Continuous Charge Distribution in Electric Field

Physics Electricity and Magnetism • Electric Fields and Charges

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

3. Continuous Charge Distribution in Electric Field

Compute electric field from continuous charge distributions using $d\mathbf{E} = k\,\dfrac{dq}{r^2}\,\hat{\mathbf{r}} = k\,dq\,\dfrac{\mathbf{r}}{\|\mathbf{r}\|^3}$ . Choose a geometry (line / ring / disk), use symmetry for closed forms when available, or evaluate the integral numerically.

Inputs support: pi, e, sqrt(), sin, cos, exp, log. Use * for multiplication. For non-uniform density you may enter expressions like 5e-6*(1+0.5*x) (for line) or 1e-6*(1-rho/R) (for disk).

Geometry + Inputs

Geometry: Method: Density: Constant: Coulomb constant $k$ (N·m²/C²): $\varepsilon_0$ (F/m): $\varepsilon_r$: Line density $\lambda$ (C/m): Perpendicular distance $r$ (m): Segment length $L$ (m): Perpendicular distance $a$ (m): Non-uniform $\lambda(x)$ (C/m), $x\in[-L/2,L/2]$: Ring charge input: Total charge $Q$ (C): Ring $\lambda$ (C/m): Ring radius $R$ (m): Axis location $z$ (m): Surface density $\sigma$ (C/m²): Non-uniform $\sigma(\rho)$ (C/m²), $\rho\in[0,R]$: Disk radius $R$ (m): Axis location $z$ (m): Numeric slices $N$ (even): Diagram + vector plot: Integral animation: Animation progress (optional): Play speed:

Tip: Use closed form for classic symmetric cases; switch to numerical to visualize the integral or handle non-uniform density.

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Steps

Enter values and click Solve.

Diagram + field vector

Pan/zoom diagram (labels kept minimal)

Hover: (x, y)=…

Scroll to zoom, drag to pan, double-click to reset. The arrow at the observation point is $\mathbf{E}$ (label shows $|E|$). The animation highlights a small $dq$ element and accumulates the integral (no $\tau$ label is shown on the plot).

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Theory: Continuous Charge Distributions in Electric Field

For a continuous distribution, you sum tiny contributions $d\mathbf{E}$ from charge elements $dq$. The most robust vector form is:

\[ d\mathbf{E} = k\,dq\,\frac{\mathbf{r}}{\|\mathbf{r}\|^3}, \qquad \mathbf{r} = \mathbf{r}_P - \mathbf{r}_{dq}, \qquad k=\frac{1}{4\pi\varepsilon_0\varepsilon_r}. \] \[ \text{Magnitude form:}\quad dE = k\,\frac{dq}{r^2},\quad r=\|\mathbf{r}\|. \]

Symmetry is the shortcut: many components cancel, leaving only one direction to integrate.

1) Infinite line charge (uniform $\lambda$)

By cylindrical symmetry and Gauss’s law, the field depends only on perpendicular distance $r$ and points radially.

\[ E(r)=\frac{\lambda}{2\pi\varepsilon_0\varepsilon_r\,r} =\frac{2k\lambda}{r}. \]

The tool can also show a numerical “truncated integral” to illustrate convergence, but the closed form is the standard result.

2) Finite line segment (centered, uniform $\lambda$)

Place the segment on the $x$-axis from $-L/2$ to $+L/2$, and evaluate at $P=(0,a)$. Horizontal components cancel by symmetry, so only $E_y$ remains.

\[ \mathbf{E}=\int_{-L/2}^{L/2} k\,\lambda\,\frac{\langle -x,\ a\rangle}{(x^2+a^2)^{3/2}}\,dx \quad\Rightarrow\quad E_x=0. \] \[ E_y=\frac{2k\lambda}{a}\sin\theta,\qquad \sin\theta=\frac{L/2}{\sqrt{(L/2)^2+a^2}}. \]

If $\lambda$ varies with position (non-uniform $\lambda(x)$), symmetry may no longer remove $E_x$, so numerical integration is used.

3) Ring on its axis

For a ring of radius $R$ in the plane, the field on the axis at distance $z$ has only an axis component (transverse parts cancel).

\[ E_z = k\,\frac{Qz}{(R^2+z^2)^{3/2}}. \]

The numeric method integrates around the ring ($dq=\lambda R\,d\phi$) to confirm the same result.

4) Disk on its axis

A disk can be treated as concentric rings. For uniform $\sigma$, the closed form on the axis is:

\[ E_z = 2\pi k\sigma\left(1-\frac{|z|}{\sqrt{z^2+R^2}}\right)\,\mathrm{sgn}(z). \]

For a non-uniform $\sigma(\rho)$, integrate rings:

\[ dq=\sigma(\rho)\,2\pi\rho\,d\rho,\qquad dE_z = k\,\frac{z}{(z^2+\rho^2)^{3/2}}\,dq. \]

Numerical notes (Simpson’s rule)

Simpson’s rule approximates an integral by fitting parabolas over subintervals. It is accurate for smooth integrands, and requires an even number of slices $N$. Increase $N$ if you see oscillations or if the density function varies rapidly.

Steps

Diagram + field vector

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