Motional Emf Calculator

Physics Electricity and Magnetism • Electromagnetic Induction and Inductance

Written by STEM Calculators Team Published January 18, 2026 Updated February 24, 2026

// topic/calculator.ejs — Motional EMF Calculator (updated: fixes fmt3(...) leaking into LaTeX + adds units on plot)

7. Motional EMF Calculator

Computes motional EMF for a straight rod moving in a magnetic field: $\varepsilon = B\,\ell\,v_\perp = B\,\ell\,v\sin\theta$ (and current $I=\varepsilon/R$ if the circuit is closed).

Units: tesla (T), meters (m), seconds (s), volts (V), ohms (Ω), amperes (A). Inputs accept 1e-3, pi, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Inputs

Magnetic field $B$ (T): Direction of $ \mathbf{B} $: Rod length $\ell$ (m): Speed $v$ (m/s): Rod motion direction: Angle $\theta$ between $ \mathbf{v} $ and $ \mathbf{B} $ (deg): Closed loop? Resistance $R$ (Ω):

In the rail-generator sketch, the rod slides on rails in a uniform $ \mathbf{B} $ (into/out of the page). Charge separation is set by $ \mathbf{F}=q(\mathbf{v}\times\mathbf{B}) $.

Ready

Steps

Enter values and click Solve.

Generator rail visual + $\varepsilon(t)$ plot

Rod motion (top) and EMF vs time (bottom)

Solve to render.

What you’re seeing

Uniform $ \mathbf{B} $ direction (⊙ out / ⊗ into page)
Rod velocity $ \mathbf{v} $ (motion on rails)
Sliding rod (length $\ell$)
Polarity from $ \mathbf{v}\times\mathbf{B} $ (which end becomes +)
Plot of $ \varepsilon(t) $ in volts (V) vs $t$ in seconds (s)

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Theory

Motional EMF (general form)

When a conductor moves through a magnetic field, charges inside experience the magnetic force $q\,\mathbf{v}\times\mathbf{B}$. This produces charge separation and an induced EMF.

\[ \varepsilon = \int (\mathbf{v}\times\mathbf{B})\cdot d\boldsymbol{\ell}. \]

Here $d\boldsymbol{\ell}$ is directed along the conductor path. The integrand is the component of $\mathbf{v}\times\mathbf{B}$ tangent to the conductor.

Straight rod in uniform $\mathbf{v}$ and $\mathbf{B}$

For a straight rod of length $l$ oriented along the unit direction $\hat{\boldsymbol{\ell}}$, we can write $\mathbf{L}=l\,\hat{\boldsymbol{\ell}}$. If $\mathbf{v}$ and $\mathbf{B}$ are uniform:

\[ \varepsilon = (\mathbf{v}\times\mathbf{B})\cdot \mathbf{L}. \]

The magnitude becomes

\[ |\varepsilon| = |\mathbf{v}\times\mathbf{B}|\,l\,|\cos\psi| = vB\sin\theta \, l\,|\cos\psi|, \]

where $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{B}$, and $\psi$ is the angle between $\mathbf{v}\times\mathbf{B}$ and the rod direction $\hat{\boldsymbol{\ell}}$.

Special case: if the rod is aligned with $\mathbf{v}\times\mathbf{B}$ and $\mathbf{v}\perp\mathbf{B}$, then $ |\varepsilon| = Blv $.

Polarity and right-hand rule

The direction of $\mathbf{v}\times\mathbf{B}$ tells you which way positive charges are pushed. The end they accumulate at becomes higher potential.

\[ \mathbf{E}_{\text{mot}}=\mathbf{v}\times\mathbf{B}. \]

Use the right-hand rule for $\mathbf{v}\times\mathbf{B}$: point index along $\mathbf{v}$, middle along $\mathbf{B}$, thumb gives $\mathbf{v}\times\mathbf{B}$.

Closed circuit current

If the rod is part of a closed circuit of total resistance $R$, the induced current magnitude is:

\[ I=\frac{\varepsilon}{R}. \]

The current direction is such that the magnetic effects of the induced current oppose the change (Lenz’s law).

Steps

Generator rail visual + \(\varepsilon(t)\) plot

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