A) Write down the microscopic Maxwell's equations for the...

a) Write down the microscopic Maxwell's equations for the fields E and B. Using these equations, show that one can always write B = ? × A E = -?A/?t - ?V , where A is the vector potential. Explain the origin of the two terms in the expression for E. b) Consider an infinite current sheet in the x–y plane at z = 0 containing uniform surface current K in the ex direction. Using the integral form of Ampère's law, or otherwise, deduce the magnetic field for z > 0 and z < 0 making clear any assumptions you make which are based on symmetry. Deduce the vector potential. c) Now consider the case where K is a time-dependent vector K(t) = K0 t n? , where n? points in the x–y plane. What is the induced electric field?

Transcript

00:01 Well, here in this question, first you ask you to demonstrate that the max equations allows you to write the magnetity and electric in this form.

00:10 Well, so you need to first write down the max three equations.

00:15 You have four max equations, right? once, e equals four power row, right? and the second is divergence being zero.

00:25 And the third is total of e equals minus c, 1, positive.

00:31 Tee, b, and then you also have this b equals 1 over c partial t e and plus 4 pi over c times current density, right? g.

00:45 Now, what you need to do, first, you note that this is zero, right? and that means, you know, if the divergence of certain vector is zero, then means that this vector can always be written as a core, right? the core of a vector is over theory, and if a vector has no divergence, then they must be called.

01:06 This is a general theorem, okay? a vector that has no divergence must be called.

01:12 So it suggests us to introduce b equals partial the core of a.

01:17 That's all the region of this expression for b.

01:21 And now if you substitute this into this expression, okay? if we substitute this into this expression, or we can actually, or in other words, or if we substitute into this, it's even better, okay? so if we plug this expression into this one, what do we get into this expression? then we'll find a partial e, because minus c, partial t, partial a, right? that's what we find, right? and then we can take this, move this to this side.

01:57 We will find the partial, the core of partial e plus over c, c, c, a equals zero, right, from this equation.

02:08 Now, the core of this vector zero, you know, if the code of a vector zero, it can only be a gradient.

02:14 So in other words, this must be gradient.

02:16 So we can set this gradient to be, to be minus data to be, to be the greater of certain scalar function five.

02:25 And from this, you see that e is simply given by minus, 5 minus over c partial ta, right? so this is, this is, this is, this is preans the, uh, the origin of the, uh, the expression for this lead field in terms of this scalable tension and vector potential, right? uh, so basically, um, the results comes from this to, uh, to, uh, to, to much way equations, right? now imagine you have an infinite current shape in the xy play, uh, with a uniform surface, uh, surface current.

02:59 You ask you to find a vector, vector potential.

03:05 You are asked you to find the back potential...

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