What is the potential inside a uniformly charged spherical shell of radius R and total charge Q?

What is the potential inside a uniformly charged spherical shell of radius R and total charge Q?

• A. V_inside = kQ/R; E_inside = 0
• B. V_inside = 0; E_inside = kQ/R^2
• C. V_inside = kQ^2/R; E_inside = 0
• D. V_inside = kQ/R; E_inside = kQ/R^2

Answer: (A)

In a uniformly charged spherical shell, the electric field inside is zero due to symmetry; any Gaussian surface inside the shell encloses no charge, so E = 0 there. If the field is zero inside, the potential must be constant throughout the interior because E = -dV/dr. This constant value has to match the potential right at the surface, since the potential cannot jump without an infinite field. The surface potential for a shell with total charge Q at radius R is V = kQ/R, where k = 1/(4πε0). So inside the shell, V is kQ/R and E is 0. The other options would imply a nonzero inside field or a dimensional form that doesn’t match a potential, which isn’t consistent with the symmetry and Gauss’s law.

In a uniformly charged spherical shell, the electric field inside is zero due to symmetry; any Gaussian surface inside the shell encloses no charge, so E = 0 there. If the field is zero inside, the potential must be constant throughout the interior because E = -dV/dr. This constant value has to match the potential right at the surface, since the potential cannot jump without an infinite field. The surface potential for a shell with total charge Q at radius R is V = kQ/R, where k = 1/(4πε0). So inside the shell, V is kQ/R and E is 0. The other options would imply a nonzero inside field or a dimensional form that doesn’t match a potential, which isn’t consistent with the symmetry and Gauss’s law.