Deep Dive: Motional Electromotive Force (Motional EMF)

Deriving the flux rule from the Lorentz force

In the main topic, we stated that the motional electromotive force (motional EMF) is $\mathcal{E}=vBh$ for a bar moving through a magnetic field. Here, we show why this is true, starting only from the Lorentz force law. The symbol $\mathcal{E}$ is the script capital E used for EMF: energy supplied by the source per unit charge, measured in volts. It is distinct from the electric-field vector $\mathbf{E}$.

A Sliding-Bar Generator: The Setup

The gold vertical conductor is the moving bar. It slides along two fixed, parallel silver rails while maintaining electrical contact with both. The rails are joined on the left through a fixed resistor $R$ and an ammeter, which complete the rectangular circuit. The crosses represent a uniform magnetic field into the screen, supplied by an external magnet (not drawn); dots represent a field out of the screen when reversed. The shaded region is the surface enclosed by the circuit, not a sheet of material. Its area is $A=hx$, so moving the bar changes the area even though $B$ stays constant.

1. The Lorentz Force on Charges

We start with the fundamental Lorentz force law. When a charge $q$ moves with velocity $\mathbf{v}$ in a magnetic field $\mathbf{B}$, it experiences a force:

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

Now, consider a conducting bar of length $h$ moving with velocity $\mathbf{v}$ perpendicular to both the bar and a uniform field $\mathbf{B}$. For rightward motion and a field into the screen, $\mathbf{v}\times\mathbf{B}$ points upward. In an isolated bar, charges separate until an electric field balances the magnetic force. With the rails and resistor connected, the bar instead drives a continuous current: up through the bar, left along the upper rail, through the load, and back along the lower rail. The top of the ideal bar is positive relative to its bottom.

2. Motional Electromotive Force (Motional EMF) as Work per Unit Charge

For a moving circuit in a static magnetic field, the motional electromotive force (motional EMF) is the loop integral of $\mathbf{v}\times\mathbf{B}$, where $\mathbf{v}$ is the local velocity of the conductor:

\mathcal{E} = \oint (\mathbf{v} \times \mathbf{B}) \cdot d\mathbf{l}

Only the bar moves; the stationary rails and resistor contribute no motional term. Integrating upward along the bar for rightward motion in a field into the screen gives:

\mathcal{E} = \int_0^h (vB) \, dl = vBh

The energy comes from the external mechanical drive, not from the magnetic field. Magnetic forces do no work on charges; the drive supplies the work needed to move the bar against magnetic drag. With negligible rail and bar resistance, the terminal voltage equals the generated EMF, and the resistor dissipates $P=\mathcal{E}I=I^2R$.

3. Connecting to the Flux Rule

Choose counterclockwise circulation as positive, so the positive surface normal points out of the screen. Write the signed field component as $B_z$; for the default inward field, $B_z=-B$. As the bar moves by $dx$, the area changes by $dA=h\,dx$, and:

\Phi_B=B_z hx, \qquad d\Phi_B=B_z h\,dx

The rate of change of magnetic flux is:

\frac{d\Phi_B}{dt}=B_z hv

The signed EMF is therefore:

$$ \mathcal{E}=-\frac{d\Phi_B}{dt}=-B_z hv $$

This is Faraday's Law. The minus sign (Lenz's Law) indicates the induced EMF opposes the change in magnetic flux.

The surface here is the surface bounded by the moving circuit. Even when $\mathbf{B}$ is constant, the loop sweeps out a changing area, so $\Phi_B = \int_{S(t)} \mathbf{B}\cdot d\mathbf{a}$ changes. Changing only the mathematical surface while keeping the same fixed boundary does not create an EMF; the physical change is the motion of the conductor and its boundary.

In a circuit, this motional EMF appears in a moving conductor such as the sliding bar in the visualization. An ordinary stationary inductor has a different source: its changing current changes the linked magnetic flux and produces the self-induced EMF $\mathcal{E}_L=-L\,dI/dt$. If the inductor itself moves through an external magnetic field, it can also have a motional contribution.

4. Motion, Voltage, and Current

Held still: the loop still encloses magnetic flux, but $v=0$, so $\mathcal{E}=0$ and $I=0$. Flux alone does not produce a current.
Moving right: the shaded area increases. For the default inward field, the current is counterclockwise and creates an outward field opposing the increase in inward flux.
Moving left: the area decreases and both the voltage polarity and current reverse. The yellow arrows indicate conventional current, opposite to electron drift.
Larger resistance: the same motion generates the same EMF, but less current flows. The readouts show magnitudes; the signs on the bar and arrows give direction.

In this demonstration an external drive moves the bar sinusoidally. It momentarily stops at each end of its travel, where the EMF is zero, and moves fastest at the midpoint, where $|\mathcal{E}|$ peaks. A 90-degree phase relationship applies to this sinusoidal variation; the general rule for any motion is $\mathcal{E}=-d\Phi_B/dt$.

The model uses ideal sliding contacts and an ideal ammeter; all resistance is in the load. Self-inductance, contact resistance, and switching transients are neglected. Adjusting or reversing $B$ selects a new constant field; it does not simulate the transient EMF caused by changing the field.