Electromotive force

Electromotive force (EMF) is the work per unit charge done by non-electrostatic forces around a closed loop, measured in volts:

$V_{\text{emf}}={\oint }_C\frac{{\mathbf{F}}_{\text{ns}}}{q}\cdot d\mathbf{l}\text{(V)},$

where ${\mathbf{F}}_{\text{ns}}/q$ is the per-charge non-electrostatic force pushing carriers around the circuit. Despite the name, EMF is not a force — it’s an energy-per-charge, with the same units as voltage. The “force” in the name is historical.

EMF is what keeps current flowing in a circuit against resistive losses. A purely electrostatic field is conservative ($\oint {\mathbf{E}}_s\cdot d\mathbf{l}=0$); it can’t drive a steady current around a loop on its own. Something else has to do positive work on the charges along part of the loop to compensate the dissipative loss along the rest. That “something else” is what we call EMF.

Sources of EMF

Several distinct physical mechanisms can produce EMF:

Chemical (battery): chemical reactions at the electrodes push ions across the cell interior, doing work against the electrostatic field that builds up. The cell voltage is the EMF.

Magnetic induction (generator, transformer): a changing Magnetic flux through a loop induces an EMF via Faraday’s law $V_{\text{emf}}=-d\Phi /dt$.

Thermoelectric (thermocouple): a temperature gradient across a junction of dissimilar metals produces a small EMF (Seebeck effect).

Photoelectric (solar cell): photons excite carriers across a junction, producing an EMF that drives current.

Mechanical (Van de Graaff generator): a moving belt physically transports charge against the field.

In each case, the physics of the non-electrostatic force is different, but the function in the circuit is the same: pump charges from low potential to high potential, doing positive work that the rest of the circuit dissipates.

EMF in Faraday’s law

For an induced EMF specifically, the rigorous statement is

$V_{\text{emf}}={\oint }_C(\mathbf{E}+\mathbf{u}\times \mathbf{B})\cdot d\mathbf{l},$

with $\mathbf{u}$ the velocity of the line element $d\mathbf{l}$ relative to the lab frame. The two pieces correspond to the two ways flux can change (Faraday’s law):

• Transformer EMF: $\oint \mathbf{E}\cdot d\mathbf{l}$ for a stationary loop in a time-varying $\mathbf{B}$.
• Motional EMF: $\oint (\mathbf{u}\times \mathbf{B})\cdot d\mathbf{l}$ for a moving loop in a static $\mathbf{B}$.

Both reduce to $V_{\text{emf}}=-d\Phi /dt$.

EMF versus voltage

Voltage (potential difference) is path-independent: $V_{ab}=V_a-V_b$ depends only on endpoints. EMF is path-dependent — it’s a line integral around a closed loop. Around a static-field loop, EMF is zero; around a loop with a battery or with changing flux, it’s nonzero.

In a circuit with a battery of EMF $\mathcal{E}$ and a resistor $R$ (terminals shorted across the battery), Kirchhoff’s loop equation:

$\mathcal{E}-IR=0⟹I=\mathcal{E}/R.$

The EMF appears with a positive sign — it’s the source of energy that the resistor dissipates. If you draw the battery with internal resistance $r$, the terminal voltage is $V=\mathcal{E}-Ir$, less than the EMF when current flows. Open-circuit, $I=0$ and $V=\mathcal{E}$.

Worked example: rotating bar in a magnetic field

A rigid bar of length $L$ rotates with angular velocity $\omega$ in a uniform $\mathbf{B}$ field parallel to the rotation axis. Each point at distance $r$ from the axis moves at $\mathbf{u}=\omega r\hat{\mathbf{\varphi }}$.

The motional EMF along the bar:

$V_{\text{emf}}={\int }_0^L(\mathbf{u}\times \mathbf{B})\cdot d\mathbf{l}={\int }_0^L\omega rBdr=\frac{1}{2}\omega B{L}^2.$

This is the principle of the homopolar generator (Faraday’s disc): a rotating conducting disc in an axial field produces a steady DC EMF between center and rim — no flux change in the usual sense, but motional EMF is real and measurable.

In a dynamical $\mathbf{E}$ field

In electrostatics, $\mathbf{E}$ is conservative, so EMF around any closed loop is zero — you can’t power a circuit with electrostatic fields alone. In dynamic problems, Faraday’s law $\nabla \times \mathbf{E}=-{\partial }_t\mathbf{B}$ produces an $\mathbf{E}$ that has a rotational component, not derivable from a scalar potential alone. The closed-loop integral of this rotational $\mathbf{E}$ is precisely the transformer EMF.

This is the deep reason batteries and generators do work: they break the conservative character of the electric field locally (inside the battery) or globally (around the generator coil), creating a net “uphill push” around the circuit.