Electric Flux and Gauss’s Theorem — Part 2: Applications: visual lesson, ISC Class 12 Physics

Lesson

Applications of Gauss’s theorem

Part 2 — three symmetric charge distributions whose fields Gauss’s theorem gives in a few lines.

In this lesson you will

Derive the field of an infinitely long straight line of charge
Derive the field of an infinite plane sheet of charge
Derive and graph the field inside, on and outside a charged thin spherical shell

Diagram

Infinitely long straight line of charge

Take a coaxial cylinder of radius $r$ and length $l$ . By symmetry $\vec E$ is radial and of the same size on the curved surface; the flat ends get no flux ( $\vec E \parallel$ the faces). $E \times 2\pi rl = \dfrac{\lambda l}{\varepsilon_0}\ \Rightarrow\ E = \dfrac{\lambda}{2\pi\varepsilon_0 r} = \dfrac{2k\lambda}{r}$

E ∝ 1/r, directed radially outward for λ > 0.

Worked example

Worked example 4: field near a charged wire

A long straight wire carries $3.0\ \mu$ C per metre. Find the field 5.0 cm from it.

$E = \dfrac{2k\lambda}{r} = \dfrac{2 \times 9 \times 10^{9} \times 3.0 \times 10^{-6}}{0.050} = 1.08 \times 10^{6}$ N C $^{-1}$ , radially outward.

long line: E ∝ 1/r
point charge: E ∝ 1/r²
infinite sheet: E constant

The more the charge is spread out, the more slowly its field falls with distance.

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