Electric Dipole Field or Potential Analyzer

Physics Electricity and Magnetism • Electric Fields and Charges

Written by STEM Calculators Team Published January 10, 2026 Updated February 24, 2026

6. Electric Dipole Field/Potential Analyzer

Computes dipole potential and field for a dipole moment $\mathbf{p}=q\,\mathbf{d}$ . Far-field (point dipole) uses $\mathbf{E}(\mathbf{r})=\dfrac{1}{4\pi\varepsilon}\dfrac{1}{r^3}\big(3(\mathbf{p}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}\big)$ and $V(\mathbf{r})=\dfrac{1}{4\pi\varepsilon}\dfrac{\mathbf{p}\cdot\mathbf{r}}{r^3}$ . Also includes an exact two-charge model for visualization and torque in a uniform external field.

Inputs support: pi, e, sqrt(), sin, cos, exp, log. Use * for multiplication.

Mode + Inputs

Field model: How to specify dipole: $\varepsilon_0$ (F/m): Relative permittivity $\varepsilon_r$: Use $\varepsilon=\varepsilon_0\varepsilon_r$: Coulomb constant $k$ (N·m²/C²): Use $k$ instead of $\varepsilon$: Dipole $p_x$ (C·m): Dipole $p_y$ (C·m): Effective separation $d$ for exact model (m): Charge magnitude $q$ (C): Separation $d_x$ (m): Separation $d_y$ (m): Query point $x$ (m): Query point $y$ (m): External uniform field $E_x$ (N/C): External uniform field $E_y$ (N/C): Plot: Plot span (m): Equipotential lines: Field arrows: Presets:

Far-field is valid when the query distance $r$ is much larger than the separation $|\mathbf{d}|$. Tip: click the canvas to set the query point (drag pans).

Ready

Steps

Enter values and click Solve.

Equipotential + field visualization

Pan/zoom canvas

Hover: (x, y)=…

Drag to pan, scroll to zoom, double-click to reset. Click to set the query point. The dipole is centered at the origin.

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Theory: Electric dipole potential, field, and torque

An electric dipole can be modeled as two charges $+q$ and $-q$ separated by a displacement vector $\mathbf{d}$. The dipole moment is \[ \mathbf{p} = q\,\mathbf{d}. \]

Far-field (point dipole) approximation

When the observation distance $r$ is much larger than the separation $|\mathbf{d}|$ (i.e., $r \gg |\mathbf{d}|$), the leading terms give:

\[ V(\mathbf{r}) = \frac{1}{4\pi\varepsilon}\,\frac{\mathbf{p}\cdot\mathbf{r}}{r^3} = k\,\frac{\mathbf{p}\cdot\mathbf{r}}{r^3}, \] \[ \mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon}\,\frac{1}{r^3}\Big(3(\mathbf{p}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}\Big) = \frac{k}{r^3}\Big(3(\mathbf{p}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}\Big), \] where $k=\dfrac{1}{4\pi\varepsilon}$ and $\hat{\mathbf{r}}=\mathbf{r}/r$.

Angle form (helps interpretation)

Let $\theta$ be the angle between $\mathbf{p}$ and $\mathbf{r}$. Then $\mathbf{p}\cdot\hat{\mathbf{r}}=|\mathbf{p}|\cos\theta$. The field has a characteristic $1/r^3$ decay and angular dependence.

\[ \cos\theta=\frac{\mathbf{p}\cdot\mathbf{r}}{|\mathbf{p}|\,|\mathbf{r}|}. \]

Exact two-charge model

If you keep the two charges at $\pm\mathbf{d}/2$, you can compute $V$ and $\mathbf{E}$ by superposition:

\[ V(\mathbf{r}) = k\left(\frac{q}{r_+}-\frac{q}{r_-}\right), \quad \mathbf{E}(\mathbf{r})=k\left(\frac{q\,\mathbf{r}_+}{r_+^3}-\frac{q\,\mathbf{r}_-}{r_-^3}\right), \] where $r_+$ is the distance to $+q$ and $r_-$ to $-q$. This is accurate even when $r$ is not large compared to $|\mathbf{d}|$.

Torque and energy in a uniform external field

In a uniform external field $\mathbf{E}_{\text{ext}}$, the dipole experiences torque and has potential energy:

\[ \boldsymbol{\tau}=\mathbf{p}\times\mathbf{E}_{\text{ext}}, \qquad U=-\mathbf{p}\cdot\mathbf{E}_{\text{ext}}. \]

The torque tends to align $\mathbf{p}$ with $\mathbf{E}_{\text{ext}}$, while the energy is minimized when aligned.

Units

\[ [\mathbf{p}] = \mathrm{C\,m},\quad [V]=\mathrm{V},\quad [\mathbf{E}] = \mathrm{N/C},\quad [k]=\mathrm{N\,m^2/C^2}. \]

Steps

Equipotential + field visualization

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