Difficulty: Intermediate
Prerequisites: Coulomb's law, electric field concepts (direction and magnitude), basic calculus (derivatives and slopes).
Big picture: Electric potential is the scalar quantity that complements the electric field vector. Where the electric field tells you the force per unit charge, the potential tells you the energy per unit charge. The two are tightly linked: the electric field is the negative gradient of the potential. This topic sits at the bridge between electrostatics (fields from charges) and circuit analysis (voltage drops), so getting comfortable with V-to-E conversions here pays off for weeks to come. You should already be comfortable with electric field lines and superposition.
The electric field points in the direction of steepest decrease in electric potential. On a V vs x graph, the E-field magnitude equals the steepness of the slope, and the sign of E is opposite to the sign of the slope. If E is zero everywhere in a region, V must be constant there (though not necessarily zero). Equipotential lines are always perpendicular to field lines, and closer spacing of either indicates a stronger field.
Electric potential (V)
The electric potential energy per unit positive test charge at a point in space, measured in volts (J/C). Think of it as a "voltage landscape" that a charge sits in: high-potential regions are hilltops for positive charges, valleys for negative ones.
Electric field (E)
The force per unit positive test charge at a point, measured in N/C or equivalently V/m. In simple terms, this is the vector that tells a positive charge which way to accelerate and how hard.
Equipotential line (equipotential surface)
A curve (or surface in 3D) along which the electric potential is constant. No work is done moving a charge along an equipotential. Think of it as a contour line on a topographic map: walking along one means you stay at the same height.
Potential difference (ΔV)
The difference in electric potential between two points, V_B minus V_A. This is what drives charge flow and determines the work done on a charge moving between those points. In everyday terms, this is "voltage" as used in circuits.
Gradient of potential
The rate and direction of the steepest spatial change in V. The electric field equals the negative gradient of the potential. In one dimension this simplifies to E_x = −dV/dx.
Conservative force
A force for which the work done moving between two points is independent of the path taken. The electrostatic force is conservative, which is precisely why electric potential (a scalar) can be defined at all.
The electric field in one dimension is E_x = −dV/dx. This means:
The magnitude of E equals the magnitude of the slope of the V(x) curve.
The steeper the curve, the stronger the field at that point.
The sign of E is opposite to the sign of the slope.
Where V(x) is steeply decreasing (large negative slope), E points in the +x direction.
Where V(x) is increasing (positive slope), E points in the −x direction.
Where V(x) is flat (zero slope), the E-field is zero at that point.
At a local maximum or minimum of V, the slope is zero, so E = 0 there.
If the electric field is zero everywhere in a region, then dV/dx = 0 everywhere in that region.
This means V must be constant throughout, but it can be any constant value, not necessarily zero.
A common trap: students assume E = 0 implies V = 0. Potential has an arbitrary reference point. Zero field only means the potential does not change from place to place.
Electric field lines and equipotential lines are always perpendicular to each other.
Field lines point from high-V regions toward low-V regions.
Where field lines are closely spaced, the E-field is strong. Where they are far apart, the E-field is weak.
Similarly, where equipotential lines are closely spaced, the potential is changing rapidly, meaning the field is strong there.
To judge field strength from a field-line diagram, look at the density of lines in the neighbourhood of each point.
The work done by an external agent to move a charge q from point A to point B is: W_ext = q(V_B − V_A) = qΔV.
For a negative charge (q < 0), moving from low V to high V gives W_ext = qΔV, which will be negative (the field does positive work, so an external agent does less).
Work depends only on the potential difference between start and end points, not on the path. This is because the electric force is conservative.
Moving a charge along an equipotential line requires zero work, regardless of the distance travelled.
Comparing work for two different paths: count how many equipotential lines you cross (i.e. what is the ΔV). If both paths cross the same ΔV, the work is the same. If one crosses a larger ΔV, that path requires more work.
E_x = -\frac{dV}{dx}The electric field component along x equals the negative derivative of the potential with respect to x. In three dimensions this generalises to E = −∇V.
W_{\text{ext}} = q \, \Delta V = q \left( V_B - V_A \right)Work done by an external agent moving charge q from A to B. Positive W_ext means the agent pushes against the field; negative W_ext means the field does the pushing.
\Delta V = - \int_A^B \mathbf{E} \cdot d\mathbf{l}The potential difference between A and B is the negative line integral of the electric field along any path from A to B. This is the formal link between the field (vector) and the potential (scalar).
The relationship between potential and field is how engineers design the insulation in high-voltage power lines: they map the voltage contours around a conductor to find where the field is strongest (equipotentials closest together) and reinforce those spots. The same principle shows up in medical defibrillators, where electrodes are positioned to create a specific potential difference across the heart muscle, and the resulting E-field is what stimulates the tissue.
