Symmetric Charge Electric Field Applicator

Physics Electricity and Magnetism • Electric Fields and Charges

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

5. Symmetric Charge E-Field Applicator

Use Gauss’s law to find electric field magnitude for highly symmetric charge distributions: $sphere$ , $infinite cylinder$ , and $infinite plane$ . The tool selects the correct inside/outside expression and shows direction (perpendicular / radial).

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

Geometry + Inputs

Geometry: Charge model: $\varepsilon_0$ (F/m): Relative permittivity $\varepsilon_r$ (optional): Use $\varepsilon=\varepsilon_0\varepsilon_r$: Surface density $\sigma$ (C/m²): Observation side: Distance $d$ from plane (m): Line density $\lambda$ (C/m): Volume density $\rho$ (C/m³): Cylinder radius $R$ (m): Distance from axis $r$ (m): Volume density $\rho$ (C/m³): Sphere radius $R$ (m): Total charge $Q$ (C): Shell radius $R$ (m): Distance from center $r$ (m): Symmetry visual: Field-line hint animation: Animation progress: Play speed:

Choose a geometry and charge model. The calculator applies the standard Gauss-law results (inside/outside) and reports magnitude and direction.

Ready

Steps

Enter values and click Solve.

Symmetry cutaway view

Pan/zoom diagram

Hover: (x, y)=…

Scroll to zoom, drag to pan, double-click to reset. Arrows indicate the field direction (away from positive charge, toward negative charge).

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

Theory: Using Gauss’s law to find $\mathbf{E}$ (symmetry cases)

Gauss’s law relates electric flux through a closed surface to the enclosed charge:

\[ \Phi = \oint \mathbf{E}\cdot d\mathbf{A},\qquad \Phi = \frac{Q_{\text{enc}}}{\varepsilon_0}. \]

The power of Gauss’s law is that symmetry can make $\mathbf{E}$ constant on the Gaussian surface (or always perpendicular), so the integral becomes algebra.

1) Infinite plane with surface charge density $\sigma$

Use a “pillbox” Gaussian surface straddling the plane. Field is perpendicular to the plane and equal magnitude on both sides.

\[ 2EA = \frac{\sigma A}{\varepsilon_0} \quad\Rightarrow\quad E = \frac{\sigma}{2\varepsilon_0}. \]

Magnitude is constant (does not depend on distance from the plane). Direction is away from the plane for $\sigma>0$, toward for $\sigma<0$.

2) Infinite line / cylinder

Use a coaxial Gaussian cylinder of radius $r$ and length $L$.

Line charge (linear density $\lambda$):

\[ E(2\pi rL)=\frac{\lambda L}{\varepsilon_0} \quad\Rightarrow\quad E(r)=\frac{\lambda}{2\pi \varepsilon_0 r}. \]

Solid cylinder (uniform volume density $\rho$, radius $R$):

\[ E(r)= \begin{cases} \dfrac{\rho r}{2\varepsilon_0}, & r Direction is radial: outward for positive charge density, inward for negative.

3) Sphere

Uniform solid sphere (density $\rho$, radius $R$):

\[ E(r)= \begin{cases} \dfrac{\rho r}{3\varepsilon_0}, & r Outside, the field matches a point charge with total $Q=\rho\frac{4}{3}\pi R^3$.

Conducting spherical shell (total charge $Q$ on surface):

\[ E(r)= \begin{cases} 0, & r Zero inside conductor: in electrostatic equilibrium, $E=0$ within the conductor material (and inside the cavity for a spherical shell).

Units

\[ [\mathbf{E}] = \mathrm{N/C},\quad [\sigma]=\mathrm{C/m^2},\quad [\lambda]=\mathrm{C/m},\quad [\rho]=\mathrm{C/m^3}. \]

Steps

Symmetry cutaway view

Rate this calculator

Related calculators