Potential Difference

Work per unit charge required to move a charge between two points of the electric field.

Conceptual context

Potential difference is one of the central quantities in electrostatics and, by extension, throughout electrical engineering. Its importance lies in the fact that it allows quantifying the energetic effect of the electric field without knowing the path a charge takes between two points. This path-independence follows directly from the conservative nature of the electric field.

Historically, the concept of potential difference arose as a generalization of mechanical work. Once it was established that the work done by the electric force moving a charge between two points was path-independent, it became possible to define a scalar function—the electric potential—whose difference between two points gave that work per unit charge directly.

This leaf focuses on the central physical question: how much work per unit charge is needed to move a charge between two points of the electric field? The answer makes potential difference the fundamental tool for analyzing circuits, capacitors, and the motion of charged particles.

🟢 Essential level

Imagine carrying a ball uphill: you need to do work. If you release it, gravity carries it downhill without any push. The electric field acts analogously with charges: between two points there is always an "energy slope" that facilitates motion in one direction and opposes it in the other.

The potential difference $V_{AB}$ between two points A and B measures exactly that "slope" per unit charge: how much work the field does when moving a positive charge from A to B. If $V_{AB}$ is positive, the field drives the charge from A to B (A is "uphill" in the energy landscape). If $V_{AB}$ is negative, the field opposes that motion.

An immediate consequence: if you move a charge along a surface where the potential is constant—an equipotential surface—the field does no work. Electric field lines are always perpendicular to equipotential surfaces.

[!NOTE] The voltage displayed on your phone charger (5 V, 12 V, 20 V) is exactly a potential difference. It indicates how much energy per coulomb of charge that charger can deliver to the device's circuit.

🔵 Formal level

Basic formulas

The potential difference between A and B is defined as the work done by the electric field per unit positive charge:

Potential difference from work

where $V_{AB}$ is the potential difference in volts, $W_{AB}$ is the work done by the electric force in joules, and $q$ is the electric charge in coulombs.

The volt is the SI unit for potential difference, equivalent to one joule per coulomb (J·C⁻¹). This definition shows that $V_{AB}$ does not depend on $q$: it is a property of the field at those two points, not of the moving charge.

Real student questions

Why does the work of the electric field not depend on the path?

Because the electric field in electrostatics derives from a scalar potential: it is the negative gradient of the potential. Mathematically, the line integral of a gradient between two points depends only on the values of the function at those points, not on the path. This property is exactly the definition of a conservative field.

Is potential difference the same as voltage?

Yes. "Voltage" and "potential difference" are equivalent terms in practice. "Voltage" is used more in electrical engineering and circuits; "potential difference" in fundamental physics. Both are measured in volts (V).

Why does the electric field point from higher to lower potential?

Because the field is the negative gradient of the potential. The gradient points in the direction of maximum increase of the function; the field, being its negative, points in the direction of maximum decrease. A free positive charge moves in the field direction and hence towards lower potentials, gaining kinetic energy and losing potential energy.

Can there be a potential difference without a charge between the two points?

Yes. Potential difference is a property of the field at those two points, not of any charges that may be present between them. The field—and hence the potential—exists even without test charges.

Final synthesis

Potential difference $V_{AB}$ quantifies the work of the electric field per unit charge when moving a charge from A to B, independently of the path because of the conservative nature of the field. In a uniform field, $V_{AB}$ is proportional to $E$ and $d_{AB}$, making the geometry of the problem the key factor. The work on any specific charge is obtained by multiplying $V_{AB}$ by $q$, connecting the field property to the measurable dynamical effect.