Electric Potential
Electric potential V is electric potential energy per unit charge and is a scalar field. For point charges, V=kq/r and contributions add algebraically. For continuous sources, V=k∫dq/r. The electric field is the downhill spatial derivative of potential: E⃗=−∇V, while potential difference comes from ΔV=−∫E⃗·dr⃗. Equipotential lines are perpendicular to E⃗, and the field points toward decreasing potential.
Topic sequence · 2 of 3
Learning objectives
Objective 1
Calculate electric potential from point charges and the current required continuous charge distributions using scalar superposition and integration.
Objective 2
Move correctly between electric potential, potential difference, electric field, and potential-energy-per-charge relationships.
Objective 3
Interpret and construct V-position graphs, equipotential maps, and field maps using derivatives, line integrals, and sign conventions.
- Topic 9.1 electric potential energy and its system meaning.
- Topics 8.3–8.4 electric fields and superposition.
- Single-variable integration, derivatives, dot products, and vector components.
Definitions first
Representations you must read and create
Representation 1
V(x) graph: its signed slope gives E_x=−dV/dx; a flat segment means E_x=0 along that direction.
Representation 2
Equipotential map: tighter spacing corresponds to larger |∇V| and therefore stronger electric field magnitude.
Representation 3
Field-vector map and equipotential map: E⃗ crosses equipotentials perpendicularly and points toward decreasing V.
Representation 4
Source diagram for continuous charge: label dq, the source-to-observation distance r, and exploit scalar symmetry before integrating.
Core relationships and calculus
Variables: V J/C = volt
Use: definition of electric potential for a sufficiently small test charge
Variables: q C; r m
Use: single point charge with zero reference at infinity
Variables: V volts
Use: multiple point charges; scalar superposition
Variables: dq C; r source-to-field-point distance
Use: continuous charge distribution
Variables: ΔU_E J; q C
Use: potential difference from potential-energy change
Variables: E_x V/m = N/C
Use: one-dimensional field component from potential slope
Variables: ∇V V/m
Use: general field from the spatial gradient of potential
Variables: dr⃗ m
Use: potential difference from electric field along any convenient path in electrostatics
Potential is often easier to integrate than electric field because V is scalar: V=k∫dq/r. The field can then be recovered by differentiation, E⃗=−∇V. Conversely, if E⃗ is known, ΔV=−∫E⃗·dr⃗. These two calculus directions—source integration and spatial differentiation/line integration—are the core of Topic 9.2.
Start from ΔU_E=−∫F⃗_e·dr⃗ and divide by the test charge q: ΔV=−∫(F⃗_e/q)·dr⃗=−∫E⃗·dr⃗. For an infinitesimal displacement, dV=−E⃗·dr⃗. Along x, dV/dx=−E_x. In three dimensions, collecting the directional derivatives gives E⃗=−∇V.
For calculus-based potential from charge distributions, the current course specifically expects the listed geometries: an infinitely long uniformly charged wire or cylinder at a distance from its central axis; a thin ring along its axis; a semicircular arc or part of one at its center; and a finite wire/line charge at a collinear point or on its perpendicular bisector. Do not expand the required integration catalog beyond these geometries merely because a more advanced electrostatics text does.
Physical examples
Example 1
At the midpoint between equal positive charges, electric fields cancel by symmetry but potentials add, so E=0 while V>0.
Example 2
At the midpoint between equal and opposite charges, potential can be zero while the electric field is nonzero. Zero potential does not imply zero field.
Example 3
Along the axis of a uniformly charged ring, every dq is the same distance from the observation point, so V=kQ/√(R²+z²).
Example 4
Moving along an equipotential gives ΔV=0 because E⃗ has no component tangent to the equipotential.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Scalar-vs-vector translation: a point can have V=0 and E≠0, or E=0 and V≠0.
Reasoning pattern 2
Graph slope: the sign and magnitude of dV/dx determine E_x through E_x=−dV/dx.
Reasoning pattern 3
Equipotential spacing: closer isolines indicate a steeper potential gradient and larger field magnitude.
Reasoning pattern 4
Integral setup: write dq from λ, σ, or the specified source geometry, express r in the integration variable, and only then integrate.
Reasoning pattern 5
Path independence: in electrostatics ΔV depends only on endpoints even though the line integral may be evaluated along any convenient path.
Deeper calculus and limiting-case check
For a ring of radius R and total charge Q, V(z)=kQ/√(R²+z²). Differentiate: E_z=−dV/dz=kQz/(R²+z²)^(3/2), exactly matching the direct field-integration result from Topic 8.4. This is a powerful cross-check between scalar potential and vector field methods.
Interactive potential-versus-field comparison
Place two point charges on the x-axis and evaluate scalar potential V and signed field component Eₓ at a point that is not on either charge.
Experimental connection
Map electric potential in a shallow conducting-water tray or conductive paper using a low-voltage DC source and voltmeter. Draw equipotential isolines, estimate field direction as perpendicular to the isolines, estimate |E|≈|ΔV/Δs| between nearby lines, and compare the inferred field map with direct qualitative expectations from electrode geometry.
Connections across Physics C
Connection 1
Topic 9.1 supplies the energy-per-charge interpretation V=U_E/q.
Connection 2
Topics 8.3–8.4 supply electric fields and source distributions; Topic 9.2 offers a scalar route that can simplify the same geometry.
Connection 3
Topic 9.3 uses qΔV to turn the potential map into particle-energy changes.
Connection 4
Later conductor and capacitor topics rely on equipotential surfaces and potential differences as organizing ideas.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Treating V as a vector and resolving it into x/y components. Potential is scalar.
Avoid this
Assuming V=0 means E=0. Only a locally flat potential in all directions implies zero field.
Avoid this
Forgetting the minus sign in E_x=−dV/dx or ΔV=−∫E⃗·dr⃗.
Avoid this
Using the angle between E⃗ and an equipotential as something other than 90° in electrostatics.
Avoid this
Integrating a continuous source without first expressing dq and the source-to-point distance r in one consistent variable.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
What is the difference between electric potential and electric potential energy?
Potential V is a property of location created by source charges and is measured in J/C. Potential energy U_E belongs to a particular charge-field system and is measured in joules.
Can electric potential be zero where electric field is not zero?
Yes. Equal and opposite source charges can give zero scalar potential at points where their field vectors reinforce.
Why are equipotential lines perpendicular to electric field lines?
If E had a component along an equipotential, moving along that line would change V. By definition ΔV is zero along an equipotential, so the tangential field component is zero.
Why is potential often easier than electric field for continuous charge?
Potential contributions are scalar, so they add algebraically; field integration requires vector components and directional cancellation.
What does a steep V(x) graph mean?
A large-magnitude slope means a large electric-field component in the opposite direction because E_x=−dV/dx.