13—Vector Calculus 2 41513.26 Prove the identity∇.(fF~) =f∇.F~+F~.∇f.
(b) Apply Gauss’s theorem to∇.(fF~)for a constantF~ to derive a result found in another problem.
13.27 The vector potential is not unique, as you can add an arbitrary gradient to it without affecting its curl.
Suppose thatB~=∇×A~with
A~=αxxyzˆ +βˆy x^2 z+γˆz xyz^2
Find a functionf(x,y,z)such thatA~′=A~+∇fhas thez-component identically zero. Do you get the sameB~
by taking the curl ofA~and ofA~′?
13.28 Take the vector field
B~=αˆxxy+βy xyˆ +γzˆ(xz+yz)
Write out the equation B~ = ∇ ×A~ in rectangular components and figure out what functions Ax(x,y,z),
Ay(x,y,z), andAz(x,y,z)will work. Note: From the preceding problem you see that you may if you wish pick
any one of the components ofA~to be zero and that will cut down on the labor. Also, you should expect that
this problem is impossible unlessB~ has zero divergence. That fact should comeoutof your calculations even if
you don’t anticipate it. Determine the conditions onα,β, andγthat make this problem solvable, and show that
this is equivalent to∇.B~= 0.
13.29 A magnetic monopole, if it exists, will have a magnetic fieldμ 0 qmˆr/ 4 πr^2. The divergence of this magnetic
field is zero except at the origin, but that means that not every closed surface can be shrunk to a point without
running into the singularity. The necessary condition for having a vector potential is not satisfied. Try to construct
such a potential anyway. Assume a solution in spherical coordinates of the formA~=φfˆ (r)g(θ)and figure out
whatf andg will have thisB~ for a curl. Sketch the resultingA~. You will run into a singularity (or two,
depending). Ans:A~=φμˆ 0 qm(1−cosθ)/
(
4 πr^2 sinθ)
(not unique)13.30 Apply Reynolds’ transport theorem to the other of Maxwell’s equations.
∇×B~=μ 0 ~j+μ 0 0∂E~
∂tDon’t simply leave the result in the first form that you find. Manipulate it into what seems to be the best form.
Useμ 0 0 = 1/c^2. Ans:
∫(
B~−~v×E/c~^2)
.d~`=μo∫(
~j−ρ~v)
.dA~+μ 0 0 (d/dt)∫
E~.dA~