Induced Electric Field Tool

Physics Electricity and Magnetism • Electromagnetic Induction and Inductance

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

8. Induced Electric Field Tool

Computes the non-conservative induced electric field from Faraday’s law: $\displaystyle \oint \mathbf{E}\cdot d\boldsymbol{\ell} = -\frac{d\Phi_B}{dt}$ . For symmetric cases, $ \mathbf{E}(r)=E_\phi(r)\,\hat{\boldsymbol{\phi}} $ (tangent circles).

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

Inputs

Geometry:

Changing magnetic field

Direction of $\mathbf{B}$:

Sets the sign convention for $B_z$.

Change:

Controls the sign of $dB_z/dt$ relative to the chosen direction.

Rate $|dB/dt|$ (T/s):

Example: $0.1\,\mathrm{T/s}$.

Initial $|B_0|$ (T):

Used only to display $B(t)$; $\mathbf{E}$ depends on $dB/dt$.

Time $t$ (s):

Displays $B(t)=B_0+(dB/dt)t$.

Disk / solenoid-like region

Region radius $R$ (m):

$dB/dt$ applies mainly for $r\le R$.

Faraday loop + observation point

Loop radius $r_{\ell}$ (m):

Computes $\varepsilon=\oint \mathbf{E}\cdot d\boldsymbol{\ell}$ on this loop.

Observation $x$ (m):

Point $P=(x,y)$ in the cross-section.

Observation $y$ (m):

Radius $r=\sqrt{x^2+y^2}$.

Visual controls

Arrow scale:

Scales the $\mathbf{E}$ arrow at $P$ (visual only).

Line density:

More circles/arrows (qualitative).

Show guides:

Shows loop and region boundary.

Pan/zoom: drag to pan • mouse wheel / trackpad / pinch to zoom • “Reset view” restores default.
Play animates point $P$ around a circle (fixed radius) to show how tangential $\hat{\boldsymbol{\phi}}$ rotates.

Ready

Steps

Enter values and click Solve.

Induced $ \mathbf{E} $ visual (closed loops)

Cross-section ($x$-$y$ plane): $B_z(t)$, Faraday loop, induced $\mathbf{E}$ lines, and $\mathbf{E}(P)$

What the vectors mean

Origin / axis (normal $\hat{\mathbf{z}}$)
Faraday loop radius $r_{\ell}$
Region boundary $r=R$ (disk mode)
Induced $\mathbf{E}$ (tangent arrows + loop lines)
Observation point $P$

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Induced electric field $ \mathbf{E}_{\text{ind}} $

When magnetic flux changes in time, the induced electric field is non-conservative (it forms closed loops). Faraday’s law in integral form is

\[ \oint_{\mathcal{C}} \mathbf{E}\cdot d\boldsymbol{\ell} = -\,\frac{d\Phi_B}{dt}, \qquad \Phi_B=\int_{\mathcal{S}} \mathbf{B}\cdot d\mathbf{A}. \]

Uniform changing $B(t)$ and circular symmetry

If $ \mathbf{B}(t)=B_z(t)\,\hat{\mathbf{z}} $ is spatially uniform over a circular region and changes at rate $ \dfrac{dB_z}{dt} $, symmetry implies the induced field is azimuthal: $ \mathbf{E}(r)=E_\phi(r)\,\hat{\boldsymbol{\phi}} $.

Using a circular Faraday loop of radius $r$, $ \oint \mathbf{E}\cdot d\boldsymbol{\ell}=E_\phi(2\pi r) $.

Two common idealized geometries

1) Uniform $dB_z/dt$ everywhere (infinite uniform $B(t)$)

\[ E_\phi(r)= -\frac{r}{2}\,\frac{dB_z}{dt}. \]

2) Uniform $dB_z/dt$ only inside radius $R$ (outside $\approx 0$)

\[ E_\phi(r)= \begin{cases} -\dfrac{r}{2}\,\dfrac{dB_z}{dt}, & r\le R,\\[6pt] -\dfrac{R^2}{2r}\,\dfrac{dB_z}{dt}, & r\ge R. \end{cases} \]

This matches the idea of a long solenoid: $B$ changes mainly inside, and the induced $ \mathbf{E} $ circulates outside.

Direction (Lenz’s law + right-hand rule)

The sign comes from the minus sign in Faraday’s law: the induced circulation opposes the change in flux.

If $B_z$ (out of the page) is increasing ($dB_z/dt>0$), then $E_\phi(r)<0$: circulation is clockwise (viewed from $+\hat{\mathbf{z}}$).
If $B_z$ is decreasing ($dB_z/dt<0$), then $E_\phi(r)>0$: circulation is counter-clockwise.

Here $ \hat{\boldsymbol{\phi}} $ is the azimuthal unit vector in cylindrical coordinates; in the $x\!-\!y$ plane, \[ \hat{\boldsymbol{\phi}}=\left(-\frac{y}{r},\ \frac{x}{r},\ 0\right),\quad r=\sqrt{x^2+y^2}. \]

Curl relation: $ \nabla\times\mathbf{E}=-\partial\mathbf{B}/\partial t $

In cylindrical symmetry the curl has a $z$-component:

\[ (\nabla\times\mathbf{E})_z = \frac{1}{r}\frac{d}{dr}\!\left(rE_\phi\right). \]

For the “inside radius $R$” model:

\[ (\nabla\times\mathbf{E})_z = \begin{cases} -\dfrac{dB_z}{dt}, & rR, \end{cases} \]

consistent with $ \partial B_z/\partial t $ being nonzero only in the interior region.

Visual tip: field lines are drawn tangent to $ \mathbf{E} $ and form closed loops. Higher $|dB_z/dt|$ makes stronger $|\mathbf{E}|$.

8. Induced Electric Field Tool

Steps

Induced \( \mathbf{E} \) visual (closed loops)

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