What is the electric potential just outside a hollow spherical shell of radius R and charge Q at a distance r > R?

What is the electric potential just outside a hollow spherical shell of radius R and charge Q at a distance r > R?

• A. V(r) = kQ/R
• B. V(r) = kQ/R^2
• C. V(r) = kQ/r
• D. V(r) = kQ/(r^2)

Answer: (C)

Outside a hollow spherical shell with total charge Q, the symmetry makes the field behave as if all the charge were at the center. The electric field magnitude is E(r) = kQ / r^2 in the radial direction. The potential difference from infinity to r is V(r) - V(∞) = -∫∞^r E·dr, and with V(∞) = 0 this gives V(r) = ∫∞^r (kQ / t^2) dt = kQ / r. So the potential decreases like 1/r with distance, matching the field of a point charge at the center. The surface value would be kQ/R, but for any distance greater than R the correct expression is kQ/r. The other forms do not have the right 1/r distance dependence outside the shell.

Outside a hollow spherical shell with total charge Q, the symmetry makes the field behave as if all the charge were at the center. The electric field magnitude is E(r) = kQ / r^2 in the radial direction. The potential difference from infinity to r is V(r) - V(∞) = -∫∞^r E·dr, and with V(∞) = 0 this gives V(r) = ∫∞^r (kQ / t^2) dt = kQ / r. So the potential decreases like 1/r with distance, matching the field of a point charge at the center. The surface value would be kQ/R, but for any distance greater than R the correct expression is kQ/r. The other forms do not have the right 1/r distance dependence outside the shell.