Optics, electromagnetic theory Definition

Electric charge

Builds on Triple integral Dirac delta function

Definition

A system of point charges in $\R^3$ is a finite list of distinct points $x_1, \dots, x_n \in \R^3$ together with real numbers $q_1, \dots, q_n$ . The number $q_i$ is the charge at $x_i$ , and $Q = q_1 + \cdots + q_n$ is the total charge. A continuous charge distribution is an integrable function $\rho \colon \R^3 \to \R$ , the charge density. The charge in a bounded region $\Omega \subseteq \R^3$ is the triple integral $Q(\Omega) = \int_\Omega \rho(y) \, dy,$ and the total charge is $Q = \int_{\R^3} \rho(y) \, dy$ .

Remarks

Example: the density $\rho = c$ on the unit ball and $\rho = 0$ outside has total charge $Q = \tfrac{4}{3} \pi c$ , the constant times the volume of the ball. Non-example: the function $\rho(y) = 1$ for all $y \in \R^3$ is not a charge distribution with a total charge, since it is not integrable over $\R^3$ . A point charge $q$ at $x_0$ is often written as the density $q \, \delta(y - x_0)$ , with the Dirac delta. Both cases are signed measures on $\R^3$ , and many texts define a charge distribution as a finite signed Borel measure.

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Tag	`QPYEYVNJ1S`
URI	`m2://QPYEYVNJ1S`
Kind	definition
Writing	W0 stub
MSC	78A30, 78A25
Defines	Electric charge
Uses	Triple integral, Dirac delta function
Changed	3 November 2026

Electric charge

Definition

Remarks

Reading path

Further reading

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