Definition:Solid Angle/Subtend

Definition

Let $S$ be a surface oriented in space.

Let $P$ be a point in that space.

The solid angle subtended by $S$ at $P$ is equal to the surface integral:

$\ds \Omega = \iint_S \frac {\mathbf {\hat r} \cdot \rd \mathbf S} {r^2}$

where:

$\mathbf {\hat r} = \dfrac {\mathbf r} r$ is the unit vector corresponding to the position vector $\mathbf r$ of the infinitesimal area element $\d \mathbf S$ at $P$

$r$ is the magnitude of $\mathbf r$

$\mathbf {\hat n}$ represents the unit normal to $\d S$.

Also presented as

The solid angle subtended by $S$ at $P$ can also be presented as:

$\ds \Omega = \iint_S \frac {\mathbf {\hat r} \cdot \mathbf {\hat n} \rd S} {r^2}$

where:

$\mathbf {\hat r} = \dfrac {\mathbf r} r$ is the unit vector corresponding to the position vector $\mathbf r$ of the infinitesimal surface $\d S$ with respect to $P$

$r$ is the magnitude of $\mathbf r$

$\mathbf {\hat n}$ represents the unit normal to $\d S$.

Sources

• 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.4$ Gauss's Law: $1.4.2$ The flux of the electric field out of a closed surface