Electric Field Point Charge Tool

Physics Electricity and Magnetism • Electric Fields and Charges

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

2. Electric Field Point Charge Tool

Compute the electric field from a single point charge: $\mathbf{E} = k\,q\,\dfrac{\mathbf{r}}{\|\mathbf{r}\|^3}$ with $\mathbf{r}=\mathbf{r}_P-\mathbf{r}_q$ and magnitude $E = k\,\dfrac{|q|}{r^2}$ . Direction is radially away for $q>0$ and toward for $q<0$.

Inputs support: pi, e, sqrt(), sin, cos, exp, log. Use * for multiplication. Scientific notation like 4e-6 works.

Inputs

Constant: Coulomb constant $k$ (N·m²/C²): $\varepsilon_0$ (F/m): $\varepsilon_r$ (relative permittivity): Point charge $q$ (C): Charge position $x_q$ (m): Charge position $y_q$ (m): Field point $x_P$ (m): Field point $y_P$ (m): Field arrow plot: Animation progress (optional): Play speed:

Optional: the animation moves a small test point along the line from the charge to the query point to illustrate how $E$ changes with distance.

Ready

Steps

Enter values and click Solve.

Field arrow plot

Charge, query point, and $\mathbf{E}$ arrow (pan/zoom)

Hover: (x, y)=…

Scroll to zoom. Drag to pan. Double-click to reset. Red charge = $+$, blue charge = $-$. The ring is the query point; the light arrow is $\mathbf{E}$ at the query point. The small moving dot/arrow is just an optional visual “test point”.

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Theory: Electric field of a point charge

A point charge $q$ creates an electric field $\mathbf{E}$ that points radially outward if $q>0$ and radially inward if $q<0$. The field at a point $P$ depends on the displacement from the charge location $\mathbf{r}_q$ to the field point $\mathbf{r}_P$.

Core formulas

\[ \mathbf{r}=\mathbf{r}_P-\mathbf{r}_q,\qquad r=\|\mathbf{r}\| \] \[ \mathbf{E}=k\,q\,\frac{\mathbf{r}}{r^3} \quad\text{where}\quad k=\frac{1}{4\pi\varepsilon_0\varepsilon_r} \] \[ E=\|\mathbf{E}\|=k\,\frac{|q|}{r^2},\qquad \hat{\mathbf{r}}=\frac{\mathbf{r}}{r} \]

In words: direction is along $\hat{\mathbf{r}}$, and magnitude scales like $1/r^2$.

Direction rule

If $q>0$: $\mathbf{E}$ points away from the charge (same direction as $\hat{\mathbf{r}}$).
If $q<0$: $\mathbf{E}$ points toward the charge (opposite $\hat{\mathbf{r}}$).
If $q=0$: $\mathbf{E}=\mathbf{0}$.

Field line density (intuition)

Field lines are a visualization: their density is proportional to the field strength $E$. Because $E\propto 1/r^2$, field lines appear denser near the charge and sparser farther away.

Common pitfall

The electric field $\mathbf{E}$ is defined independent of any test charge: the force on a test charge $q_{test}$ would be $\mathbf{F}=q_{test}\mathbf{E}$, but this tool computes $\mathbf{E}$ itself.

Steps

Field arrow plot

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