Students often assume that if the electric field is zero, the potential must also be zero. This is wrong. Zero field means constant potential, but that constant can be any value.
Students confuse the direction of E with the direction in which V increases. E points in the direction V decreases most rapidly, the opposite direction.
Students sometimes think that a charge moving perpendicular to field lines must be doing work. If the motion is along an equipotential (perpendicular to field lines), no work is done.
Students mix up the sign convention for work. W_ext = qΔV is the work done by an external agent. The work done by the electric field itself is −qΔV. Keeping track of which agent you mean is critical for getting signs right.
⚠️ Given a V(x) graph, expect to be asked which point has the largest E-field. Look for the steepest slope.
⚠️ Expect questions asking the direction of E at a point on a V(x) curve. Remember: positive slope means E in the −x direction; negative slope means E in the +x direction.
⚠️ "E = 0 in a region, what can you say about V?" appears frequently. The answer is V is constant, not zero.
⚠️ Questions comparing the work to move a charge along two different paths are testing whether you know that work depends only on ΔV, not on the path or on the field strength along the way.
⚠️ Diagram-based questions with field lines and equipotential lines test your ability to identify where the field is strongest (lines densest) and which direction it points.
True or false: If the electric potential is zero at a point, the electric field must also be zero there.
False. V = 0 says nothing about the slope of V. The field depends on how V changes, not its value.
True or false: Moving a charge along an equipotential line requires no work.
True. ΔV = 0 along an equipotential, so W = qΔV = 0.
Fill in the blank: The electric field points in the direction of ______ potential.
Decreasing.
True or false: If E = 0 everywhere in a region, V could be 5 V throughout that region.
True. E = 0 means V is constant. That constant can be any value.
Fill in the blank: On a V vs x graph, the E-field magnitude at a point equals the ______ of the curve at that point.
Magnitude of the slope (absolute value of dV/dx).
Q: You are given a V vs x graph showing a curve with points A, B, C, D. Point B has the steepest downward slope. At which point is the E-field magnitude greatest, and why?
A: Point B. The E-field magnitude is |dV/dx|, so the point with the steepest slope (largest |dV/dx|) has the strongest field. A steep downward slope gives a large positive E_x.
Q: On the same V vs x graph, point D sits where the curve is rising (positive slope). What is the direction of the E-field at D?
A: The E-field at D points in the −x direction. Since E_x = −dV/dx and dV/dx > 0, E_x is negative, meaning the field points toward −x.
Q: If the electric field is zero everywhere inside a hollow conducting shell, what can you say about the potential inside?
A: The potential is constant everywhere inside the shell. It equals the potential on the inner surface of the shell, but it is not necessarily zero.
Q: A field-line diagram shows solid field lines and dashed equipotential lines. At point C, the field lines are widely spaced, and at point A, they are closely packed. Where is the E-field weakest?
A: At point C. The density of field lines is proportional to the field strength. Widely spaced lines indicate a weak field.
Q: You move a negative charge from point A to point B, crossing two equipotential lines. You also move it from point C to point D, crossing two equipotential lines with the same spacing. Compare the work done in each case.
A: The work is the same for both paths. Work depends on ΔV (how many equipotential lines you cross and their spacing), not on the path taken or the local field strength along the way. Same ΔV means same work.
Q: A charge is moved from a region where V = 10 V to a region where V = 4 V. The charge is q = −2 µC. What is the work done by the external agent?
A: W_ext = qΔV = (−2 × 10⁻⁶ C)(4 − 10) V = (−2 × 10⁻⁶)(−6) = +12 × 10⁻⁶ J = +12 µJ. The external agent does positive work, pushing the negative charge from higher to lower potential.
This material connects directly to capacitance: a capacitor stores energy by maintaining a potential difference between two conductors, and the E-field between the plates is determined by ΔV/d. Understanding how V and E relate is also the foundation for Kirchhoff's voltage law in circuits, where the sum of potential differences around a closed loop is zero. The concept of conservative forces and path-independent work reappears in magnetic fields, where you will learn that the magnetic force does no work on moving charges, a sharp contrast with the electric case studied here.
Electric potential, voltage, potential difference, electric field, E-field, V vs x graph, equipotential lines, equipotential surfaces, field lines, gradient of potential, dV/dx, conservative force, work done by electric field, work-energy theorem electrostatics, PHYS 212, University Physics E&M, Coulomb potential, scalar potential, voltage landscape, potential energy per unit charge, line integral of E-field.