Electric Potential
Electric potential is what electric field looks like once you stop caring about direction. Where field problems require adding vectors component by component, potential problems add plain numbers — which makes multi-charge and continuous-distribution problems dramatically easier, as long as you only need the potential itself rather than the field at that point. The two are linked by E=−dV/dx: field is the (negative) slope of potential, the same relationship force has to potential energy in mechanics.
What you'll learn
- Calculate the electric potential energy of a system of point charges.
- Calculate electric potential due to one or more point charges using superposition of scalars.
- Relate electric field and potential using E=−dV/dx and V=−∫E·dl.
- Calculate the electric potential of a continuous charge distribution via integration.
- Apply energy conservation to find the speed of a charged particle after moving through a known potential difference.
- Interpret equipotential surfaces and their geometric relationship to field lines.
1. Potential Energy and Potential
The electric potential energy of two point charges is U=kq₁q₂/r — positive for like charges (you'd have to do work to push them together), negative for opposite charges (they'd release energy coming together). This is exactly analogous to gravitational PE, just with a sign that can go either way depending on the charges involved.
Electric potential V is potential energy per unit charge: V=U/q₀=kq/r for a point charge. Critically, V is a property of the *location*, existing whether or not any particular charge sits there — the actual energy a specific charge q would have is U=qV.
2. Superposition and the E-V Relationship
Potential from multiple charges adds as plain numbers: V=Σkqᵢ/rᵢ, with no components to track — a significant simplification over the vector sum needed for field. This is why the midpoint between an equal and opposite charge pair has exactly zero potential, even though the field there is nonzero: the two contributions +kq/r and −kq/r cancel as scalars, term by term.
Field and potential are linked by E=−dV/dx: the field points in the direction V decreases fastest, with magnitude equal to how steeply it's decreasing. Given a full expression for V(x), differentiating (with a minus sign) recovers E(x) directly.
3. Energy Conservation with Charged Particles
Moving a charge q through a potential difference ΔV changes its potential energy by ΔU=qΔV. If nothing else does work on it, energy conservation (ΔKE=−ΔU) lets you find a final speed directly from the potential difference alone — no need to know the field's exact shape along the path, since potential (unlike work done against a general force) doesn't care about the route taken, only the endpoints.
For a single elementary charge accelerated through V volts, the energy gained is, by definition, V electron-volts (eV) — a convenient unit in atomic and particle contexts that sidesteps converting to joules until (if) it's actually needed.
Key equations
- U = kq₁q₂/r — The energy stored in the configuration of two point charges — positive if the charges have the same sign (energy required to have pushed them together), negative if opposite (energy released).
- V = kq/r — Potential due to a single point charge, referenced to V=0 at infinity. Superposition for multiple charges is plain scalar addition: V=Σkqᵢ/rᵢ.
- E = −dV/dx — Field is the negative spatial rate of change of potential — E points in the direction V decreases fastest, and is zero wherever V is momentarily flat (a local max, min, or plateau of V), not just where V itself happens to be zero.
- ΔU = qΔV — Energy conservation for a charge q moving through a known potential difference — combine with ΔKE=−ΔU to find a final speed.
- V = ∫ k dq/r — Potential of a continuous distribution — the scalar counterpart of Unit 201's field integral, and usually noticeably easier since there's no direction to track.