What Are Bound Charges in a Dielectric Material?

Bound charges are the effective electric charges associated with polarization inside a material. They are not usually extra electrons or ions that have been added to the material. Instead, they describe how the positive and negative charges in its atoms or molecules shift slightly relative to one another, creating electric dipoles.

The key quantity is the polarization $\mathbf{P}$: electric dipole moment per unit volume. Once $\mathbf{P}$ is known, you can calculate the macroscopic bound charge it represents.

The two bound-charge formulas

In a region where polarization changes smoothly, the bound volume-charge density is

$$\rho_b=-\nabla\cdot\mathbf{P}.$$

Here, $\rho_b$ is measured in coulombs per cubic metre. A nonzero divergence means that the dipole moments do not balance locally: more dipole ends of one sign than the other remain in that small region.

At a material surface, the bound surface-charge density is

$$\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}},$$

where $\hat{\mathbf{n}}$ is the unit normal pointing outward from the material. Surface density is measured in coulombs per square metre. The sign depends on the direction of polarization relative to the outward normal.

These formulas are the standard macroscopic description of bound charge in electrostatics. They apply when the material can be represented by a polarization field; at sharp boundaries, the surface formula accounts for charge concentrated at the boundary.

Why polarization creates charge

Imagine a small piece of polarized material as many tiny dipoles. Each dipole has a negative end and a positive end. Inside a uniformly polarized bulk, neighboring dipoles’ ends cancel: the positive end of one is next to the negative end of another. But that cancellation may fail at the material’s outer surface, leaving a net surface charge.

If the polarization varies from place to place, the cancellation can also fail within the material. The divergence formula captures that leftover volume charge. This picture explains why uniform polarization gives no bound volume charge in the interior but can still produce bound surface charge.

Worked example: polarization that increases across a slab

Consider a slab occupying $0\le x\le L$, with cross-sectional area $A$. Let its polarization point in the $x$ direction and increase linearly:

$$\mathbf{P}(x)=kx\,\hat{\mathbf{x}},$$

where $k$ is a constant. The divergence is $\nabla\cdot\mathbf{P}=k$, so the volume charge density is

$$\rho_b=-k.$$

The total volume bound charge is therefore $Q_{b,\mathrm{vol}}=-kLA$. At the face $x=L$, the outward normal is $+\hat{\mathbf{x}}$, giving $\sigma_b=kL$. That face carries surface charge $Q_{b,\mathrm{surf}}=kLA$. At $x=0$, $\mathbf{P}=0$, so there is no bound surface charge there. The other faces have normals perpendicular to $\mathbf{P}$ and also have zero surface charge.

The total bound charge is $-kLA+kLA=0$. The negative volume charge is balanced by positive charge on the end face. This cancellation is a useful check: for a finite polarized object, the net bound charge is zero when all its volume and surface contributions are included.

What happens at an interface?

For a boundary between two polarized regions, take a unit normal $\hat{\mathbf{n}}$ pointing from region 1 into region 2. The net bound sheet charge at the interface is

$$\sigma_b=(\mathbf{P}_1-\mathbf{P}_2)\cdot\hat{\mathbf{n}}.$$

This follows by adding the outward-normal contributions from both sides. If region 2 is vacuum, then $\mathbf{P}_2=0$, and the formula reduces to the surface-charge formula for the material. If the normal or the ordering of the regions is reversed, keep the convention consistent; the physical charge does not change.

Bound charge and free charge

Bound charge represents the response of the material’s dipoles. Free charge is charge treated as independently supplied or mobile in the macroscopic model, such as charge delivered by a battery or placed on an electrode. “Free” does not mean that the charge cannot move; it distinguishes that charge from the polarization charge.

This distinction appears in the electric displacement field:

$$\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}, \qquad \nabla\cdot\mathbf{D}=\rho_f,$$

where $\rho_f$ is free-charge density. Gauss’s law for the electric field still includes both types of charge: $\nabla\cdot\mathbf{E}=(\rho_f+\rho_b)/\varepsilon_0$.

Common mistakes

• Using $+\nabla\cdot\mathbf{P}$: The bound volume density has a minus sign: $\rho_b=-\nabla\cdot\mathbf{P}$.
• Ignoring the surface: A uniform polarization has zero divergence, but usually has nonzero bound charge on surfaces whose normals have a component along $\mathbf{P}$.
• Forgetting the normal direction: The surface formula uses the outward normal of the material. Reversing it reverses the sign.
• Calling all charge in a dielectric bound: Free charge may also be present. Keep it separate when using $\mathbf{D}$ and Gauss’s law.
• Confusing density with total charge: Integrate $\rho_b$ over volume and $\sigma_b$ over surface area to get charges in coulombs.