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A sphere of radius R has a charge density that varies as the square of the distance from its center: rho = A*r², with A a positive constant. Find the magnitude of the electric field at a point r = R/2 from the center.

1. A/(4*pi*eps0)
2. A*R³/(40*eps0)
3. A*R³/(24*eps0)
4. A*R³/(5*eps0)

Correct answer: A*R³/(40*eps0)

Solution

Enclosed charge within radius r: q(r) = integral[0..r] A*r'² * 4*pi*r'² dr' = 4*pi*A*r⁵/5. Gauss: E*4*pi*r² = q/eps0, so E = A*r³/(5*eps0). At r = R/2: E = A*(R/2)³/(5*eps0) = A*R³/(8*5*eps0) = A*R³/(40*eps0).

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