The potential difference in a circuit is what causes current to flow through the circuit. The larger the potential difference, the faster the current will flow and the higher the current. The potential difference is the measure of the difference in voltage between two distinct points in a closed circuit. Potential difference is also known as voltage, electric potential difference, or electromotive force (usually with an implied difference or gradient). This measure also is the energy per unit charge that is required to move a charged particle from one point to another. In this way we represent electric potential energy in terms of joules (J) and electric charge in terms of coulombs (C) giving the following relationship:

We measure the potential difference in volts (which has units of joules per coulomb). The voltage in an electrical circuit controls the current in combination with the resistance. The voltage can be looked at as exerting an electric force on a positive charge (or a negative charge in the opposite direction) over a certain distance.

We can examine this further by unpacking energy (in joules) as a force (newtons) over a distance (meters). In this way, the voltage is a certain amount of energy proportional to the charge, and the energy can be viewed as a certain amount of force exerted over a distance (e.g. from point a to b).

Typically electric potential difference will be presented in the context of an electrical circuit. In the simplest circuit, there will likely be a battery connected to a resistor with wire. The battery will have some electrical energy between the negative terminal and the positive terminal. This electric potential difference then essentially exerts a force on the charges in the wire causing a movement of charge (called current). The resistor simply resists this movement of charges and regulates the current.

These three measurements and properties of a circuit are related through Ohm’s Law:

where ‌I‌ is the current flowing through the circuit, ‌V‌ is the electric potential difference across the battery, and ‌R‌ is the resistance of the resistor(s) in the circuit.

Current can seem like a somewhat arbitrary measurement just meant to facilitate a relationship between voltage and resistance, but it actually describes a very real effect. Current is in units of amperes which is equivalent to coulombs per second (essentially amount of charge passing by a point per second). Amperes are actually a standard SI unit, so it is a bit circular to define them with coulombs, but it provides the best conceptual explanation of current.

Conventional current describes the theoretical movement of positive charges throughout a circuit, but positively charged protons cannot easily move across a wire, so the actual movement of charges takes place with electrons. The electron current flows the opposite direction of conventional current, but both measurements describe the same movement of energy and relationship with electric potential energy.

When we look at electric potential (or voltage) independent of moving charges, we can still say something meaningful about the electric potential of stationary charges.

An electric charge produces an electric field through the relationship:

where the total charge, ‌Q,‌ is in Coulombs, ‌k‌ is the Coulomb constant, and the distance from the charge, ‌r,‌ is in meters. This formula tells you the strength of the electric field a certain distance away from the charge. When we look at this field strength, we can also draw a relationship to the electrical energy of this E field which can tell us about the electric potential.

We can generalize the electric potential energy of a charge by using an arbitrary charge as our measurement and then removing it later. Through this process we can ultimately find that the electric potential due to a point charge is:

This is very similar to the relationship for the electric field, but the electric potential can be seen to only decrease proportionally to the distance, while the electric field decreases proportionally the distance squared.

Today, electronics and circuits are built into nearly every aspect of life. Our electrical outlets, car batteries, phones, and lights all rely on voltage to drive a current and transfer electric power. Circuits in the real world will use even more components like capacitors (two metal plates with a voltage between them) and inductors to create desired effects.

The electric field produced by moving charges with an electric potential difference is also closely related to the electromagnetic spectrum and things like radio and satellite because moving electric charges produce a magnetic field.