Faraday's Induced Emf Calculator

Physics Electricity and Magnetism • Electromagnetic Induction and Inductance

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

1. Faraday’s Induced EMF Calculator

Computes induced emf using Faraday’s law $\varepsilon = -N\,\dfrac{d\Phi_B}{dt}$ , where (uniform field model) $\Phi_B = B\,A\cos\theta$ . You can compute an average emf over an interval, or enter time-dependent expressions and plot $\Phi_B(t)$ and $\varepsilon(t)$ .

Units: tesla (T), square meters (m²), seconds (s), webers (Wb), volts (V). Inputs accept 1e-3, pi, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). In function mode, you may use the variable t (seconds). Use * for multiplication.

Inputs

Preset: Mode: Turns $N$:

Endpoints (linear change over the interval)

Start time $t_0$ (s):

Interval start.

End time $t_1$ (s):

Interval end (must be $t_1>t_0$).

$B_0$ (T):

Start magnetic field magnitude.

$B_1$ (T):

End magnetic field magnitude.

$A_0$ (m²):

Start loop area.

$A_1$ (m²):

End loop area (set equal to keep area constant).

$\theta_0$:

Start angle between $\mathbf{B}$ and area normal.

$\theta_1$:

End angle (set equal to keep angle constant).

Angle unit:

Used for $\theta_0,\theta_1$.

Functions (use t in seconds)

Start time $t_0$ (s):

Interval start.

End time $t_1$ (s):

Interval end (must be $t_1>t_0$).

$B(t)$ (T):

Example: 0.5*(1 - t/0.2)

$A(t)$ (m²):

Example: 0.1 (constant area)

$\theta(t)$:

Example: 0 or 2*pi*60*t

Angle unit:

Used for $\theta(t)$.

Samples:

220 points (more = sharper spikes, slower).

Graph + animation

Playback speed:

Controls cursor speed only.

Show grid:

Axes + grid in the plot.

Zoom (time):

Mouse wheel also zooms horizontally.

Drag the graph to pan in time • mouse wheel/trackpad to zoom time • “Reset view” restores default.
Play moves a cursor along the computed curves and shows instantaneous values.

Ready

Steps

Enter values and click Solve.

Flux + emf plot

Top: $ \Phi_B(t) $ (Wb) • Bottom: $ \varepsilon(t)=-N\,d\Phi_B/dt $ (V)

What you’re seeing

Magnetic flux $ \Phi_B(t) $ (Wb)
Induced emf $ \varepsilon(t) $ (V)
Cursor (instantaneous readout while playing)

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Faraday’s Induced EMF Calculator — Theory

Magnetic flux

Magnetic flux through a loop is the surface integral $ \Phi_B = \int \mathbf{B}\cdot d\mathbf{A} $. For a uniform field over a flat loop, a common model is \[ \Phi_B = B\,A\cos\theta, \] where $A$ is the loop area and $\theta$ is the angle between $\mathbf{B}$ and the loop’s area normal.

Units: $ \Phi_B $ is measured in webers (Wb), where $1\ \mathrm{Wb} = 1\ \mathrm{T\cdot m^2}$.

Faraday’s law and Lenz’s law (the minus sign)

Faraday’s law states that a changing magnetic flux induces an emf: \[ \varepsilon = -N\,\frac{d\Phi_B}{dt}, \] where $N$ is the number of turns in a coil.

The negative sign is Lenz’s law: the induced effect (current/emf direction) opposes the change in flux. In practical problems, the sign depends on the chosen orientation for the loop’s normal.

Average emf over an interval

If you only know the flux at two times $t_0$ and $t_1$, an average emf is \[ \varepsilon_{\text{avg}} = -N\,\frac{\Delta \Phi_B}{\Delta t} = -N\,\frac{\Phi_B(t_1)-\Phi_B(t_0)}{t_1-t_0}. \] This is especially useful when $B$, $A$, or $\theta$ change approximately uniformly.

What can change?

Changing $B(t)$: e.g., switching a magnetic field on/off (transformer-like situations).
Changing $A(t)$: e.g., a sliding bar changing loop area (motional induction setups).
Changing $\theta(t)$: e.g., a rotating loop (generator), where $\theta=\omega t$.

In a rotating-loop generator with constant $B$ and $A$, \[ \Phi_B(t)=B A\cos(\omega t),\qquad \varepsilon(t)=N B A \omega \sin(\omega t), \] so the emf oscillates sinusoidally and peaks when the flux is changing fastest.

Worked example (from the calculator prompt)

A loop has area $A=0.1\ \mathrm{m^2}$. The magnetic field changes from $B_0=0.5\ \mathrm{T}$ to $B_1=0\ \mathrm{T}$ in $\Delta t=0.2\ \mathrm{s}$. Assume $\theta=0^\circ$ (so $\cos\theta=1$) and $N=1$.

Flux change: \[ \Phi_B(t_0)=B_0A=0.5\times 0.1=0.05\ \mathrm{Wb},\quad \Phi_B(t_1)=0, \] \[ \Delta\Phi_B=\Phi_B(t_1)-\Phi_B(t_0)=0-0.05=-0.05\ \mathrm{Wb}. \] Average emf: \[ \varepsilon_{\text{avg}}=-\frac{\Delta\Phi_B}{\Delta t} =-\frac{-0.05}{0.2}=0.25\ \mathrm{V}. \]

Interpreting the plot

The calculator plots $\Phi_B(t)$ and $\varepsilon(t)$. Whenever $\Phi_B(t)$ changes rapidly, the slope $d\Phi_B/dt$ becomes large in magnitude, and the emf curve shows sharp peaks (“spikes”).

Steps

Flux + emf plot

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