Electric Potential

What this unit covers

The topics below follow the published Physics C: E&M course framework for Unit 2. This unit is worth 15–21% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Electric Potential Energy12 min · 3 objectivesExplain why the electrostatic force is conservative and define potential energy from work · Derive U = kq₁q₂/r for a pair of point charges by integrating the Coulomb force · Apply energy conservation to the motion of charges
• Electric Potential from the Field14 min · 3 objectivesDefine potential and compute potential differences from V = −∫E·dl · Derive V = kq/r for a point charge and build V for rings and discs by integrating dV = k·dq/r · Exploit the scalar nature of potential in superposition problems
• Equipotential Surfaces12 min · 3 objectivesDescribe equipotential surfaces and their perpendicularity to the field · Relate equipotential spacing to field strength and compute work from W = qΔV · Explain why a conductor in equilibrium is a single equipotential volume
• From V to E: the Gradient13 min · 3 objectivesObtain field components from the potential using Eₓ = −dV/dx · Recover the fields of a point charge and a charged ring by differentiating V · Interpret graphs of V(x): slope gives −Eₓ and flat regions mean zero field

Formulas in Unit 2

Every term in Unit 2

All 19 terms we publish for Electric Potential, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Electric potential of a point charge

V = kq/r, taking zero at infinity. A scalar, so potentials from several charges add algebraically.

Potential from a distribution

V = ∫k dq/r. Easier than the field integral because there are no vector components to resolve.

Relation between field and potential

V = −∫E·dl and E = −∇V, which in one dimension is E = −dV/dx.

Why field is the negative gradient

A positive charge accelerates toward lower potential, so the field points down the potential hill.

Equipotential surfaces

Always perpendicular to field lines. No work is required to move a charge along one, since ΔV is zero.

Potential energy of a charge pair

U = kq₁q₂/r, including sign. Opposite charges have negative U, which becomes more negative as they approach.

Potential energy of a charge configuration

Sum kq_iq_j/r_ij over all distinct pairs. Counting a pair twice is the standard error.

Potential of a charged conductor

Constant throughout the conductor and on its surface, since the internal field is zero. The surface is an equipotential.

Field and potential can be independently zero

At the midpoint between equal opposite charges, V = 0 but E ≠ 0; at the midpoint between equal like charges, E = 0 but V ≠ 0.

Work and potential difference

W = qΔV, and W = −ΔU. A positive charge released in a field moves toward lower potential.

Why potential is easier than field

V is a scalar, so contributions add without components. Compute V by integration, then get E by differentiating.

Sign of the potential integral

V_b − V_a = −∫E·dl from a to b. The minus sign is the difference between a right and a wrong answer.

Potential inside a conducting sphere

Constant and equal to the surface value kQ/R, since the interior field is zero and no work is done moving within it.

Potential inside a uniformly charged insulating sphere

Rises toward the center, reaching kQ(3R² − r²)/2R³ — unlike a conductor, where it is flat.

Potential from a graph of E vs x

The change in potential is the negative area under the field-position graph.

Field from a graph of V vs x

E is the negative slope. A flat V means zero field, and steeply falling V means a strong field in the positive direction.

Energy of assembling charges

Bring charges in one at a time from infinity, summing the work at each step. Each pair is counted once.

Accelerating a charge through a potential difference

qΔV = ½mv², which is how electron guns and accelerators set particle speeds.

Electron volt

The energy gained by one elementary charge through one volt, 1.6 × 10⁻¹⁹ J. A convenient unit at atomic scales.

What examiners penalize here

• Sign conventions to lock in for the exam: ΔV = −∫→E·d→l (potential drops along the field), U = qV and ΔU = qΔV (signs of q included), and W_field = −ΔU. A positive charge released from rest falls toward *lower* V; a negative charge toward *higher* V. Nearly every potential FRQ point hinges on one of these signs.
• Memorize the inverse pair and their graphical readings: V_b − V_a = −∫ₐᵇ →E·d→l (area under an Eₓ graph, negated) and Eₓ = −dV/dx (slope of a V graph, negated). AP free-response loves handing you one graph and demanding the other — check your minus sign at a point where you know which way the field must push a positive charge.

Practice Physics C: E&M

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 6 units of AP Physics C: E & M

1. Unit 1 · Electric Charges, Fields, and Gauss’s Law
2. Unit 2 · Electric Potential
3. Unit 3 · Conductors, Capacitors, and Dielectrics
4. Unit 4 · Electric Circuits
5. Unit 5 · Magnetic Fields
6. Unit 6 · Electromagnetic Induction