The Field of a Polarized Object

We now define Polarization (\(\vec{P}\)) as the dipole moment per unit volume. Our goal is to calculate the electric potential produced by a polarized object without summing up billions of individual atomic dipoles.

Bound Charges

It turns out that the field produced by a polarized object is mathematically identical to the field produced by a specific distribution of Bound Charges.

Instead of dealing with \(\vec{P}\) directly in an integral, we can replace the polarized material with two equivalent charge distributions:

\(\sigma_b\)

Surface Bound Charge:
\(\sigma_b = \vec{P} \cdot \hat{n}\)

If the polarization points out of the surface, positive charge accumulates there.

\(\rho_b\)

Volume Bound Charge:
\(\rho_b = -\nabla \cdot \vec{P}\)

If the polarization is uniform (\(\nabla \cdot \vec{P} = 0\)), this term vanishes. Only non-uniform polarization creates net charge inside the volume.

V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \left( \oint_{\mathcal{S}} \frac{\sigma_b}{r} da' + \int_{\mathcal{V}} \frac{\rho_b}{r} d\tau' \right)

Deep Dive: Deriving Bound Charges

Physical Interpretation of Bound Charges

Are these "bound charges" real? Yes. They are perfectly genuine accumulations of charge.

Ideally, a dipole consists of a \(-q\) and a \(+q\). In a polarized material, the atomic dipoles align head-to-tail.

Inside the volume, the head of one dipole (\(+\)) sits next to the tail of its neighbor (\(-\)), canceling out.
At the surface, there is no neighbor to cancel the charge. The "heads" of the outermost layer form a sheet of positive charge.

Interactive: Toggle "Show Cancellation" to see how the internal charges neutralize, leaving only the surface charge \(\sigma_b\).

The Field Inside a Dielectric

Can we use these formulas for the field inside the material? It seems problematic: inside the material, you are bumping into individual atoms, where the field fluctuates wildly.

Important Distinction:

The "Field" we talk about in dielectrics is the Macroscopic Field. It is the average field over a region large enough to contain many atoms but small compared to the object.

The Macroscopic field "smoothes out" the atomic spikes. Amazingly, the macroscopic field produced by the atomic dipoles is exactly equal to the field produced by the bound charges \(\sigma_b\) and \(\rho_b\), both inside and outside the object.