Solve the Debye-Hckel equation for circular pipes whose walls carry a surface potential of (phi_{0}). Show that

Solve the Debye-Hückel equation for circular pipes whose walls carry a surface potential of \(\phi_{0}\). Show that the potential is given by

\[\phi=\phi_{\mathrm{s}} \frac{I_{0}(\kappa r)}{I_{0}(\kappa R)}\]

where \(I_{0}\) is the modified Bessel function of the first kind.

From this derive an expression for the charge distribution function \(ho_{\mathrm{c}}\) in a circular channel. This expression is useful to analyze electro-osmotic flow in a circular channel.