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HARVARD MATH 21A - Supplement on Integration and Electro-Magnetism

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Math 21a Supplement on Integration and Electro-MagnetismThe Math 21a Supplement on Electricity and Magnetism presented the equations thatdescribe all known electric and magnetic phenomena. In particular, recall that Maxwell’s equationsdescribe electric fields by a vector valued function of space and time, E(t, x), while magnetic fieldsare described by a similar object, B(t, x). Then, Maxwell postulates that only such pairs (E, B)which obey the constraints• div E = ρ,• div B = 0,• ∂∂t E = curl B - j,• ∂∂t B = - curl E ,(1)appear in nature. Here, the function ρ and the vector valued function j (both functions of time andspace) are determined by the distribution and velocities of the various charged materials or particlespresent. The pair (ρ, j) must also satisfy a constraint, which is ∂∂t ρ + div j = 0 .(2)The purpose of this supplement is illustrate various important ramifications of (1) and (2)which follow from applications of either Stokes’ Theorem or the Divergence Theorem.a) A short reviewBefore getting to the purpose at hand, I will remind you of two definitions. For the first,suppose that v is a vector valued function on R3. (In the examples below the vector valuedfunctions under consideration will be E, B and j, thus they may also depend on time.) Now,suppose that S is a surface in R3 with a chosen normal direction, n. For example, S could be thewhole surface of a sphere and n the outward pointing normal, or S could be something much moreintricate. With regard to the normal vector n, keep in mind that most surfaces have two possiblenormal directions which point opposite.The first definition to review is that of the flux of v through S in the direction of the normalvector n. In particular, remember that this flux is defined to be the value of the surface integralS∫v•n dA .(3)Why call (3) ‘flux’? Here is one reason: The value of (3) is evidently proportional to the averageamount that v points outward from S along the normal n. Thus, if v is the velocity vector of amoving fluid at each point, then the positivity of (3) indicates that there is a net flow of fluidthrough S in the direction n. On the other hand , if (3) is negative, then the net flow of fluidthrough S is in the direction of -n. So, when v is the velocity of a fluid, the integral in (3) acts justlike flux should act. When v represents some other physical quantity, the corresponding integralin (3) is still called a flux integral. The second definition in this review requires the specification of a closed path γ in R3. Thatis, γ is some (possibly very twisty) path whose endpoint coincides with its starting point. Havingchosen a direction around γ, we introduced the path integralγ∫v•dx .(4)In the physics literature, this integral is called the circulation of v around γ. (If v is supposed torepresent the velocity vector of a moving fluid, then positivity of (4) indicates that the fluid issomewhat whirl-pool like and with a net rotation around γ. On the otherhand if (4) is negative,then the fluid should also be somewhat whirl-pool like, but rotating in the opposite sense to whichγ is traversed. This is, I believe, the historical basis for using the term ‘circulation’ for the integralin (4).)b) Charges, currents and fluxWith the preceding definitions understood, return now to Equations (1) and (2). Inparticular, consider first (2). The function ρ in (2) is supposed to give the density of electriccharges at each point in space and at each time. This means no more nor less then the assertion thatthe integral over any volume V of ρ gives the total charge in V. Let QV(t) denote this total charge.Since ρ is a function of t, so is the charge in V. In particular, QV(t) can change with time if there isa net movement of charge into or out of V. Since the current density j describes moving charges,there should be some relationship between the time derivative of QV(t) and the flux of j across theboundary surface S of the volume V. And, with the help of the Divergence Theorem, (2) predictsjust such a relationship. To see how this comes about, first invoke the definition of QV as theintegral of ρ over V to write ∂∂t QV = ∫∫∫V ∂∂tρ .(5)Next, invoke (2), to rewrite the right hand side above as∫∫∫V ∂∂tρ = - ∫∫∫V div j .(6)Finally, invoke the divergence theorem to rewrite the right hand side of (6) as- ∫∫∫V div j = - ∫∫S j•n dA ,(7)where n is the outward pointing normal vector to S. Together, these manipulations of the righthand side of (5) produce the equality∂∂t QV = - ∫∫S j•n dA .(8)This equation asserts that the rate of change of the total charge in V is minus the outward flux ofcurrent through the boundary of V.c) Electric fluxesReturn now to Equation (1). In particular, the Divergence Theorem can be used with thefirst line in (1) to equate the total charge in V, our QV, with the flux of the electric field E throughthe surface. Indeed, integrate both sides of (1) over the volume V. The integral of the right handside gives QV by definition. Meanwhile, according to the Divergence Theorem,∫∫∫V div E = ∫∫S E•n dA .(9)Thus, the first of the Maxwell equations implies thatQV(t) = ∫∫S E•n dA .(10)Here is an example: In the previous supplement, I asserted that E = q (4π)−1 |x|−3 x and B= 0 is a solution to (1) outside of a ball centered at the origin. Here, q can be any constant. I alsostated that q can be interpreted as the net electric charge in the ball. This interpretation of theconstant q can be verified by computing the integral on the right hand side of (10) in the case whereS is a sphere of radius sufficiently large to enclose the ball of charge. Indeed, take R to be such aradius. The outward pointing normal vector n to such a sphere is n = |x|−1 x, and soE•n = q (4π)−1 |x|-4 x•x = q (4π)−1 R−2(11)since x•x = |x|2 and |x|2 = R2 is the equation for the sphere S. Thus, the integrand in (10) isconstant in this case, and so the value of (10) is equal to this constant integrand times the area ofthe sphere. The latter is 4π R2, so the value for QV in (10) is, indeed, the number q.By the way, note that in general, (10) expresses QV(t) in terms of the electric flux through asurface S, while (8) expresses the time derivative of QV(t) in terms of the flux of the currentthrough S. Are these two expressions always compatible? They


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HARVARD MATH 21A - Supplement on Integration and Electro-Magnetism

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