Uniformly charged sphere
(E2.3.1) E2.3 · Application of Gauss's Law to Various Charge Distributions →Outside, it acts exactly like a point charge at the centre. Inside a solid insulating sphere of radius a, the field rises linearly from zero at the centre.
| field outside the sphere | N/C | |
| field inside the sphere | N/C | |
| total charge on the sphere | C | |
| radius of the sphere | m | |
| distance from the centre | m | |
| Coulomb constant | 8.99×10⁹ N·m²/C² (=1/4πε₀) | |
| field outside, inside the sphere | N/C |
Charge spread uniformly through the volume. The inside branch fails for a conductor.
The linear inside branch assumes charge spread through the volume — inside a hollow conducting shell the field is zero instead, since there is no enclosed charge.
Outside (r > a), draw a gaussian sphere enclosing all the charge:
Inside (r < a), only the charge within radius r counts. With uniform density it scales as the volume:
Apply the law again on a sphere of radius r:
Solve — the r³ beats the r², leaving a field proportional to r:
Q = 6.0 nC, a = 10 cm. Outside, at r = 20 cm:
Inside, at r = 5.0 cm:
Both branches give the same value at r = a — the check every such problem must satisfy.
The linear inside branch applies to charge spread through a volume. Inside a hollow conducting shell the field is zero instead, because there is no enclosed charge at all.