Download Homework No. 10 (Spring 2015) PHYS 420: Electricity and Magnetism II

Homework No. 10 (Spring 2015)
PHYS 420: Electricity and Magnetism II
Due date: Wednesday, 2015 May 6, 4.30pm
1. (20 points.) Using Maxwell’s equations, without introducing potentials, show that the
electric and magnetic fields satisfy the inhomogeneous wave equations
1 ∂2
1
1 1 ∂
2
−∇ + 2 2 E(r, t) = − ∇ρ(r, t) −
J(r, t),
(1a)
c ∂t
ε0
ε0 c2 ∂t
1 ∂2
2
(1b)
−∇ + 2 2 B(r, t) = µ0 ∇ × J(r, t).
c ∂t
2. (20 points.) Consider the retarded Green’s function
1
1
′
′
′
′
G(r − r , t − t ) =
δ t − t − |r − r | .
|r − r′ |
c
(2)
(a) For r′ = 0 and t′ = 0 show that
1 r
.
G(r, t) = δ t −
r
c
(b) Then, evalauate
Z
(3)
∞
dt G(r, t).
(4)
−∞
(c) RFrom the answer above, what can you comment on the physical interpretation of
∞
dt G(r, t).
−∞
3. (20 points.) Using the identity
δ(F (x)) =
X δ(x − ar )
,
dF r dx x=a r
(5)
where the sum on r runs over the roots ar of the equation F (x) = 0, evaluate
δ(ax2 + bx + c).
(6)
4. (20 points.) Using the identity
δ(F (x)) =
X δ(x − ar )
,
dF r dx x=a r
(7)
where the sum on r runs over the roots ar of the equation F (x) = 0, evaluate
δ(sin x).
1
(8)
5. (20 points.) A charged
particle with charge q moves on the z-axis with constant speed
p
v, β = v/c, γ = 1/ 1 − β 2 . The scalar and vector potential generated by this charged
particle is
1
q
p
,
2
2
4πε0 (x + y ) + γ 2 (z − vt)2
ẑ
q
p
cA(r, t) = βγ
.
4πε0 (x2 + y 2) + γ 2 (z − vt)2
φ(r, t) = γ
(a) Using
E = −∇φ −
∂
A,
∂t
A = ∇ × A,
(9a)
(9b)
(10a)
(10b)
evaluate the electric and magnetic field generated by the charged particle to be
E(r, t) = γ
xî + y ĵ + (z − vt)k̂
q
,
4πε0 [(x2 + y 2 ) + γ 2 (z − vt)2 ] 32
cB(r, t) = βγ
q
−y î + xĵ
.
4πε0 [(x2 + y 2) + γ 2 (z − vt)2 ] 23
(11a)
(11b)
(b) Evaluate the electromagnetic momentum density for this configuration by evaluating
G(r, t) = ε0 E(r, t) × B(r, t).
2
(12)