What is Flux of the field? — Physics

Definition 26.9 University Physics — Year 1 · Chapter 26 — Electrostatics: Field and Gauss’s Law

Cut a surface $S$ into small elements of area $\dd S$ , each with a unit normal $\vect n$ (for a closed surface, the outward normal). The flux of $\vect E$ through $S$ is the sum

\Phi = \sum \vect E\cdot\vect n\,\dd S = \sum E_n\,\dd S \qquad (\mathrm{V}\,\mathrm{m}):

the field’s normal component weighted by area — the “number of field lines” crossing $S$ , counted positively when they cross along $\vect n$ .

The field of a charged sphere against distance from its centre: linear inside and in 1/r2 outside when the charge fills the volume (solid); zero inside and the same outside when it sits on the surface (dashed). Outside, nothing distinguishes the sphere from a point charge at its centre. — The field of a charged sphere against distance from its centre: linear inside and in $1/r^2$ outside when the charge fills the volume (solid); zero inside and the same outside when it sits on the surface (dashed). Outside, nothing distinguishes the sphere from a point charge at its centre.

Superposition for two opposite sheets. Left: the uniform fields of each sheet alone (red from the positive one, blue toward the negative one). Right: their sum, / _0 in the gap and zero elsewhere. — Superposition for two opposite sheets. Left: the uniform fields of each sheet alone (red from the positive one, blue toward the negative one). Right: their sum, $\sigma/\varepsilon_0$ in the gap and zero elsewhere.

Examples

Example 26.13 (Two planes: the plane capacitor)

Two parallel planes carrying $+\sigma$ and $-\sigma$ : each creates $\sigma/2\varepsilon_0$ pointing away from it (toward it for the negative one). Between the planes the two fields add to $\sigma/\varepsilon_0$ , directed from $+$ to $-$ ; outside they cancel. With $\sigma = 1\,\text{µ}\mathrm{C}/\mathrm{m}^{2}$ , $E = 10^{-6}/8.85 \times 10^{-12} = 1.1 \times 10^{5}\,\mathrm{V}/\mathrm{m}$ between, zero outside: the field of a charged capacitor is confined to its gap (Chapter 27).

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Examples

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