Difficulty: Intermediate | Prerequisites: Coulomb's law, electric field, Gauss's law basics
Electric potential is the energy landscape that charges move through: positive charges roll "downhill" from high V to low V, and the work done by the electric field equals the charge times the potential drop. Equipotential lines are contour lines on this landscape, always perpendicular to field lines. When a conductor (a metal slab or sphere) sits in an external field, charge redistributes on its surfaces to kill the field inside, and the charge on each face is determined by superposition and the requirement that E = 0 in the metal.
Electric potential (V)
The electric potential energy per unit charge at a point in space. In simple terms, it measures how much energy a positive test charge would have at that location, per coulomb.
Potential difference (ΔV)
The difference in electric potential between two points: ΔV = V_B – V_A. Think of it as the "voltage drop" that drives charge from one point to another.
Electric potential energy (U)
The energy stored in a configuration of charges by virtue of their positions. For a pair of point charges: U = kq₁q₂/r.
Work done by the electric field (W)
W = qΔV (where ΔV is the drop in potential along the path). The field does positive work when it moves a positive charge from higher to lower potential.
Equipotential lines (surfaces)
Contours of constant electric potential. They are always perpendicular to field lines, and no work is done moving a charge along one.
Superposition of electric fields
The total electric field at any point is the vector sum of the fields produced by each source individually. For infinite charged planes, each contributes σ/(2ε0) on each side.
Induced charge
Charge that rearranges on a conductor's surfaces in response to an external field or nearby charge. The total charge on the conductor does not change, but it separates to cancel the field inside.
Infinite conducting plate
An idealised plane of charge extending forever. Each face produces a uniform field of σ_face / (2ε0) pointing away from (if positive) or toward (if negative) the face.
Metal slab in an external field
A thick conductor placed between other charge sources. Charge redistributes onto its two faces so that E = 0 inside, with the charge on each face determined by superposition.
The work done by the electric field on a charge q moving from A to B is W = q(V_A – V_B).
Positive work means the field pushes the charge in its direction of motion. A positive charge moving from high V to low V gains kinetic energy.
The work depends only on the endpoints, not the path (the electrostatic field is conservative).
If two pairs of points have the same potential difference, the field does the same work moving the same charge between either pair.
Field lines point from high potential to low potential.
Equipotential lines are perpendicular to field lines everywhere.
Points on the same equipotential line are at the same potential. Moving a charge along an equipotential requires zero work.
When field lines converge (get closer together), the field is stronger there, and equipotential lines are more closely spaced.
Comparing potential at two points: follow the field lines. If the field points from B toward C, then V_B > V_C (potential decreases in the direction of the field).
Single infinite plate with surface charge density σ
Produces a uniform field E = σ / (2ε0) pointing away from the plate (if σ > 0) on each side.
The field is the same magnitude everywhere, regardless of distance.
Two parallel plates, each with charge density σ_p
Use superposition: add the fields from each plate as vectors.
Between the plates, the fields from the two plates may add or partially cancel depending on the signs of the charges.
Outside both plates, the fields from the two plates partially cancel or add.
Metal slab placed near charged plates
The slab is a conductor: E must be zero inside it.
Charge redistributes onto its left and right faces.
The charge on each face is found by requiring E = 0 inside. Using superposition of all sources (both plates plus both slab faces), set the total field inside the slab to zero.
For the exam problem with two plates at σ_p = −3 µC/m² and a slab with total σ_s = +8 µC/m²: the right-face charge is σ_R = (σ_s/2) + σ_p, found by superposition.
Electric field at the origin (between the two plates, to the left of the slab)
Superpose contributions from all sources. The two plates are symmetric about the origin, so their fields at the origin add. The slab's two faces also contribute.
The answer involves |σ_s + 2σ_p| / (2ε0) or |σ_s – 2σ_p| / (2ε0), depending on geometry.
Potential energy of a positive charge moving between points
Moving from B (between the plates) to A (between the plates, closer to a negative plate): if the field at that location points from B toward A, the field does positive work and potential energy decreases.
If the field points from A toward B, the charge must move against the field, so potential energy increases.
An uncharged metal sphere brought near a charged surface experiences induction: charge on the near side is opposite in sign to the external charge, and the far side has the same sign.
The attraction between the near side and the external charge is always stronger than the repulsion from the far side (because the near side is closer).
Result: an uncharged conductor is always attracted to a nearby charged surface, regardless of the sign of the external charge.
