Field of a spherical shell

The uniformly charged spherical shell (or hollow sphere) is one of the most elegant results in electrostatics: outside it behaves like a point charge, inside the field is exactly zero.

Principle — Field of the spherical shell

A conducting spherical shell of radius $R$ with charge $Q$ generates: $\begin{aligned} E &= k\,\frac{Q}{r^2} & &(r > R) \quad \text{like a point charge} \\ E &= 0 & &(r < R) \quad \text{zero field inside} \end{aligned}$ The potential inside is constant and equal to $V = kQ/R$.

Key formula

Spherical shell of radius $R$: $\ev{E = 0 \quad (r < R), \qquad V = \frac{kQ}{R} \quad (r \leq R)}$ The internal potential is constant, equal to the value at the surface.

Why is $E = 0$ inside? By spherical symmetry, an internal Gaussian surface (a sphere of radius $r < R$) encloses no charge, since all the charge $Q$ sits on the outer shell. Gauss’s theorem then gives

$\Phi = \frac{Q_{\text{int}}}{\varepsilon_0} = 0$

and, since by symmetry the field should be the same at every point on the Gaussian surface, the only possibility compatible with zero flux is $E = 0$ at every internal point.

Note

The spherical shell underlies the Faraday cage: a hollow conductor completely shields its interior from external electric fields. This principle is exploited in electrical measurement laboratories, in microwave oven shielding, and in precision electronics.

Field of the uniformly charged spherical shell. The field lines emerge only outward, as from a point charge. Inside there are no lines: $E = 0$. The internal potential is constant and equal to the value at the surface.

Collegamenti

Argomenti: Campo elettrico e potenziale Concetti: Teorema di Gauss · Conduttori in equilibrio elettrostatico Oggetti: Sfera carica

Esercizi collegati: Two connected conducting spheres · Shield and charge in a hollow conductor · Draw the equipotentials