Electric Field of an Infinite Sheet
Gauss's law and infinite planes
The electric field generated by a uniformly charged infinite plane is a classic example in electrostatics and an elegant demonstration of Gauss's law. Because an infinite sheet extends without bound, the field it produces is perpendicular to the surface and has constant magnitude at every point, independent of distance. This surprising result emerges from symmetry. Choosing a pillbox-shaped Gaussian surface that straddles the sheet eliminates flux through the sides, leaving only the top and bottom caps. Gauss's law, , then simplifies to the familiar relation , where σ is the surface charge density and ε₀ the permittivity of free space.
Entering a value for surface charge density produces the corresponding electric field magnitude. Conversely, supplying the electric field allows the calculator to compute the surface charge density. The relation works for both positive and negative charge densities, with the field pointing away from positively charged sheets and toward negatively charged ones. The form enforces that exactly one of the two inputs is provided; leaving both blank or filling both yields a helpful reminder to choose a single quantity.
Using the infinite-sheet approximation
The independence of distance means this idealized scenario differs from the behavior of finite plates, which exhibit edge effects causing field lines to bulge outward near the borders. However, near the center of a very large but finite sheet, the infinite-plane approximation holds remarkably well. Engineers rely on it when modeling the field near large charged metal plates or capacitor electrodes when plate spacing is small compared with their dimensions. Between two oppositely charged infinite sheets the magnitude doubles to because the fields add in the region between them.
To appreciate the scale, consider a sheet with . The resulting electric field is about 56.5 N/C. Such fields are common in laboratory demonstrations of electrostatic phenomena.
Example conversions
The table below offers sample conversions between surface charge density and electric field magnitude. These values can guide expectations in laboratory settings.
| σ (C/m²) | E (N/C) |
|---|---|
| 1×10-9 | 0.0565 |
| 1×10-6 | 56.5 |
| 5×10-6 | 282.7 |
| 1×10-5 | 565.0 |
Because the electric field is directly proportional to surface charge density, scaling σ by a factor simply scales the field by the same factor. This linearity holds as long as air remains insulating; at very high fields, breakdown occurs and the surrounding medium becomes conductive, limiting the maximum achievable field in practice.
Infinite-sheet reasoning also supports boundary conditions in electromagnetism. Crossing a surface with surface charge changes the normal component of the electric field by . This insight underpins analyses of capacitors, dielectric slabs, and shielding materials. Analogous mathematics appears in fluid dynamics and heat conduction, where uniformly distributed sources over a plane produce constant gradients.
Continue exploring electrostatics with the parallel plate capacitance calculator, compare geometries using the line charge electric field calculator, or analyze potentials with the electric field energy density tool.