Electromotive Force

From Fields to Circuits

Electromagnetic fields vary throughout space, but an ordinary circuit can often be described using only a current in each branch and a voltage across each element. This is the lumped-element approximation: the circuit is small compared with the relevant electromagnetic wavelength, changes are slow enough that propagation delays across it are negligible, and charge in the wires redistributes quickly. Under these low-frequency conditions, field laws become simpler circuit rules.

The reduction starts by replacing the current-density field with the total current through a wire's cross-section:

I=\int_A \mathbf{J}\cdot d\mathbf{a}

At a junction, this current balance is called Kirchhoff's Current Law (KCL), named after physicist Gustav Kirchhoff. If current enters faster than it leaves, charge piles up at the node:

\sum I_{\mathrm{in}}-\sum I_{\mathrm{out}}=\frac{dQ_{\mathrm{node}}}{dt}

In an ordinary steady circuit, the node does not store a growing amount of charge, so KCL becomes the familiar form $\sum I_{\mathrm{in}}=\sum I_{\mathrm{out}}$. The current can split between branches, but the total leaving the junction must match the total entering it.

This is the first circuit rule we need: a steady junction cannot keep storing a growing amount of charge. The next concept adds the material relation that tells us how a particular resistor responds to the current flowing through it.

Deep Dive: Why Current Cannot Pile Up

1. Ohm's Law & Resistance

To maintain a current flow, a "push" is required to overcome the internal resistance of a material. In most substances, the current density $\mathbf{J}$ is directly proportional to the force per unit charge $\mathbf{f}$.

Microscopic Ohm's Law: For most purposes, the driving force is the electric field $\mathbf{E}$, leading to: $\mathbf{J} = \sigma \mathbf{E}$ where $\sigma$ is the conductivity of the material. Its reciprocal, $\rho = 1/\sigma$, is the resistivity.
Macroscopic Ohm's Law: For a specific object (like a wire), the total current $I$ is proportional to the potential difference $V$: $$$ V = IR $$$ where $R$ (resistance) is a function of the material's conductivity and the geometry of the arrangement.
Joule Heating: As charges collide with the material, work done by the electrical force is converted into heat. The power delivered is: $$$ P = VI = I^2R $$$

Deep Dive: The Drude Model

2. Electromotive Force (EMF)

A battery or generator supplies energy to charges so that current can keep going around a complete circuit. The source may use chemistry, as in a battery, or mechanical motion, as in a generator. The quantity that measures this source action is called electromotive force, abbreviated EMF.

The symbol $\mathcal{E}$ is the script (calligraphic) capital E used for EMF; it is often called “epsilon” in this context. It means energy supplied by the source per unit charge around the loop: $1\,\mathrm{V}=1\,\mathrm{J/C}$. It is not the electric field $\mathbf{E}$ and it is not a mechanical force, despite the word “force” in its name.

To write the definition, let $\mathbf{E}$ be the electrostatic electric field and let $\mathbf{f}_s$ be the non-electrostatic source force per unit charge inside the battery or generator. The total force per unit charge is $\mathbf{f}=\mathbf{E}+\mathbf{f}_s$.

\mathcal{E}\equiv\oint \mathbf{f}\cdot d\mathbf{l} =\oint(\mathbf{E}+\mathbf{f}_s)\cdot d\mathbf{l} =\oint\mathbf{f}_s\cdot d\mathbf{l}

Why does the last equality hold? An electrostatic field is conservative, so its closed-loop integral is zero: $\oint\mathbf{E}\cdot d\mathbf{l}=0$. Therefore, the EMF is the closed-loop work per unit charge supplied by the non-electrostatic source force $\mathbf{f}_s$.

$\mathcal{E}$ vs $V$ $\mathcal{E}$ describes energy supplied by the source per charge around the loop. $V$ is a potential difference between two points, usually measured across a circuit element or source terminals.

$\mathbf{E}$ vs $\mathcal{E}$ Bold $\mathbf{E}$ is the electric-field vector. Script $\mathcal{E}$ is a scalar loop quantity: the EMF, measured in volts.

$\mathbf{f}$ vs $\mathbf{f}_s$ $\mathbf{f}$ is total force per charge; $\mathbf{f}_s$ is the chemically or mechanically generated source push per charge.

Open circuit: when $I=0$, an ideal source has terminal voltage $V=\mathcal{E}$. Inside a real source, $\mathbf{f}_s$ balances the internal electrostatic field.
Closed circuit: when current $I$ flows through internal resistance $r$, the terminal voltage is reduced to $V=\mathcal{E}-Ir$.

Deep Dive: Batteries & Sources

3. Motional Electromotive Force (Motional EMF)

Start with the physical setup. Two fixed, parallel conducting rails are joined on the left through a resistor $R$ and an ammeter. A vertical conducting bar of length $h$ touches both rails through sliding contacts and forms the right side of the circuit. An external agent moves the bar to the right with velocity $\mathbf{v}$ through a uniform, externally supplied, static magnetic field $\mathbf{B}$. The rails, resistor, and ammeter remain stationary.

When the circuit is closed, the path for conventional current is up through the moving bar, left along the upper rail, down through the ammeter and resistor, and right along the lower rail back to the bar. The field and motion determine the direction; the resistor determines how much current flows.

Established law: every charge in the moving bar obeys the Lorentz force law $\mathbf{F}_{mag}=q(\mathbf{v}\times\mathbf{B})$. This identifies the physical cause: the force pushes positive charges in one direction and negative charges in the other.

\mathbf{F}_{mag}=q(\mathbf{v}\times\mathbf{B})

Definition: motional EMF is the source push per unit charge produced because the conductor moves through the field. In this geometry, only the moving bar contributes a motional term:

\mathcal{E}=\oint (\mathbf{v}\times\mathbf{B})\cdot d\mathbf{l} =\int_0^h (vB)\,dl=vBh

Result for the sliding bar: the moving bar is a voltage source with $|\mathcal{E}|=Bh|v|$. This result comes before the current calculation. If the circuit is open, the bar has a voltage but $I=0$.

Then close the circuit: the resistor sets the current, $I=\mathcal{E}/R$, and the load receives $P=I^2R$. The energy comes from the external mechanical drive; the magnetic force redirects charges but does no work by itself.

Equivalent flux description: the shaded loop area is $A=hx$. With a signed field component $B_z$, $\Phi_B=B_zA=B_zhx$, so $$\mathcal{E}=-\frac{d\Phi_B}{dt}=-B_zhv.$$ The flux rule gives the same result after the moving boundary has been accounted for. Holding the bar still leaves flux in the loop, but $d\Phi_B/dt=0$, so there is no motional EMF.

One flux rule, two mechanisms: flux can change because the circuit moves, because the magnetic field changes, or because both change. This example isolates the motional mechanism.
Eddy currents: a solid piece of metal can contain circulating induced currents, producing magnetic braking as mechanical energy becomes heat.

Deep Dive: Motional Electromotive Force (Motional EMF)