A sphere of radius r1=30 cm has a charge density variation p0(r/r1) where p0 =200pC/m^3. The value of total charge on the sphere is:

 

Total Charge on Sphere (Variable Density)

Given

• Radius R = 30 cm = 0.3 m
• Charge density: ρ(r) = 200 (r / R) pC/m³

Step 1: Convert to SI Units

ρ(r) = 200 × 10^-12 × (r / R)

Step 2: Use Charge Formula

Q = ∫₀ᴿ ρ(r) · 4πr² dr

Step 3: Substitute

Q = ∫₀ᴿ (200 × 10^-12 × r/R) · 4πr² dr

Step 4: Simplify

Q = (200 × 10^-12 × 4π / R) ∫₀ᴿ r³ dr

Step 5: Integrate

∫ r³ dr = r⁴ / 4

Q = (200 × 10^-12 × 4π / R) × (R⁴ / 4)

Step 6: Simplify

Q = 200π × 10^-12 × R³

Step 7: Substitute R = 0.3

R³ = (0.3)³ = 0.027

Q = 200π × 10^-12 × 0.027

Q = 5.4π × 10^-12

Final Answer:
Q ≈ 1.7 × 10^-11 C
Or, 17 pC
Option A