2.3 Electric Potential Due to a System of Charges

Superposition of Potential

Electric potential is a scalar, so the potential from a system of point charges is found by simple algebraic addition: V = kΣ(qᵢ/rᵢ), where rᵢ is the distance from each charge qᵢ to the point of interest. Unlike electric field, there are no components or directions to worry about — just add signed values.

Continuous Charge Distributions

For continuous charge, replace the sum with an integral: V = k∫(dq/r). Break the object into infinitesimal charge elements dq, express dq in terms of a density (λ, σ, or ρ) and a geometric variable, then integrate over the whole distribution. Choosing the right coordinate system simplifies the setup significantly.

Potential vs. Reference Point

Potential is defined relative to a reference, usually V = 0 at infinity. Since V is scalar and path-independent, you can compute total potential at a point without worrying about the path taken — a major simplification compared to computing electric field, which requires vector addition.

Key terms

Potential due to a system of point charges

V = kΣ(qᵢ/rᵢ) — scalar sum of each charge's contribution.

Potential due to continuous charge distribution

V = k∫(dq/r), integrating infinitesimal charge elements over the distribution.

Why is potential easier to combine than electric field?

Potential is a scalar quantity, so contributions add algebraically without needing vector components.

Reference point for potential

Usually chosen so that V = 0 at infinity, far from all charges.

Potential from equal and opposite charges at equal distance

Net potential is zero, since scalar contributions +kq/r and −kq/r cancel.

Try an AP-style question

Charges +2 nC and −2 nC are 2 m apart. What is the potential at the midpoint?

1. −18 V
2. 18 V
3. 36 V
4. 0