Welcome back to Electrodynamics and Its Applications. My name is Seungbum Hong, and to my right side we have Melodie Glasser. So we will start with discussing about what it means to understand physics. As you can see in this slide, the physicist or scientist needs a facility in looking at problems from several points of view. An exact analysis of real physical problem is quite complex. And any particular physical situation may be too complicated to analyze directly by solving differential equation, right? However, one can still get a very good idea of the behavior of a system if one has some feel for the character of the solution in different circumstances. So if we recap our first lecture, we used the concept of a field and then, in order to understand electric field or magnetic field, we used two characteristics, namely flux and circulation. And in that way we were able to understand the big picture of Maxwell equations. There is only one precise way of presenting the laws, like Maxwell equations, that is by means of differential equations, as we just learned before. And if I quote Dirac, he said, I understand what an equation means if I have a way of figuring out the characteristics of its solution without actually solving it. So in our lecture the overarching approach will be to take, first a complete loss, so we will not sacrifice any reality in the equations. And then step back and apply them to simple situations step by step, developing the physical ideas as we go along. So the basic tools that we will learn today will be the scalar and vector fields, temperature and heat flow, as well as basic operational rules. So, Melodie, I guess you have learned this in other classes as well, right? So this will be a recap of what you probably have already learned but in a different flavor. We will not only memorize the formula or equations here, we will try to visualize them and try to understand in a physical sense. So for two vectors which has direction and magnitude, if you do multiplication, you can do it in two ways. The first one is called dot product, which will give a scalar value, and the other one's called cross product, which will give you another vector that is perpendicular to both A and B, and represent the volume of this parallelogram that is drawn here. And the detailed equations are written here. So, as you can see, knowing these two operations we can discuss some of the characteristics of these operations. So the first one is if you do cross product on two identical factors, it will give you 0. And if you do cross and dot product in series like this, and if one of them has the same vector inside, then it will also give you 0. And A dot B cross C will be the volume of this parallelogram defined by A vector, B vector, and C vector. And if you do cross product two times on three vectors, this will be the formula that you want to use in our lecture, okay? So for the dot product, as you can see here, it is the projection of one vector onto the other. And that multiplication of the magnitude of those two that will result in the maximum value if those two vectors are aligned, right? If you have perpendicular relationship, that will be 0, okay? Another two important equalities from the calculus that we will use a lot of time is this decimal change of a function. That is described in the 3 dimensional world by Cartesian numbers x, y, z, is equal to the linear combination of each increments along each axis. Which can be represented by the slope along that axis times the small change in that axis. Also, for the second partial derivative, changing the order of two parameters like x and y will not influence the value of your second partial derivatives. So these two, I hope you will memorize and use it in some situation that we will discuss later, okay? So, let's recap what field is. So, Melodie, can you explain to us what field is and what types of field we have? >> So, the field is a set of values which describe the system, and there are the scalar field which have just quantities, numbers, and you can see that in the temperature map here. And then the vector field also has magnitude and direction. So if you look at this velocity map on the turning cylinder you can see how fast each point is moving and which direction each point is moving. >> Perfect, and sometimes, as Melodie explained to us, we can have correlation between those two. Like in the case of temperature field, it's a scalar field, but based on that scalar field, by looking at the gradient of the temperature, we can build up the heat flow vector field. So you can see sometimes you can correlate the scalar and vector fields. Now, in this lecture, because electrodynamics is pretty abstract, we're going to use a lot of analogy. Between electrodynamics and heat transport, because heat transport is much easier to understand. Okay, in order to do that we need to define some of the terminologies. So let's first define heat flow vector, h, this is a small letter h, which points in the direction of flow, as you can see in this picture, and has a magnitude equal to the amount of thermal energy that passes per unit time and per unit area. So that's the flux in transport, right? Through an infinitesimal surface element at right angles to the direction of flow. So the equation here tells you the h factor has the direction of e sub f, which is the unit vector of the flow, f stands for flow. And