PHY 5346
HW Set 1 Solutions – Kimel
3. 1.4 Gauss’s Law:
∫ E⃗ ⋅ da⃗ = Qenclosed
0
a) Conducting sphere: all of the charge is on the surface σ = Q
4πa2
E4πr2 = 0, r < a
E4πr2 = Q
0
, r > a
E = 0, r < a
E⃗ =
Q
4π 0r2
r̂, r > a
b) Uniform charge density: ρ =
Q
4
3 πa
3 , r < a, ρ = 0, r > a.
E4πr2 = Qr
3
0a3
→ E = Qr
4π 0a3
,
r < a
E4πr2 = Q
0
→ E =
Q
4π 0r2
,
r > a
c) ρ = A rn
Q = 4π ∫ r2drArn = 4πAan+3/n + 3 → ρ = n + 3Q
4πan+3
rn, r < a
ρ = 0, r > a
E4πr2 = n + 3Q
4π 0an+3
4πrn+3/n + 3 → E =
Q
4π 0r2
rn+3
an+3
, r < a
E =
Q
4π 0r2
,
r > a
1. n = −2.
E =
Q
4π 0ra
,
r < a
E =
Q
4π 0r2
, r > a
2. n = 2.
E = Qr
3
4π 0a5
,
r < a
E =
Q
4π 0r2
,
r > a