## Thursday, September 29, 2011

### Capacitor problem

Problem:
A capacitor is constructed from two square metal plates of area $L^2$ separated by a distance $d$. One half of the space between the plates is filled with a substance of dielectric constant $(\kappa_1)$. The other half is filled with another substance with constant $(\kappa_2)$. Calculate the capacitance of the device assuming that the free space capacitance is $C_0$.

Solution:
Given
$A = L^2$
$d_1 = d_2 = \frac{d}{2}$
Dielectric constant $\kappa_1$ and $\kappa_1$
$C_0$

Find $C$

Formula
In series $C = \frac{A\epsilon_1\epsilon_2}{\epsilon_{2}d_1+\epsilon_{1}d_2}$
$\epsilon = \epsilon_{0}\kappa$
$C_0 = \epsilon_{0}\frac{A}{d}$

Calculation
$C = \frac{A\epsilon_1\epsilon_2}{\epsilon_{2}d_1+\epsilon_{1}d_2} = \frac{A\epsilon_0\kappa_1\epsilon_0\kappa_2}{(\epsilon_0\kappa_2+\epsilon_0\kappa_1)d/2}$
$C = 2\frac{A\epsilon_0}{d}\frac{\kappa_1\kappa_2}{(\kappa_2+\kappa_1)}$
$C = 2C_0\bigg(\frac{\kappa_1\kappa_2}{\kappa_2+\kappa_1}\bigg)$