the magnitude is delta J over delta a, where delta J is the flux, the amount of thermal energy that passes per unit time, and delta a will be the unit area. So, you can see delta J is the thermal energy that passes per unit time through the surface element delta a. And ef is a unit vector in the direction of flow. So, if we do some mathematical manipulation, delta J over delta a2, if you look at these two surfaces That has some relationship of angle as described here is tilted then you see delta j will not change but delta a will change, right? So delta j over delta A2 will be delta J over delta A1 times cosine theta and we can use the dot product which we just learned, where the h heat flow vector and n it's the surface normal vector, all right? So that will be h.n. So the heat flow per unit time and per unit areas through any surface element Whose unit normal is n will be given by h.n, okay? So Melody, if this is the vector that we're going to use, when will this be maximized? >> When the two vectors are align in the same direction. >> Perfect, yes, when they are align in the same direction this will be maximize, when they have a perpendicular relationship it will be minimize. All right, so now we will cover some of the derivatives of the field in order to understand the characteristics of our field. According to our mathematics, the Taylor Series is one of the popular series that we learn in high schools or universities. And the reason why we learn that is the following. So if you have an arbitrary function f(x). And if you know the values of this function at one point, let's say 0 and the derivative, first relative and the second derivative And so on and so forth to a higher order term. Then without moving your point from 0 you can predict all the values that the function is that function of x. So how is that possible? It is possible because derivatives here shows the relationship between neighboring points. So knowing the relationship between neighboring points you can extend that relationship to cover all space all space in broad. So that's the power of derivatives. And most of the time as you may learn the higher order term or contribute less and less to the whole function. So knowing the first order or second order might be sufficient. So in this class, we're going to only cover the first order derivatives and second order derivatives to understand the characteristics of our vector field. So that's why we will start with the first order derivative. So how shall we take the derivative of temperature with respect to position will be the related question here. >> Okay. >> And one guess might be, how about round T over round x, round T over round x? Note that this is partial derivative and it's not the direct derivative, and Melody, if we do a round T over round x, will that it be a vector field? >> No. >> Or a scalar field? >> No. >> And why is that the case? >> Because if you shift the coordinate system you won't get a similar value- >> Exactly. >> For your equation. >> So the important thing here is scalar and vectors are inherent upon the choice of the coordinates. So we will learn more about how we can prove one field is a scalar or a vector field using the operation that we just learned, namely the dot product operation. So it is true only if, when we rotate the coordinate system the components of the vector transform among themselves in the correct way. So this will be another way to prove whether it is a vector field. So ask a question who's answer is independent of the coordinate system and try to express the answer in an invariant form. So here is an example, if S, which is a scalar, = A.B and if A and B are vectors, S is a scalar. Likewise, if A is a vector, S a scalar, and there are 3 numbers, B1, B2 and B3 that satisfied the relationship A x B1 + A Y B2 + A 2 B3 this is the operation of dot product equals S then B1, B2, B3 are the components Bx, By, Bz of some vector B. So this is quite an abstract idea, so let's take a look at some specific example to understand what this means. So, in this example we are going to prove that round T over round x, round T over round y, round T over round z. Constitutes a vector, okay? In order to do that, let's take a look at this picture. So, think about a contingent coordinates, where you have x and y and z. In perpendicular fashion and imagine you have two points that are very close to each other, P1 and P2. And connect those points with an arrow which will be the relative position vector of P2 with respect to P1 and think of this as a diagonal vector for a box that is drawn as a dotted line here, okay? In this case we already learned temperature is a scalar field, all right? And let's think of the temperature at P1 and P2. And also lets think about the temperature difference of delta T between those two points. Because T2 a scalar and T1 is scalar. Subtraction of those two, delta T will be also scalar. It's inherent upon the choice of the coordinates, right? And when T1 nad T2 are temperatures that P1 and P2 are separated by the small interval delta R which is the relative position vector then we can use delta x, delta y, delta z. And we also know that the position vector is a vector field, all right? So using the equation that I asked you to memorize where the change of a function can be described by linear combination of the change along each axis, which is here. Then using this formula and