Electric field inside a uniformly charged solid sphere of radius R and total charge Q at distance r < R from the center is given by which expression?

Electric field inside a uniformly charged solid sphere of radius R and total charge Q at distance r < R from the center is given by which expression?

• A. E(r) = Q r / (4π ε0 R^3)
• B. E(r) = Q r / (4π ε0 R^2)
• C. E(r) = Q / (4π ε0 r^2)
• D. E(r) = Q r^2 / (4π ε0 R^3)

Answer: (A)

Inside a uniformly charged solid sphere, the electric field increases linearly with distance from the center. This comes from Gauss’s law: choose a Gaussian surface a sphere of radius r (with r < R). The field on that surface is radial and uniform, so E(r)·4πr^2 = Q_enc/ε0. The charge enclosed is proportional to the volume inside radius r, so Q_enc = Q (r^3/R^3). Substituting gives E(r)·4πr^2 = [Q (r^3/R^3)]/ε0, which leads to E(r) = Q r / (4π ε0 R^3). So the field inside is proportional to r, with the proportionality factor set by the total charge and the sphere radius.

Inside a uniformly charged solid sphere, the electric field increases linearly with distance from the center. This comes from Gauss’s law: choose a Gaussian surface a sphere of radius r (with r < R). The field on that surface is radial and uniform, so E(r)·4πr^2 = Q_enc/ε0. The charge enclosed is proportional to the volume inside radius r, so Q_enc = Q (r^3/R^3). Substituting gives E(r)·4πr^2 = [Q (r^3/R^3)]/ε0, which leads to E(r) = Q r / (4π ε0 R^3). So the field inside is proportional to r, with the proportionality factor set by the total charge and the sphere radius.