Read Gauss's Law for a Non-Uniform Charged Sphere
Spherical symmetry lets you use Gauss's law even when charge density varies with radius. The key is to integrate only the charge enclosed by the Gaussian sphere of radius r.
Electricity questions depend on identifying charge, a conducting path and the condition that controls current or magnetic effect.
Separate charge, current and the path between them
Charge is a property · current needs a conducting path
Charge is a property; current needs a conducting path. Decide first whether the question concerns charging, resistance, current or earthing.
Inside the distribution
Compute Q_enclosed(r) = integral from 0 to r of rho(r') 4 π r'² dr'. Then E(r) 4 π r² = Q_enclosed/ε₀.
Outside the distribution
For r greater than the outer radius, only total charge matters: the field behaves as if all charge were concentrated at the center.
Use boundary checks
At the surface, the inside and outside expressions should agree if there is no surface charge.
OlympIQ exam lens
- ✓Test symmetry before using Gauss's law.
- ✓Integrate rho times the spherical shell volume.
- ✓Check r = 0, surface, and large-r behaviour.
For non-uniform density, Q_enclosed is not rho times 4/3 π r³ unless rho is constant.
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