replacing f by T, temperature, then you can understand the change in temperature will be equal to round T over round x times delta x + round T over round y times delta y + round T over round z delta z. When these change becomes infinitesimally small, approaching zero, right? Then, as you look into this equation, this is the operation of dot product between the relative position vector and the three numbers that I just wrote here, okay? Will be dot product operations. So, we know delta R is vector, we know delta T's scalar, therefore, these three numbers constitute a vector. So that's the end of the proof. So this might be a very easy and convenient way to prove that three numbers, whether three numbers constitute a vector or not, but another way We will shortly discuss after thinking about the meaning of the equation that we just discussed. So del-T is called gradient of T or del-T, and mathematically is equal to round T over round x, round T over round y, round T over round z. Because it's a vector, you have a magnitude and direction, magnitude and direction. And we will think about what magnitude and direction of this vector mean, okay? This is the neat form to abbreviate what we just discussed, the change in delta temperature between those two positions will be the dot product between the del-T, the vector that we just proved to be a vector and relative position vector. All right, seeing this equation, now let's think about what the del-T really means. So Melodie, from this equation what can we know about del-T? >> The direction of the vector. >> And what is the direction of this vector? >> The direction of the vector is the maximin point. >> Yes, maximum uphill direction of temperature. So if you follow along this direction, which means delta R is aligned with del-T, then del-T will be maximized. If you follow perpendicular to the temperature change, the del-T, then there will be no temperature change. So that will be the direction of the isotherm, isothermal line, right? So if the equation says that the difference in temperature between two nearby points is the dot product of the gradient of T and the vector displacement between the points. So let's prove del-T over del-X, del-T over del-Y del-T over del-Z is a vector in a different way. We just prove it using the dot product operation. And the rationale that if the three numbers fulfill the dot product operation, then it should be a vector. So we just discussed that we shall show that the components of any three numbers should transform in just the same way that components of unknown vector, something like R, which is position vector two under rotation of the coordinate system. So you may recall the equations to describe the change in the coordinates when you rotate the frame in the counterclockwise direction by angle theta. In that case, the new coordinate system which is x prime and y prime in this case will be described like this, x prime equals x cosine theta plus y sine theta, y prime equals minus x sine theta plus y cosine theta. And because we have rotated this frame above the axis z, the z will not change. So it will be in variance so z prime will be equal to z, okay? If we do some transformation, we can also express our x and y in terms of x prime and y prime using these kind of formula. And looking at the picture and using the geometry that you learned from us, you will be able to understand why these equations are written in this particular way. Now, let's take a look at the position here which is P1. And also, let's take a look at P2 which is in the neighborhood of P1. And let's assume for simplicity that we are moving the point only along the x direction, so the P2 will have a new coordinate which is x plus delta x,y,z. So y position and z position will be the same. Only x position will be different, pi delta x. So in a prime system which is the frame that has been rotated counterclockwise by angle theta, P1 wil be x prime, x, y prime, and z prime. P2, in this way, you will see when you rotate it, you will add the y-component to that system. So we will have x prime plus delta x prime, y prime plus delta y prime, and z prime, okay? And let's take a look at how the temperature difference will be expressed in a different way. So for the prime system, temperature change delta T will be linear combination of the change along x prime axis and y prime axis. So this will be the equation you want to use, and if you look at the picture closely, you will understand that this will be rho T over round x prime, cosine theta times delta x because delta x prime equals delta x cosine theta from this picture, right? And delta y prime equals minus delta x sine theta, from this picture. So you can replace those two by this one. And if you compare two equations above, we see that round T over round X, which will be round T over round X delta X, will be equal to round T over round X-prime cosine theta minus round T over round Y-prime sine theta. So let's take a look at how x prime and y prime which is the position vector transform under the rotation. So you see they are one-to-one corresponding to each other. You see cosine theta minus sine theta which is cosine theta minus theta times the x prime coordinate and y prime coordinate. which is x prime coordinate of gradient and y prime coordinate of gradient. In this case, we can prove del-T is definitely a vector field derived from the scalar field.
