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Mathematics for Physics

29 Charge Density

A measure of how electric charge is distributed in space that is used to write continuous charge as charge per unit length, area, or volume.

Line charge density. Line charge density is charge per unit length along a thin distribution. This principle is used to describe charge on a wire or a thin rod.

The line charge density is

\[ \lambda = \dfrac{dq}{d\ell} \]

where

  • \(\lambda\) is the line charge density.
  • \(dq\) is an infinitesimal charge.
  • \(d\ell\) is an infinitesimal length.

Surface charge density. Surface charge density is charge per unit area on a thin sheet. This principle is used to describe charge on a conductor surface or a plate.

The surface charge density is

\[ \sigma = \dfrac{dq}{da} \]

where

  • \(\sigma\) is the surface charge density.
  • \(dq\) is an infinitesimal charge.
  • \(da\) is an infinitesimal area.

Volume charge density. Volume charge density is charge per unit volume. This principle is used to describe a continuous charge distribution filling a region.

The volume charge density is

\[ \rho = \dfrac{dq}{d\tau} \]

where

  • \(\rho\) is the volume charge density.
  • \(dq\) is an infinitesimal charge.
  • \(d\tau\) is an infinitesimal volume.

The integral for total charge. The total charge is the integral of the appropriate density over the distribution. This principle is used to recover \(Q\) from \(\lambda\), \(\sigma\), or \(\rho\).

The total charge from a volume density is

\[ Q = \displaystyle\int\rho\,d\tau \]

where

  • \(Q\) is the total charge.
  • \(\rho\) is the volume charge density.
  • \(d\tau\) is the volume element.