B and E if you've seen my previous video on Maxwell's equations you'll recall that e represents the electric field that we're trying to study and B represents the magnetic field that we're trying to study both E and B the electric field and magnetic field are vector fields that's why they have arrows on the top of them to signify that they're vector fields now when we say vector fields what we actually mean is that at every point in the region of space being studied where the e field and the B field exist each of this field has a vector value in other words there's a vector or an arrow that can be used to represent the value of this field a field or B field at every point in space in the case of the electric field the vector at each point represents the force that would be felt by a positive test charge placed at that point in other words if we were to take a positive charge and place it at that particular point at the base of the arrow the force experienced by the charge would be towards the right and it'll be a fairly large force whereas if we were to place it here instead then it would experience a small force to the left the B field or the magnetic field is similar but it represents the force experienced by a North Pole of a magnet placed in the magnetic field if the explanation was a bit too complicated then definitely check out the previous video I went into a lot more detail and did it a little bit more slowly as well however let's carry on so now we can see that we're integrating something to do with the B field and the e field on either side of the equation so let's clarify what we mean by DL and D s if we think back to our explanation about integral signs we can see that they look a little bit like the D X's which is what we used to label the bases of the trapezium in this case we're not talking about using trip easier to work out the area of anything anymore but we are using DL and D s exactly like we use DX they are the little L of something over which we'll be integrating now that sounds a little bit vague but before we look at D L and D s specifically we need to realize that both DL and D s are also vectors they have arrows over the top of them and the dot that's been pointed out on either side of the equation doesn't just represent a normal multiplication it represents a sort of vector multiplication known as the scalar product or dot product and the reason it's known as a scalar product even though we're dealing with vectors is because when you multiply these two vectors in a special way you get a scalar value and it preserves the relationship between the two vectors that we multiply together let's not go into too much detail about that here but then on the left hand side we've got the electric field multiplied by this DL vector and then we integrate this whole expression so what do we mean by DL well DL is a very small vector that lies on the perimeter of a surface that we shall call s now the surface that we call s is going to become very important relatively soon so let's keep an eye on it but in the meantime let's say that this surface s is a circle and on its perimeter we are talking about little elements DL that go all the way around the surface s in other words DL is representing a little length of the perimeter of the surface and so what we're doing here on the left-hand side is to multiply the electric field by the DL vector which tells us how much the electric field is pointing in the direction of the DL vector at any particular point so let's say that at this point the electric field is pointing in the direction of the DL vector well in this case the electric field and the DL vector are going to multiply and because they're aligned this means that the contribution of the electric field is going to be large at this point so we're looking at the contribution of the electric field at every single DL along the perimeter of a circular surface this means we're adding up the behavior of the electric field along the entire length of the circle and the reason that we're talking about this circle in the first place is because that circle will become very important pretty soon now let's quickly move on to D s and D s is a similar kind of vector but this D S represents an area specifically every single D s is a vector that is perpendicular to a little area element inside the circle that we're considering in other words if we take a very tiny area which we'll call da for the area then D s is the vector that's perpendicular to that that represents this area and the reason that we use a vector to represent an area is because that's one way of simplifying the description of an area instead of thinking about an area as two dimensional surface we can now just think of it as one vector so what we're doing here is to multiply this area vector by the magnetic field now in other words what we're looking at is the contribution of the magnetic field within every single little area element and then once we add it all up we add up all the contributions of the magnetic field to the little area elements within the circle that we're looking at and therefore we're looking at the total contribution of the magnetic field within the circle now there's a complicated reason for why we're looking at the electric field along the perimeter of the circle and we're instead looking at the magnetic field going through the circle and hence why we're using DL when talking about the electric field and D s when talking about the magnetic field but the idea is on the left hand side we've got the contribution of the electric field along the perimeter of the circle and on the right we've got the contribution of the magnetic field through the circle and it's important by the way that all of the details that add up are around the perimeter of the surface that we're considering on the right hand side in other words when we add up all the little DA's or DSS depending on how you want to think about it the area that we find will have the perimeter that we get when we add up all of the deals together now there's a little bit of an elephant in the room we haven't talked about the D divided by DT thing on the right-hand side of the equation well all this means is that we're finding the rate of change of whatever is inside the integral think about it this way you know when you find the velocity of an object we're trying to find the distance it travels divided by the time taken for that object to travel that distance in other words the velocity of an object is Delta X that's the change in