Pre-Quantum Electrodynamics

• PM 10.5,6
• Gr 4.2.2

Surface charge and volume charge from polarization.

Consider an initially neutral small volume \({\cal V}\) with closed surface \({\cal S}\). When the external electric field is applied, the material in \({\cal V}\) will distort, and some charges will move across \({\cal S}\). The leftover charge in \({\cal V}\) is simply the bound charge, and is the negative of the charge that has moved across the boundary \({\cal S}\), \[ \int_{\cal V} d\tau ~\rho_b = -\oint_{\cal S} d{\bf a} \cdot {\bf P} \] Using the divergence theorem, and since this must hold for any volume, we recover \(\rho_b = - {\boldsymbol \nabla} \cdot {\bf P}\).

Surface charge: draw pictures of material near its boundary, with cancellations everywhere except on boundary.

Volume charge: drawing of diverging polarization.