Electric Charges and Fields
Using Gauss's law
Find fields of symmetric charge distributions.
Gauss's law becomes a quick tool for finding E when the charge is spread symmetrically. The trick is to choose a closed surface (a Gaussian surface) that matches the shape, so that E is the same size everywhere on it and easy to take out of the flux.
| Charge shape | Gaussian surface | Field |
|---|---|---|
| Infinite line, λ per metre | cylinder around the line | E = λ / (2πε₀r) |
| Infinite plane sheet, σ per m² | small cylinder through the sheet | E = σ / (2ε₀) |
| Thin spherical shell, charge q | sphere through the point | outside: kq/r²; inside: 0 |
Line charge: E falls as 1/r. Example: λ = 2 × 10⁻⁷ C/m at r = 0.1 m: E = 2kλ/r = 2 × 9 × 10⁹ × 2 × 10⁻⁷ / 0.1 = 3.6 × 10⁴ N/C.
Plane sheet: E is the same at every distance: it does not depend on r at all. The field lines are parallel and evenly spaced.
Two parallel sheets with +σ and –σ: their fields add between the plates and cancel outside. So E = σ/ε₀ between and 0 outside. This is the parallel plate capacitor of the next chapter.
Spherical shell:
- Outside, it acts as if all its charge sat at the centre: E = kq/r².
- Inside, a Gaussian sphere encloses no charge, so E = 0 everywhere inside.
This is why a car or an aircraft is a safe place in lightning: the field inside a closed metal body is zero. It is called electrostatic shielding.
The method only works with symmetry. For an odd-shaped charge, Gauss's law is still true, but it cannot give E by itself.
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