Relation between surface charge and electrostatic potential

Consider a small pill box through the surface, one end of which is in the inner side of particle, where there is no charge, and the other end lies into the solution.

If the particle and the solution, as a whole are electrically neutral, then:

(10.11)

now, using Poisson's equation:

(10.12)

we get :

	(10.13)

Assuming the potential to be low, we can use the Debye-Hückel Model for the potential, 

Therefore,

(10.14)

but if potential is high then we will have to use Chapman’s equation as in this case DH approximation will not be valid.

Also,

Therefore, the surface charge density becomes,

(10.15)