A sphere of radius r₁ = 30 cm has a charge density variation ρ(r) = ρ₀ (r / r₁) where ρ₀ = 200 pC/m³. Find the total charge on the sphere.
26.
A sphere of radius r₁ = 30 cm has a charge density variation ρ(r) = ρ₀ (r / r₁) where ρ₀ = 200 pC/m³. Find the total charge on the sphere.
Options:
- 17 pC
- 8.5 pC
- 34 pC
- 200 pC
Solution
The charge element is:
dQ = ρ dV
For a spherical shell:
dV = 4πr² dr
Substitute ρ(r):
dQ = ρ₀ (r / r₁) × 4πr² dr
Total charge:
Q = ∫₀ʳ¹ ρ₀ (r / r₁) 4πr² dr
Q = (4πρ₀ / r₁) ∫₀ʳ¹ r³ dr
Q = (4πρ₀ / r₁) [r⁴ / 4]₀ʳ¹
Q = πρ₀ r₁³
Now substitute values:
r₁ = 0.3 m
Q = π × 200 × (0.3)³ pC
Q = π × 200 × 0.027 pC
Q = 5.4π pC
Q ≈ 17 pC
Final Answer: 17 pC