A spherically symmetric charge distribution has charge density rho(r) = rho0*(3/4 - r/R) for r R, where r is the distance from the centre O. Find the electric field at a point P located inside the distribution at distance r (r

Question

A spherically symmetric charge distribution has charge density rho(r) = rho0*(3/4 - r/R) for r <= R, and rho = 0 for r > R, where r is the distance from the centre O. Find the electric field at a point P located inside the distribution at distance r (r < R).

Accepted Answer

(rho0*r/(4*epsilon0)) * (1 - r/R). By spherical symmetry, Gauss's law gives E(r) = q_enc/(4*pi*epsilon0*r²). Integrate the given density to get the enclosed charge: q_enc = integral₀^r rho0(3/4 - r'/R) 4 pi r'² dr' = 4 pi rho0 [ (3/4)(r³/3) - (1/R)(r⁴/4) ] = 4 pi rho0 [ r³/4 - r⁴/(4R) ] = pi rho0 r³ (1 - r/R). Then E = q_enc/(4 pi epsilon0 r²) = rho0 r (1 - r/R)/(4 epsilon0).

A spherically symmetric charge distribution has charge density rho(r) = rho0*(3/4 - r/R) for r <= R, and rho = 0 for r > R, where r is the distance from the centre O. Find the electric field at a point P located inside the distribution at distance r (r < R).

Solution

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