Motional EMF

Motional electromotive force (motional EMF) is the voltage induced when a conductor moves through a magnetic field. Charges in the conductor experience the magnetic part of the Lorentz force, separate along the conductor, and create an electric field that produces a measurable potential difference (voltage) between the ends. This is the same physical idea used inside many generators: mechanical motion is converted into electrical energy.

What this calculator computes

This calculator uses the standard motional-EMF magnitude formula for a straight conductor moving in a uniform magnetic field:

• ε = induced EMF (volts, V)
• B = magnetic flux density (tesla, T)
• L = effective conductor length cutting across the field (meters, m)
• v = conductor speed (meters per second, m/s)
• θ = angle between the velocity direction v and the magnetic field direction B (degrees in this calculator)

The sin(θ) term comes from the fact that only the component of velocity perpendicular to the magnetic field contributes. If the conductor moves parallel to the field lines, it does not “cut” field lines and the induced EMF magnitude goes to zero.

Where the formula comes from (brief derivation)

For a charge q moving with the conductor at velocity v through a magnetic field B, the magnetic force is:

F_mag = q(v × B)

In a straight rod, this force drives positive and negative charges toward opposite ends, creating an internal electric field E. At equilibrium, electric and magnetic forces balance in magnitude:

qE = qvB sin(θ) ⇒ E = vB sin(θ)

The potential difference between the ends of a rod of length L is then:

ε = EL = BLv sin(θ)

This is also consistent with Faraday’s law when the moving conductor forms part of a closed circuit (for example, a sliding rod on rails), because the motion changes the area of the loop and therefore changes magnetic flux through the loop.

How to use the calculator

1. Enter B in tesla (T).
2. Enter the effective conductor length L in meters (m).
3. Enter the speed v in meters per second (m/s).
4. Enter the angle θ (degrees) between the direction of motion and the magnetic field direction. Use 90° for perpendicular motion.
5. Click Compute to get the induced EMF in volts (V).

Interpreting the result

• The calculator returns the magnitude of motional EMF. In real circuits, the polarity/sign depends on direction (right-hand rule) and on how you define the terminals.
• If you are using this in a circuit with resistance R, a rough current estimate is I ≈ ε/R (ignoring internal resistances and back-EMF effects).
• Increasing any of B, L, or v increases EMF linearly; changing angle changes EMF through sin(θ).

Worked example

Problem: A 0.50 m rod moves at 3.0 m/s through a uniform 0.80 T magnetic field. The angle between v and B is 60°. Find the motional EMF magnitude.

• B = 0.80 T
• L = 0.50 m
• v = 3.0 m/s
• θ = 60° ⇒ sin(60°) ≈ 0.866

ε = (0.80)(0.50)(3.0)(0.866) ≈ 1.04 V

Interpretation: Under these conditions, the rod develops about 1.0 volt between its ends. If it completes a circuit, current will flow with a direction set by Lenz’s law and the geometry.

Quick reference: sin(θ) for common angles

Assumptions and limitations

• Uniform magnetic field: The formula assumes B is uniform over the region the conductor spans. Strong spatial variation can change the effective EMF.
• Straight conductor and simple geometry: L is the effective length that cuts across field lines. Curved conductors or complex motion may require integrating along the path.
• Rigid motion at a single speed: Acceleration, vibration, or changing speed can make the induced voltage time-dependent.
• Neglects edge effects and induced-field feedback: At higher currents, the induced current’s magnetic field and circuit dynamics (self-inductance) can alter conditions and reduce agreement with the simple model.
• Magnitude only: Direction/polarity is not computed. Use the right-hand rule (v × B) and Lenz’s law to determine sign for a chosen terminal convention.
• SI units expected: Inputs are assumed in tesla, meters, and m/s, with the output in volts.

References (for further reading)

• Halliday, Resnick, Walker, Fundamentals of Physics (sections on motional EMF and Faraday’s law)
• Purcell & Morin, Electricity and Magnetism (Lorentz force and electromagnetic induction)