PHY 5346
HW Set 4 Solutions – Kimel
2. 2.9 The system is pictured below
a) We have treated this probem in class. We found the charge density induced was
σ = 30E0 cosθ
We also note the radial force/unit area outward from the surface is σ2/20. Thus the force on the
right hand hemisphere is, using x = cosθ
Fz = 1
20
∫ σ2 ẑ ⋅ da⃗ = 1
20
30E0 22πR2 ∫
0
1
x3dx = 1
20
30E0 22πR2/4 = 94
π 0E0
2R2
An equal force acting in the opposite direction would be required to keep the hemispheres from
sparating.
b) Now the charge density is
σ = 30E0 cosθ +
Q
4πR2
= 30E0x +
Q
4πR2
= 30E0 x +
Q
12π 0E0R2
Fz = 1
20
∫ σ2 ẑ ⋅ da⃗ = 1
20
30E0 22πR2 ∫
0
1
x x +
Q
12π 0E0R2
2
dx
Thus
Fz = 94
π 0E0
2R2 + 1
2
QE0 +
1
320πR2
Q2
An equal force acting in the opposite direction would be required to keep the himispheres from
separating.