Proof of charge neutrality inside a homogeneous conductor

We show that no charges exist inside the conductors $A$ and $B$. With (8) and (28) we obtain from (26) and (27) for the electric field

$$\vec E = \left(Q - \frac{1}{e} \left(\frac{\partial \mu}{\partial... ... \frac{1}{e} \left(\frac{\partial \mu}{\partial n}\right)_T \vec\nabla n \, .$$ (31)

We denote the coefficient of $\vec\nabla T$ on the r.h.s. by $Q^\prime$. Taking the divergence and neglecting a spatial variation of this coefficient and of the derivative $(\partial\mu/\partial n)_T$, which may result from the temperature variation, we obtain, using the Maxwell equation

$$-e \delta n(\vec r)/\varepsilon_0= \rm {div}\,\, \vec E (\vec r)$$ (32)

the following equation for the charge density $-e\delta n (\vec r)$:

$$-e \delta n ( \vec r) = \varepsilon_0 \left( Q^\prime \Delta T(\... ...e} \left(\frac{\partial \mu}{\partial n}\right)_T \Delta n (\vec r) \right)\,.$$ (33)

$\Delta$ stands for the Laplacian. Under stationary conditions as assumed the heat conduction equation

$$C \dot T - \lambda \Delta T =0,$$ (34)

where $C$ denotes the specific heat per volume, leads to

$$\Delta T (\vec r) = 0\, ,$$ (35)

which reduces eqn. (33) to the homogeneous differential equation

$$\Delta n(\vec r) = \xi^{-2} \delta n(\vec r)\, ,$$ (36)

where

$$\xi = \left(\frac{\varepsilon_0}{e^2} \left(\frac{\partial \mu}{\partial n}\right)_T \right)^{1/2}$$ (37)

is a screening length of atomic dimension. Inside a conductor the solution of eqn. (36) decays to zero exponentially within a distance from the surface given by $\xi$. (We note the following analogy: In the London theory of supraconductivity an equation of the same form as eqn. (36) describes the expulsion of the magnetic field from the interior of a superconductor (Meissner-Ochsenfeld effect) [3].) On a macroscopic scale, therefore, the condition of charge neutrality

$$\delta n (\vec r) = 0\, .$$ (38)

holds. The charges generating the thermoelectric potential must be located exclusively at surfaces and interfaces!