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PHY 218: Electricity and Magnetism II
Prof. S. Teitel stte@pas.rochester.edu ---- Spring 2015

Problem Set 1

Due Wednesday, February 4, in lecture

• Problem 1 [10 points]

In lecture, when we derived the following expression for the energy stored in a configuration of magnetostatic currents,

Wmag = (1/2)∫ d3j(r)⋅A(r)

we assumed that the vector potential A was in the Coulomb gauge, i.e. that A = 0.

Show that in fact this expression is correct even when A is not in the Coulomb gauge.

Hint: Suppose A is in the Coulomb gauge. Construct another vector potential by taking A'(r) ≡ A(r) + χ(r), where χ(r) is any scalar function. Note, this A'(r) is a good vector potential, i.e. it gives the same B = ∇×A' as does A, however A' does not in general satisfy A' = 0. Show that Wmag = ∫ d3j(r)⋅A'(r) still gives the correct energy stored in the configuration.

• Problem 2 [10 points]

Suppose we have a situation where we have a known, time varying, magnetic field B(r, t), and there is no net charge, i.e. ρ(r) = 0. Write an integral expression that gives the resulting electric field E(r, t) in terms of the known B(r, t).

Hint: The relevant Maxwell Equations are: E(r, t) = 0 and ×E(r,t) = − ∂B(r,t)/∂t. Think about another situation in which you have seen a similar set of equations!

• Problem 3 [10 points]

Use the formalism of the Levi-Civita tensor εijk to derive the following vector identities:

a) ×(fg) = f(×g) − g×(f)    where f is any scalar function and g is any vector function

b) ×(×A) = −∇2A +(A)

• Problem 4 [10 points]

Griffiths 4th ed. problem 7.10

• Problem 5 [10 points]

Griffiths 4th ed. problem 7.16

• Problem 6 [10 points]

Griffiths 4th ed. problem 7.17

• Problem 7 [10 points]

Griffiths 4th ed. problem 7.22