The potential due to a point charge falls off with distance as which function?

The potential due to a point charge falls off with distance as which function?

• A. 1/r^2
• B. 1/r
• C. r
• D. constant

Answer: (B)

Electric potential from a point charge falls off inversely with distance. The field from a point charge spreads over a sphere, giving E ∝ 1/r^2. The potential V is related to the field by E = -dV/dr for a radial field. So dV/dr = -(kQ)/r^2, and integrating yields V(r) = kQ/r + C. With the usual reference V → 0 as r → ∞, C = 0, so V(r) ∝ 1/r. This is why the potential decreases with distance proportional to 1/r, unlike the field which decreases as 1/r^2.

Electric potential from a point charge falls off inversely with distance. The field from a point charge spreads over a sphere, giving E ∝ 1/r^2. The potential V is related to the field by E = -dV/dr for a radial field. So dV/dr = -(kQ)/r^2, and integrating yields V(r) = kQ/r + C. With the usual reference V → 0 as r → ∞, C = 0, so V(r) ∝ 1/r. This is why the potential decreases with distance proportional to 1/r, unlike the field which decreases as 1/r^2.