Difficulty: Intermediate | Prerequisites: Electric field concepts, Coulomb's law, work-energy theorem from mechanics
Big picture: Electric potential is the energy landscape of the electric field. Where the field tells you the force on a charge, the potential tells you the energy. This topic introduces the idea of assembling charge configurations (how much work it takes to bring charges from infinity to their positions), the relationship between field and potential (E = −dV/dx), and how to read potential plots. These ideas feed directly into capacitance and circuits, which dominate the second half of the course.
The electric potential at a point is the work per unit charge required to bring a test charge from infinity to that point. The work to assemble a configuration of point charges is the sum of kqᵢqⱼ/rᵢⱼ over all unique pairs. The electric field points from high potential to low potential, and a positive charge feels a force in the direction of decreasing potential.
Electric potential (V)
The potential energy per unit charge at a point in space, measured in volts (V = J/C). V = kq/r for a single point charge at distance r. In simple terms, potential is how much "electrical height" a point has; charges "roll downhill" from high potential to low, the same way objects fall from a high shelf to the floor.
Potential energy of a pair of charges
U = kq₁q₂/r₁₂. This is positive for like charges (you must do work to push them together) and negative for opposite charges (they attract, so you must do work to pull them apart).
Work to assemble a charge configuration
The total potential energy stored in a configuration of charges, equal to the sum of kqᵢqⱼ/rᵢⱼ over every unique pair (i, j) with i < j. This is the work an external agent must do to bring the charges from infinity to their final positions, one at a time.
Equipotential surface
A surface on which the potential is the same everywhere. The electric field is always perpendicular to equipotential surfaces. No work is done moving a charge along an equipotential.
Relationship between E and V
E = −dV/dx (in one dimension). The field points in the direction of steepest decrease in potential. In simple terms, if you plot V along a line, the field at any point is the negative slope of that plot.
Four charges are placed at the following positions: q at (0, 0), q at (0, a), q at (a, 0), and 2q at (a, a).
You bring the charges in one at a time from infinity.
The first charge costs zero work (nothing else is present).
Each subsequent charge interacts with every charge already in place.
The total work is the sum over all six unique pairs:
(0,0)↔(0,a): charges q and q, distance a → kq²/a
(0,0)↔(a,0): charges q and q, distance a → kq²/a
(0,0)↔(a,a): charges q and 2q, distance a√2 → 2kq²/(a√2) = √2 kq²/a
(0,a)↔(a,0): charges q and q, distance a√2 → kq²/(a√2) = kq²/(√2 a)
(0,a)↔(a,a): charges q and 2q, distance a → 2kq²/a
(a,0)↔(a,a): charges q and 2q, distance a → 2kq²/a
Total: W = (kq²/a)[1 + 1 + √2 + 1/√2 + 2 + 2] = (kq²/a)[6 + √2 + 1/√2]
Simplify: √2 + 1/√2 = √2 + √2/2 = 3√2/2 = 3/√2.
So W = (q²/4πε₀a)(6 + 3/√2).
Along a horizontal dotted line passing through the four charges described above, the potential varies.
Near each positive charge, the potential spikes upward (V → +∞ as you approach a positive point charge).
Between and far from the charges, the potential is lower but positive (all charges are positive, so V is positive everywhere on the line).
Point M at (a/2, a/2) is equidistant from some of the charges. The potential there is the sum of kqᵢ/rᵢ from each charge.
The correct sketch shows positive peaks near each charge location along the line, with the potential dipping between them but remaining positive throughout (since all charges are positive and 2q contributes a larger peak).
The electric field at M is E = −∇V, which points in the direction of steepest potential decrease.
At M = (a/2, a/2), by the arrangement of charges (three charges of q at three corners and 2q at the fourth corner), the charge configuration is not symmetric about M.
The 2q charge at (a, a) is the dominant source. It pulls the potential higher on the upper-right side of M.
The field at M therefore has a component pointing away from (a, a), toward lower potential, which is generally toward the lower-left.
