What is the electric potential at distance r from a point charge q (with V(∞) = 0)?

What is the electric potential at distance r from a point charge q (with V(∞) = 0)?

• A. V(r) = (1/(4π ε0)) q / r
• B. V(r) = (1/(4π ε0)) q r
• C. V(r) = (1/(4π ε0)) r / q
• D. V(r) = (1/(4π ε0)) q r^2

Answer: (B)

Electric potential from a point charge falls off with distance as 1/r, and the zero of potential is set at infinity. The electric field of a point charge is E(r) = (1/(4π ε0)) q / r^2 directed radially. The potential difference is V(r) − V(∞) = −∫∞^r E·dr. With V(∞) = 0, this becomes V(r) = −∫∞^r (1/(4π ε0)) q / r^2 dr = (1/(4π ε0)) q / r. So the potential at distance r is V(r) = (1/(4π ε0)) q / r. If q is positive, V is positive; if q is negative, V is negative. Forms that grow with r or involve r^2 would not go to zero at infinity and thus cannot satisfy V(∞) = 0.

Electric potential from a point charge falls off with distance as 1/r, and the zero of potential is set at infinity. The electric field of a point charge is E(r) = (1/(4π ε0)) q / r^2 directed radially. The potential difference is V(r) − V(∞) = −∫∞^r E·dr. With V(∞) = 0, this becomes V(r) = −∫∞^r (1/(4π ε0)) q / r^2 dr = (1/(4π ε0)) q / r. So the potential at distance r is V(r) = (1/(4π ε0)) q / r. If q is positive, V is positive; if q is negative, V is negative. Forms that grow with r or involve r^2 would not go to zero at infinity and thus cannot satisfy V(∞) = 0.