A conductor moving through a magnetic field can develop an electromotive force even when the magnetic field itself is not changing in time. This is motional EMF.

The mechanism is the magnetic part of the Lorentz force. A charge $q$ moving with velocity $\mathbf v$ through a magnetic field $\mathbf B$ experiences

$$\mathbf F_B=q\mathbf v\times\mathbf B.$$

In a conducting rod, mobile charges are pushed toward opposite ends. Charge separation continues until an electric field builds up strongly enough to balance the magnetic force.

Sliding rod in a uniform magnetic field

Consider a straight conducting rod of length $L$ moving with speed $v$ perpendicular both to its length and to a uniform magnetic field $B$.

For a positive charge in the rod, the magnetic-force magnitude is

$$F_B=qvB.$$

At electrostatic balance inside the moving rod,

$$qE=qvB,$$

so

$$E=vB.$$

The potential difference between the rod's ends is therefore

$$\boxed{\mathcal E=BLv}.$$

The sign and which end is at higher potential follow from the direction of $\mathbf v\times\mathbf B$.

Example

A rod of length

$$L=0.40,\mathrm m$$

moves at

$$v=3.0,\mathrm{m/s}$$

through a uniform field

$$B=0.50,\mathrm T,$$

with all three directions mutually perpendicular. Then

$$\mathcal E=(0.50)(0.40)(3.0)=0.60,\mathrm V.$$

So the moving rod develops an EMF of magnitude

$$\boxed{0.60,\mathrm V}.$$

A moving rod that closes a circuit

Suppose the rod slides on conducting rails and completes a closed loop of total resistance $R$. The induced current is

$$I=\frac{\mathcal E}{R}=\frac{BLv}{R}.$$

That current experiences a magnetic force opposing the motion. Its magnitude on the rod is

$$F=ILB =\frac{B^2L^2v}{R}.$$

Mechanical work is therefore required to keep the rod moving at constant speed.

The mechanical power supplied is

$$P_{\rm mech}=Fv =\frac{B^2L^2v^2}{R}.$$

The resistor dissipates

$$P_{ m elec}=I^2R =\left(\frac{BLv}{R}\right)^2R =\frac{B^2L^2v^2}{R}.$$

Thus

$$\boxed{P_{\rm mech}=P_{\rm elec}},$$

showing explicitly how mechanical energy is converted into electrical and thermal energy.

Connection with magnetic flux

As the rod moves, the loop area changes. If the enclosed area is $A=Lx$, then the magnetic flux is

$$\Phi_B=BA=BLx.$$

Therefore

$$\left|\frac{d\Phi_B}{dt}\right| =BL\frac{dx}{dt} =BLv,$$

which is exactly the motional EMF found from the Lorentz force.

This agreement is important: the flux rule and the magnetic force on charges are two consistent ways to analyze this moving-circuit case.

For a general moving conducting path, the magnetic contribution to the EMF can be written

$$\mathcal E_{\rm motional}=\int (\mathbf v\times\mathbf B)\cdot d\boldsymbol\ell.$$

Motional EMF is the operating principle behind many electrical generators: mechanical motion changes how conductors move through magnetic fields, producing electrical energy.