A positive test charge at M experiences a force in the direction of the field, so the force points "down and left" (toward lower potential, away from the 2q charge).
Potential from a point charge:
V = kq / r
Potential energy of a charge pair:
U = kq₁q₂ / r₁₂
Work to assemble N charges:
W = Σ (over all pairs i < j) kqᵢqⱼ / rᵢⱼ
Field-potential relationship (one dimension):
E = −dV/dx
Force on a charge in an electric field:
F = qE
A positive charge is pushed from high V to low V.
The work to assemble charges is the energy stored in any static charge configuration, from the binding energy of ionic crystals (like table salt) to the energy stored in a capacitor. Understanding this energy is how engineers calculate how much a capacitor can deliver in a defibrillator pulse or a camera flash.
The field-potential relationship is the foundation for voltage in circuits. A battery maintains a potential difference, and current flows from high potential to low potential through the circuit, doing work on resistors and other components along the way.
Students often confuse potential (V, a scalar, in volts) with potential energy (U, in joules). Potential is per unit charge; potential energy is the actual energy for a specific charge: U = qV.
When computing the assembly work, students sometimes count pairs twice or miss a pair. For N charges there are N(N−1)/2 unique pairs. With four charges, that is six pairs.
Students sometimes think the force on a charge at a point is proportional to the potential at that point. The force depends on the gradient of the potential (how fast V changes), not on V itself. A region of high but constant potential has zero field and zero force.
A common error in reading potential plots: students assume the potential must cross zero between two positive charges. It does not. Two positive charges produce positive potential everywhere; V dips between them but never reaches zero at a finite distance.
⚠️ The "work to assemble" question is algebra-heavy. List all pairs systematically (a table helps) and compute each distance carefully. Diagonal distances across a square of side a are a√2.
⚠️ Sketch questions about potential along a line are common. Remember: potential diverges (goes to ±∞) at point charges, and between same-sign charges it dips but does not cross zero.
⚠️ The force direction at a point is determined by E = −∇V, not by V itself. On a potential sketch, the force on a positive charge points in the direction where V is decreasing most steeply (downhill on the plot).
⚠️ If all charges are positive, V is positive everywhere. The only way to get V = 0 at a finite point is to have a mix of positive and negative charges.
True or false: The electric potential due to a single positive point charge is positive everywhere and decreases with distance.
Fill in the blank: The number of unique pairs among four charges is ________.
True or false: A positive charge placed at a point of high potential, where the potential is locally constant (flat), experiences zero net force.
Fill in the blank: The electric field points in the direction of ________ potential.
True or false: The work required to assemble a configuration of all positive charges is positive.
Q: Four charges q, q, q, and 2q are at the corners of a square of side a. What is the work required to assemble them from infinity?
A: W = (q² / 4πε₀a)(6 + 3/√2). You compute this by summing kqᵢqⱼ/rᵢⱼ over all six pairs, using distance a for adjacent corners and a√2 for diagonal corners.
Q: Along a line passing through or near several positive charges, can the electric potential ever be negative?
A: No. The potential from each positive charge is positive (V = kq/r > 0 for q > 0), and by superposition, the sum is always positive when all charges are positive.
Q: A positively charged particle is placed at point M in the region of four positive charges (three q and one 2q). In what direction does the force on it point?
A: Down and to the left, away from the 2q charge. The force on a positive charge points in the direction of decreasing potential, which is away from the region dominated by the strongest nearby charge.
Q: Why does the potential diverge (go to infinity) as you approach a point charge?
A: Because V = kq/r, and as r → 0, V → ∞ for a positive charge (or −∞ for a negative charge). In reality, point charges are an idealisation; real charge distributions have finite size.
Electric potential connects directly to capacitance: C = Q/V. The potential difference across a capacitor's plates determines how much charge it stores at a given voltage, which is the subject of the next topic in this midterm. The work-energy ideas here also connect to the energy stored in a capacitor, U = ½CV², and to the concept of voltage in circuits (Kirchhoff's voltage law says the sum of potential drops around a loop is zero).
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