Electric Potential (Voltage) Explained — V = W/q

Potential is energy per charge

Gravity has an everyday analogy. A rock on a high shelf has more gravitational potential energy than one on the floor — and the energy difference depends on the rock's mass. Divide by the mass and you get something that belongs to the location, not the rock: the gravitational potential.

Electric potential works the same way. A charge $q$ sitting in an electric field has electric potential energy $U$. Divide by the charge and you get a property of the location in the field:

$$V = \frac{U}{q}$$

Voltage at a point is the energy per coulomb — it tells you how much work it takes to push one unit of positive charge there, per coulomb. A battery's terminal voltage is the energy it hands to each coulomb it pushes around the circuit.

Potential difference is what matters

Like altitude, electric potential only has meaning relative to a reference. A mountain's height is measured from sea level; voltage is measured between two points. A voltmeter reads the potential difference ΔV between its two probes, not an absolute value.

This is why birds can sit safely on a high-voltage line: both feet are at nearly the same potential, so almost no current flows through the bird. Danger comes from touching two points at very different potentials. When engineers say a wire is at '0 V' (ground), they have simply chosen it as the reference level — the sea level of the circuit.

Voltage and field: two views of the same landscape

Electric potential $V$ and electric field $E$ describe the same situation in two languages. The field is the slope of the potential: it points downhill from high potential to low potential, and its strength is the steepness.

$$E = -\frac{\Delta V}{\Delta x}$$

The minus sign says the field points in the direction of decreasing potential. Positive charges are pushed downhill (toward lower potential); negative charges are pushed uphill. A strong field is a steep potential drop — that is why thousands of volts across a millimeter-long spark gap produce enormous fields.

Equipotentials: contour maps of voltage

An equipotential surface is a surface where the voltage is the same everywhere — the voltage version of a contour line on a topographic map. Two facts make them powerful:

• No work is needed to move a charge along an equipotential (no energy change, so no work).
• Field lines are always perpendicular to equipotentials, pointing from high to low potential.

Around a single positive point charge, equipotentials are spheres (circles in a flat diagram) and field lines radiate straight outward, crossing them at right angles. Between two parallel plates the equipotentials are flat planes parallel to the plates, evenly spaced because the field is uniform.

Worked example: energy from voltage

A 9 V battery moves $4\ \mathrm{C}$ of charge from its negative to its positive terminal. How much work does the battery do?

The work to move charge $q$ through a potential difference $\Delta V$ is:

$$W = q\,\Delta V = (4\ \mathrm{C})(9\ \mathrm{V}) = 36\ \mathrm{J}$$

The battery converts $36\ \mathrm{J}$ of chemical energy into electric potential energy. Every coulomb carries $9\ \mathrm{J}$ as it flows out, which it can dump into the load as heat, light, or motion — that is the bookkeeping behind $P = IV$.

Voltage and current: cause and flow

Potential difference is the push and current is the flow. In a resistor, the two are proportional: $\Delta V = IR$ (Ohm's law). A bigger voltage across the same resistor pushes more charge through per second.

But voltage and current are different quantities. A static spark can be at thousands of volts yet carry almost no current (tiny charge), while a car battery at only 12 V can deliver hundreds of amps for a moment. Voltage measures energy per charge; current measures charge per second. Multiplying them gives power: $P = IV$, energy per second.

Variables and units

Common mistakes

Real-world applications

• Battery and wall-socket voltage ratings tell you the energy per coulomb available to drive devices.

• High-voltage transmission lines reduce current for the same power, cutting resistive heating losses in the grid.

• Medical defibrillators charge to a known voltage so the delivered energy per charge is precisely controlled.