Magnetic Dipole Moment Analyzer

Physics Electricity and Magnetism • Magnetic Fields and Sources

Written by STEM Calculators Team Published January 17, 2026 Updated February 24, 2026

5. Magnetic Dipole Moment Analyzer

Compute dipole moment: $\boldsymbol{\mu} = I\,A\,\hat{\mathbf{n}}$ , dipole far-field: $\mathbf{B}_{\text{dip}}(\mathbf{r})=\dfrac{\mu_0}{4\pi r^3}\left(3(\boldsymbol{\mu}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\boldsymbol{\mu}\right)$ , torque: $\boldsymbol{\tau}=\boldsymbol{\mu}\times\mathbf{B}_{\text{ext}}$ , and energy: $U=-\boldsymbol{\mu}\cdot\mathbf{B}_{\text{ext}}$ .

Inputs support numbers like 1e-3 and expressions like pi, sqrt(2), sin(), cos(), tan(), log(), ln(), abs(). Use * for multiplication.

Loop + Field Inputs

Loop shape: Current $I$ (A):

Circle parameters

Radius $r$ (m):

Area $A=\pi r^2$.

Rectangle parameters

Side $a$ (m):

Area $A=a\,b$.

Side $b$ (m):

Use meters.

Dipole direction (loop normal)

Normal $n_x$:

Will be normalized automatically.

Normal $n_y$:

Example: $(0,0,1)$.

Normal $n_z$:

Right-hand rule: curl fingers with current.

External magnetic field (torque/energy)

$B_x$ (T):

$ \mathbf{B}_{ext}=(B_x,B_y,B_z)$.

$B_y$ (T):

Set 0 if none.

$B_z$ (T):

If $|\mathbf{B}_{ext}|=0$, torque is zero.

Observation point for dipole field (far-field)

$x$ (m):

Point $\mathbf{r}=(x,y,z)$.

$y$ (m):

Try $(0,0,z)$ for axial.

$z$ (m):

Far-field: $r$ should be $\gg$ loop size.

Diagram controls

Yaw (deg): Arrow scale (visual): Dipole field pattern: Compass needle torque sim:

Play animates (1) a marker moving along the loop (current direction) and (2) a compass needle rotating toward the projected $\mathbf{B}_{ext}$.

Ready

Steps

Enter values and click Solve.

Vectors + torque diagram

Loop (projected) + vectors

What each vector means

$ \boldsymbol{\mu}=I\,A\,\hat{\mathbf{n}} $ (loop dipole moment)
$ \mathbf{B}_{ext} $ (external field)
$ \boldsymbol{\tau}=\boldsymbol{\mu}\times\mathbf{B}_{ext} $ (torque)
$ \mathbf{r} $ (from dipole to observation point)
$ \mathbf{B}_{dip}(\mathbf{r}) $ (far-field dipole field)
Current direction marker on loop (Play)

Dipole field pattern (shape)

Field lines in a plane containing $ \boldsymbol{\mu} $

Schematic dipole field lines (direction only).

Compass needle torque sim (illustrative)

Needle aligns toward $ \mathbf{B}_{ext} $ (projection)

Press Play to see rotation.

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Theory: Magnetic Dipole Moment and Dipole Fields

A small current loop behaves like a magnetic dipole. Its dipole moment vector points along the loop’s normal (right-hand rule: curl fingers with the current; thumb gives the direction of the normal).

1) Dipole moment of a current loop

For a loop carrying current $I$ with area $A$, the dipole moment is

\[ \boldsymbol{\mu} = I\,A\,\hat{\mathbf{n}}, \]

where $ \hat{\mathbf{n}} $ is the unit normal to the loop. Units: $ \mathrm{A\cdot m^2} $.

2) Torque and potential energy in an external field

In an external magnetic field $ \mathbf{B}_{ext} $, the dipole experiences a torque

\[ \boldsymbol{\tau}=\boldsymbol{\mu}\times\mathbf{B}_{ext}, \]

tending to align $ \boldsymbol{\mu} $ with $ \mathbf{B}_{ext} $. The potential energy is

\[ U=-\boldsymbol{\mu}\cdot\mathbf{B}_{ext}=-|\boldsymbol{\mu}|\,|\mathbf{B}_{ext}|\cos\theta, \]

where $\theta$ is the angle between $ \boldsymbol{\mu} $ and $ \mathbf{B}_{ext} $.

3) Far-field dipole magnetic field

At observation point $ \mathbf{r} $ (measured from the dipole), the ideal dipole field is

\[ \mathbf{B}_{dip}(\mathbf{r}) =\frac{\mu_0}{4\pi r^3}\left(3(\boldsymbol{\mu}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\boldsymbol{\mu}\right), \qquad \hat{\mathbf{r}}=\frac{\mathbf{r}}{r}. \]

This is a far-field approximation (best when $r$ is much larger than the loop size). The constant is $ \mu_0 = 4\pi\times 10^{-7}\ \mathrm{T\cdot m/A} $.

Useful special cases (dipole aligned with the observation axis)

Axial line ($\mathbf{r}$ along $\boldsymbol{\mu}$): \[ |\mathbf{B}|=\frac{\mu_0}{4\pi}\frac{2|\boldsymbol{\mu}|}{r^3}. \]
Equatorial plane ($\mathbf{r}\perp \boldsymbol{\mu}$): \[ |\mathbf{B}|=\frac{\mu_0}{4\pi}\frac{|\boldsymbol{\mu}|}{r^3}. \]

What the calculator visualizes

$\boldsymbol{\mu}$, $\mathbf{B}_{ext}$, and $\boldsymbol{\tau}$ vectors (direction-focused)
Observation vector $\mathbf{r}$ and the far-field $\mathbf{B}_{dip}(\mathbf{r})$
A schematic dipole field-line pattern in a plane containing $\boldsymbol{\mu}$
A compass needle animation that rotates toward the projected $\mathbf{B}_{ext}$