distance divided by delta T that's the time interval over which that change in distance occurs in this case we're doing exactly the same thing but a little bit more rigorously and a little bit more calculus Li if that's a word which it's not basically the D in the numerator is the same thing as the Delta in the Delta X part of the velocity equation and the DT is the same thing as the delta T in other words we're finding the rate of change of this whole integral or how fast they're integral changes over time and the negative sign basically just flips the sign is just the negative value of this rate of change and by the way the phrase rate of change simply means how quickly or slowly something is changing so on the right hand side we're finding the negative of how quickly the integral is changing where the integral is the contribution of the magnetic field within the area of the circle that we're thinking about now that all sounds very complicated but this is where it gets simpler what if I was to tell you that the perimeter the circle that we were considering is a loop of wire and that wire can carry a current what if I also tell you that the left-hand side of the equation gives us the EMF or voltage generated in that coil of wire this is because the voltage in a circuit is very closely related to the electric field in that circuit after all electric fields are created by electric charges and a voltage is just a measure of how electric charges want to move kind of like how gravitational fields are a measure of how objects with mass want to move in that gravitational field right so now we're thinking about a coil of wire and on the left hand side we have the EMF generated in that coil of wire and we're talking about magnetic fields going through that wire and more specifically the rate of change of the total contribution of the magnetic field within that wire this might be sounding a little bit familiar to some of you right now this equation is a very thorough and very glorified way of encoding the effects of electromagnetic induction that's the process where a changing magnetic field through a conducting coil causes an EMF to be generated within that coil some of you might even have done an experiment where you take a bar magnet and you move it in and out of a conducting piece of wire and if that wire forms a closed loop then we can see that there's a current passing through the wire that current flows because we generate a voltage or EMF across the coils when we move the bar magnet in or out of the coil in other words a changing magnetic field through the area of the coil causes an EMF across the coil so this equation once again is essentially talking about electromagnetic induction what's also accounting for all of the effects that could possibly come about because of it and the way it does this is by being very mathematically rigorous we've used integrals and vectors and all this kind of neat but complicated stuff to describe something which a high school we would have just learned as you move a magnet in and out of a coil and we see any MF generated and this stuff is actually quite tricky to understand so if there's something that I haven't fully explained properly then let me know in the comments down below as well and I'll try and clarify and as always if I've got something wrong let me know as well but yeah so because this Maxwell equation is describing the effect of electromagnetic induction this particular equation is known as the Maxwell Faraday equation because Faraday's law of electromagnetic induction if you don't know what that is then definitely check out some other YouTube videos about it it's actually a really cool effect and if you don't know about it already then I highly recommend learning about it in high school we basically learned that the faster that the magnetic flux that's the amount of magnetic field passing through a certain area changes the larger the EMF less generated in other words the quicker we move our bar magnet in and out of the coil the larger the EMF that's generated and it's important to note that it's a change in magnetic flux through the area of the coil that makes the difference not just putting a magnetic field through the coil that in itself is not enough that magnetic field must change and there are different ways to do this you could of course move the magnet in and out of the coil but you could also change the area of the coil or rotate the coil and whatever we choose to do the right-hand side of the equation is accounting for the fact that the magnetic field through the coil is changing that's why we've got the D by DT term because it's calculating the rate of change of that magnetic field it's also accounting for the vector nature of the magnetic field because the amount of magnetic field passing through an area that's perpendicular to it is going to be different to the magnetic field passing through that same area if the area is slightly rotated relative to the magnetic field and of course there's a lot more intricate details that I haven't gone into but we don't need to know about those the fact is that we now have a better understanding hopefully of this Maxwell equation or rather the Maxwell Faraday equation and with all of that being said guys I hope you enjoyed this video once again if there's something you don't understand let me know in the comments below if you've got any other misconceptions about physics then let me know in the comments as well and if I can clarify anything or if I've got anything wrong tell me down below also I know I've been away a little bit I've only been uploading once every three weeks or so but you know work life is busy and I've been playing a lot of badminton and doing a bit of music on the side as well and so if enjoying life a little bit so I'm gonna try and come back to YouTube because I miss doing it but I hope you guys are ok with having a video every two or three weeks in the meantime and hopefully I'll be producing more videos because like I said I genuinely really miss making videos but again that depends on work commitments and anyway I'm gonna stop rambling thank you so much for watching and I'll see you next time buh buh buh buh bye [Music]

Channel: Parth G

Views: 168,639

Rating: 4.9722199 out of 5

Keywords: Maxwell's Equations, Electromagnetic Induction, Electric Field, Magnetic Field, Electromagnetism, Integration, Calculus, Physics, Parth G, Vector, Vector Calculus, Let There Be Light

Id: 6Aab3k2nsOY

Length: 19min 57sec (1197 seconds)

Published: Tue Jul 23 2019