DOC PREVIEW
HARVARD MATH 21A - Supplement on Electricity and Magnetism

This preview shows page 1 out of 3 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Math 21a Supplement on Electricity and MagnetismVector fields on R3 play a central role in Maxwell’s theory of electricity and magnetism. Inparticular, the electric field in space is described in Maxwell’s theory by a vector valued function ofspace and time, E(t, x). This vector provides the direction and magnitude of the electric field atany given point and at any given time. Likewise, the magnetic field is also described in Maxwell’stheory by a vector valued function of space and time, B(t, x). (Why not the letter ‘M’ for magneticfield? I am not sure. In any event, the traditional letter is ‘B’.)Maxwell proposed a set of equations which he postulated constrain the possible vectorvalued function pairs (E, B) which can arise as real world electric and magnetic fields. These arethe famous Maxwell equations. (Actually, various portions of these equations were written downby others prior to Maxwell’s culminating contribution.) These equations involve the operations ofcurl and divergence in a fundamental way. In any event, here are Maxwell’s equations in avacuum (no charged, polarizable or magnetically susceptible materials or particles present):• div E = 0,• div B = 0,• ∂∂t E = curl B,• ∂∂t B = - curl E .(1)Here, I have written these equations in units where various natural constants (such as the speed oflight) are equal to 1. In a physics book, these equations generally appear with various naturalconstants whose values depend on the particular choice of units of measurement.By the way, please take note of the evident symmetric treatment of the electric and magneticfields here. When there are charged particles present, these equations are modified in a way whichbreaks the symmetry between E and B. For example, if charged particles are present, then (1) ismodified as follows: First, the distribution of the charged particles in space determines a function,ρ(t, x), which measures the charge density at the point x and at time t. Thus, the triple integral ofρ over any volume V gives the total charge in V at the given time. Second, the particles may be inmotion, and a moving charge produces what is called a current. For a single moving charge, thisis a vector function of time which is proportional to the velocity vector of the particle. The currentdue to an ensemble of moving charges (such as electrons moving down a wire) is described by avector valued function, j(t, x), which measures the current density at the point x at time t. Thus,the triple integral of a component of j over a volume V is meant to give the component of thecurrent due to all of the moving charges in V.Note that the function ρ and the vector valued function j are not completely independent ofeach other. Indeed, since the triple integral of ρ measures the total charge at a given time in aregion, then this integral will change if charges move in or out of the region. Thus, the change ofρ with respect to time has something to do with the net motion charge across the boundary of theregion (in versus out). Meanwhile, moving charges determine the vector valued function j. Thus,we are led to the conclusion that the change of ρ with time must have something to do with j.And, this conclusion is born out by a more careful analysis. In particular, this more carefulanalysis leads to the following constraint on the possible pairs (ρ, j) which can arise in nature: ∂∂t ρ + div j = 0 .(2)We shall see in a subsequent supplement how this equation allows one to calculate the time rate ofchange of charge in a region from the behavior of j on the region’s boundary.With the function ρ and the vector valued function j understood, here are Maxwell’sequations in the presence of charges:• div E = ρ,• div B = 0,• ∂∂t E = curl B - j,• ∂∂t B = - curl E .(3)Here are some simple examples: First, suppose that a metal ball has total charge q, that thecharges are uniformly distributed in a layer near the surface of the ball, and that all of the chargesare stationary so that ρ is independent of time and j = 0. If the electric and magnetic fields areassumed to be independent of time also, then the relevent solution to (3) outside the ball (where ρ= 0 too) has the magnetic field B = 0, and the electric fieldE = q 143π||x x .(4)Thus, E points radially outward from the ball and |E| falls off with distance from the center of theball as the reciprical of the square of the distance. I leave it to you to verify that div E = 0 and alsothat curl E = 0. Note that (4) is not correct inside the ball where the formula for E is somewhatmore complicated.For a second example, suppose that electrons are moving at constant speed along acylindrical wire. Here, lets suppose that the z-axis is the center of the wire, that the wire iselectrically neutral, and that the current density, j, depends neither on time, nor on the angle in thex-y plane. If the electric and magnetic fields are also assumed to be independent of time, then therelevant solution to (3) outside the wire (where j = 0 also) has E = 0 and, at a point withcoordinates (x, y, z), hasB = σ 1222π+()xy (-y, x, 0) ,(5)where σ is value of the double integral of the z-component of j over the disk cross section of thewire. (The latter disk can be thought of as residing in the x-y plane; it is the intersection of thecylindrical wire with the x-y plane.) I leave it to you to check that div B = 0 and also that curl B =0.For the third and final example, suppose that there are no charges and no currents, so that(1) is relevant. Here is a solution to (1):• E = sin(t - z) (1, 0, 0) .• B = cos(t - z) (0, 1, 0).(6)Note that E and B both depend on postion and time. For example, the places where E = 0 move intime. For example, at t = 0, E is zero where z = 0, but not so at most later times. Indeed, theplaces where E = 0 are given byz = t + 2π n ,(7)where n can be any integer. In a very real sense, the electric field in (6) is propagating like amoving wave up the z-axis. And, so is the magnetic field. Such a wave is called an ‘electro-magnetic wave’. Radio waves, light waves, gamma rays are all examples of electro-magneticwaves.Once again, I leave it to you to verify that all of the equations in (1) are obeyed by


View Full Document

HARVARD MATH 21A - Supplement on Electricity and Magnetism

Documents in this Course
PDE’s

PDE’s

2 pages

PDE's

PDE's

2 pages

Review

Review

2 pages

intro

intro

2 pages

curve

curve

2 pages

mid1

mid1

7 pages

p-1

p-1

6 pages

contour

contour

2 pages

practice1

practice1

10 pages

diffeq1

diffeq1

6 pages

TRACES

TRACES

2 pages

PDE's

PDE's

2 pages

Review

Review

108 pages

GRAPHS

GRAPHS

2 pages

Review

Review

4 pages

VECTORS

VECTORS

2 pages

Load more
Download Supplement on Electricity and Magnetism
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Supplement on Electricity and Magnetism and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Supplement on Electricity and Magnetism 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?