Work done by the field on charge q
W = q(V_A - V_B) = -q \Delta VElectric field of a single infinite plane with surface charge density σ
E = \frac{\sigma}{2\varepsilon_0}This field is uniform (distance-independent) and points away from the surface if σ > 0.
Electric field between two infinite parallel plates with equal and opposite charge (±σ)
E_{\text{between}} = \frac{\sigma}{\varepsilon_0}Outside both plates, E = 0 (the fields cancel).
Potential energy of a pair of point charges
U = \frac{k q_1 q_2}{r}Kinetic energy gained by a charge accelerated through potential difference ΔV
\frac{1}{2}mv^2 = |q| \, |\Delta V|Students often think work depends on the path taken. It does not (for electrostatic fields). Only the potential difference between the start and end points matters.
Students often think that points on different equipotential lines could be at the same potential. They cannot. Each equipotential line represents a single value of V.
Students often assume an uncharged conductor near a charged surface feels no force. It does, because of induced charge separation. The attraction side is always closer and wins.
Students often forget that for infinite plates, the field from a single plate is σ/(2ε0), while the field between two opposite plates is σ/ε0 (double, because both plates contribute). Mixing these up is one of the most common errors.
⚠️ Comparing work done along two different paths (A→D vs B→C): if both pairs cross the same number of equipotential lines, the work is the same.
⚠️ "Is V at C greater than, equal to, or less than V at B?" Follow the field lines. Potential drops in the direction of E.
⚠️ Superposition problems with multiple infinite plates plus a conducting slab: set E = 0 inside the slab to find the charge on each face. This is heavily tested.
⚠️ Uncharged conductors near charged surfaces: the answer is always attraction, never repulsion or zero force.
⚠️ Moving a positive charge against the field increases its potential energy. Moving it with the field decreases it.
True or false: The electric field always points from low potential to high potential. (False. It points from high to low.)
Fill in the blank: The work done by the electric field on a charge q moving through a potential drop ΔV is W = ______ . (qΔV, where ΔV is the drop, i.e. V_initial – V_final.)
True or false: An uncharged metal sphere is repelled by a nearby positive plate. (False. It is attracted, due to induced charge.)
Fill in the blank: The field from a single infinite plane of charge density σ is E = ______ on each side. (σ / 2ε0.)
True or false: Moving a charge along an equipotential line requires the field to do work. (False. No work is done along an equipotential.)
Q: A positive charge is moved from point A to point D, crossing two equipotential lines. A second positive charge is moved from B to C, also crossing two equipotential lines. How does the work done by the field compare?
A: The work is equal. Work depends only on the potential difference between endpoints, and crossing the same number of equipotential lines (with the same spacing) means the same ΔV.
Q: Field lines point to the right. Is the potential at point C (further right) greater than, equal to, or less than the potential at point B (further left)?
A: Less than. Potential decreases in the direction of the electric field.
Q: Two infinite plates at x = –a and x = +a each carry σ_p = −3 µC/m². A metal slab of total charge σ_s = +8 µC/m² is placed with its left face at x = 3a. What is the charge on the slab's right face?
A: σ_R = (σ_s / 2) + σ_p = (8/2) + (−3) = +1 µC/m². (Use superposition of all sources, set E = 0 inside the slab, and solve for the face charges.)
Q: A positive charge moves from point B (between the plates) to point A (also between the plates, closer to the negative plate on the left). Does its potential energy increase, decrease, or stay the same?
A: It depends on the field direction at that location. If the net field points from B toward A, the field does positive work and U decreases. If it points from A toward B, U increases. For this geometry with negative plates and a positive slab to the right, the field between the plates points to the right (toward the slab), so moving left (B to A) means moving against the field, and U increases.
Q: An uncharged metal sphere is placed near a positively charged plate. What happens?
A: The sphere is attracted to the plate. Negative charge is induced on the near side (closer to the plate, stronger attraction) and positive charge on the far side (weaker repulsion). The net force is attractive.
The work-energy theorem for charges connects directly to capacitor energy storage (U = ½CV²) and to the acceleration of particles through potential differences, which appears in the X-ray tube problem. Superposition of infinite-plane fields is the foundation of the parallel-plate capacitor model, which dominates the rest of the course.
Equipotential-line reasoning returns in circuit analysis: a wire is an equipotential, and voltage drops across components are potential differences.
Electric potential, voltage, potential difference, work done by electric field, equipotential lines, equipotential surfaces, field lines perpendicular to equipotentials, superposition of E fields, infinite charged plates, conducting slab, induced charge, uncharged conductor attraction, PHYS 212 Exam I, electrostatics, University of Illinois, electricity and magnetism