Electrodynamics (I): Homework 1
Due: October 2, 2014
Exercises in Griffiths
1.6, 1.12, 1.13, 1.22, 1.26, 1.31, 1.40 (hand-in 5%), 1.54, 1.56, 1.57 (hand-in 10%), 1.61
(hand-in 25%), 1.62 (hand-in 25%)
Ex.1 hand-in
10% Prove the following vector indenties:
~×B
~ · (∇ × C)
~ =B
~ · (A
~ · ∇)C
~ −A
~ · (B
~ · ∇)C
~
(a) 5% A
(b) 5%
R
~ × ∇) × A
~ = H d~r × A
~
∂S
S (dS
~ Here S is an open surface bounded by
for any continuous and differentiable vector field A.
the closed curve ∂S.
~ r), show that dA
~ ≡ A(~
~ r + d~r) − A(~
~ r) = (d~r · ∇)A
~ for
(c) 5% For any vector field A(~
~ only depends on magnitude of ~r and has only θ
infinitesimal d~r. Furthermore, if A
~ = Aθ (r)θ̂, show that ∇ · [A(~
~ r) × ~r] = 0.
component, i.e., A
Ex.2 hand-in
~ · ∇)A
~+A
~ × (∇ × A).
~
(a) 5% If A2 = x − y + 2z, find the magnitude of the vector (A
~ r) = A
~ 0 ei~k·~r where A
~ 0 and ~k are constant vectors. Find
(b) 5% Given a vector field A(~
~ r)] and ∇ × [∇ × A(~
~ r)] in terms of A
~ 0 , ~r, and ~k.
∇[∇ · A(~
(c) 5% It is known that the point (2, −1, 2) lies in the intersecting curves of two
surfaces:x2 + y 2 + z 2 = 9 and z = x2 + y 2 − 3. Find the angle between the aboves surfaces
at the pointc (2, −1, 2).
(d) 5% Suppose that n̂ is a unit vector which is equally likely to lie any direction in space.
Find the average values of (~a · n̂)(~b · n̂) and (~a × n̂) · (~b × n̂) in terms of ~a and ~b. Here ~a and
~a are constant vectors.
Ex.3 hand-in 10%
~ = (2yz, −x − 3y + 2, x2 + z), find the surface integral R R ∇ × A
~ · n̂ds
Consider a vector field A
over the surface in the first octant formed by the intersection of two cylinders x2 + y 2 ≤ a2
and x2 + z 2 ≤ a2 .
1