5) In class we had this slide that was not complete, please find the parameters mentioned in the green
box written in red.
\nabla^2V = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial V}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial V}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 V}{\partial \phi^2} = 0
Since the dielectric is homogeneous, the symmetry of this problem implies that
V only varies with r. Thus, Laplace's equation becomes:
$\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial V}{\partial r} \right) = 0 \implies r^2 \frac{\partial V}{\partial r} = A \implies \frac{\partial V}{\partial r} = \frac{A}{r^2} \implies V = -\frac{A}{r} + B$
applying the boundary conditions:
$V(r=a) = -\frac{A}{a} + B = V_o$
$V(r=b) = -\frac{A}{b} + B = 0$
$\left\{ \begin{array}{l} A = \frac{V_o}{\frac{1}{a} - \frac{1}{b}} \\ B = \frac{V_o}{b(\frac{1}{a} - \frac{1}{b})} \end{array} \right.$
$\implies V(r) = V_o \frac{\frac{1}{r} - \frac{1}{b}}{\frac{1}{a} - \frac{1}{b}} (for: a < r < b)$
HW:
Using the
expression we
found for V find:
E, D, \rho, Q
