Can the Inner Surface Charge Density Be Negative?

Yes—if the cavity contains a positive charge, the induced charge density on the inner surface is negative. For a single positive point charge inside a spherical cavity, it is negative at every point on the wall. But the local sign is not determined by Gauss’s law alone: if the cavity contains charges of both signs, the density can be negative in some regions and positive in others.

What Gauss’s law tells you

In electrostatic equilibrium, the electric field inside the conducting material is zero. Draw a Gaussian surface entirely within that material, enclosing the cavity. Gauss’s law then gives zero net enclosed charge, so

$$Q_{\text{inner}}=-Q_{\text{cavity}},$$

where $Q_{\text{cavity}}$ is the total charge inside the cavity and $Q_{\text{inner}}$ is the total induced charge on its wall. This is a statement about the total charge, not the density at each point. A negative total does not, by itself, rule out some positively charged patches.

A positive point charge in a spherical cavity

Let the cavity be a sphere of radius $R$, and place a point charge $q>0$ a distance $a<R$ from its center. Let $\theta$ be the angle between the point on the inner surface and the direction from the center to the charge. The induced surface charge density is

$$\sigma(\theta)=-\frac{q}{4\pi R}\,\frac{R^2-a^2}{\left(R^2+a^2-2Ra\cos\theta\right)^{3/2}}.$$

The denominator and $R^2-a^2$ are positive, so $\sigma(\theta)<0$ everywhere. The density is most negative at the point closest to the charge, where the electric field is strongest. If the charge is at the center ($a=0$), the result reduces to the uniform density

$$\sigma=-\frac{q}{4\pi R^2}.$$

The formula applies whether the conducting shell is grounded or isolated: changing the conductor’s potential by a constant does not change the electric field or the induced density on the inner surface.

Mixed-sign charges can give both signs

The density responds linearly to the charges in the cavity. For example, put a charge $+q$ at the center of a spherical cavity and a charge $-q/2$ at distance $R/2$ from the center. The central charge contributes a uniform density $-q/(4\pi R^2)$. The off-center negative charge contributes positive density. At the point on the wall closest to that negative charge, its contribution is $3q/(4\pi R^2)$, so the total density there is positive:

$$\sigma_{\text{near}}=\frac{2q}{4\pi R^2}>0.$$

At the opposite point, the negative charge contributes only $q/(9\cdot 4\pi R^2)\approx 0.111q/(4\pi R^2)$, leaving the total density negative. Thus this configuration has both positive and negative patches, even though its net cavity charge is $q/2$ and its total induced inner charge is $-q/2$.

What if the cavity is empty?

If there are no charges inside a closed cavity, the electric field in it is zero in electrostatic equilibrium, and the inner surface has no induced charge. Charges elsewhere on the conductor or outside it do not create an electric field in the empty cavity. This is electrostatic shielding; it does not mean that the outer surface must be uncharged.

Common mistakes

• Treating the total charge as a local rule. Gauss’s law fixes the integral of $\sigma$ over the whole inner surface, not its value at every point.
• Assuming the density must be uniform. A spherical boundary does not guarantee uniform charge density; an off-center charge breaks the symmetry.
• Confusing inner and outer surfaces. The relation $Q_{\text{inner}}=-Q_{\text{cavity}}$ concerns the inner surface. The conductor’s remaining charge is distributed on its outer surface.
• Ignoring the charge signs inside the cavity. A single positive point charge gives negative density everywhere, but a mixture of positive and negative charges can yield either sign locally.