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A solid sphere of radius R carries a non-uniform volume charge density given by ρ(r) = (Q/π R⁴) r, where r is the distance from the centre and Q is the total charge. If point P lies inside the sphere at a distance r1 from the centre, what is the magnitude of the electric field at P?

1. Q/(4π ε₀ r1²)
2. Qr1²/(4π ε₀ R⁴)
3. Qr1²/(3π ε₀ R⁴)
4. 0

Correct answer: Qr1²/(4π ε₀ R⁴)

Solution

Enclosed charge q(r1) = integral of (Q/pi R^4) r * 4 pi r^2 dr from 0 to r1 = Q r1^4 / R^4. Then E = q/(4 pi e0 r1^2) = Q r1^2 / (4 pi e0 R^4).

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