I am having an issue with the question S1. My solution is slightly different to the one on the paper and was just wondering if anyone could tell me where I am going wrong.
Q: A parallel plate capacitor a composite dielectric. A thin sheet of dielectric of permittivty $\epsilon_1$ and a thickness $t_1$ is placed on top of a second thin dielectric sheet of permittyivty $\epsilon_2$ and thickness $t_2$. On top and bottom are parallel conducting plates of area S. How that the value of the capacitance is given by.
$$C=\frac{S}{\frac{t_1}{\epsilon _{1\:}}+\frac{t_2}{\epsilon _2}}[1]$$
My solution is as follows:
Outline of equations used for solution
$$D=\epsilon E\:\left[2\right]$$
where $\epsilon=\epsilon_0 \epsilon_1$
$$V=\frac{D}{\epsilon }t \space [3]$$
$$Q=CV \space [4]$$
$$D=\frac{Q}{A} \space [5]$$
My workings
$$V_T=V_1+V_2 \space [6]$$
$$V_T=\frac{D}{\epsilon _0\epsilon _{1\:}}t_1+\frac{D}{\epsilon \:_0\epsilon \:_{2\:}}t_2\:\left[7\right]$$
$$V_T=\frac{D}{\epsilon _0}\left(\frac{t_1}{\epsilon 1}+\frac{t_2}{\epsilon _2}\right)\:\left[8\right]$$
$$V_T=\frac{Q}{A\epsilon _0}\left(\frac{t_1}{\epsilon 1}+\frac{t_2}{\epsilon _2}\right)\:\left[9\right]$$
$$\frac{\left(V_T\epsilon _{0\:}A\right)}{Q}=\left(\frac{t_1}{\epsilon 1}+\frac{t_2}{\epsilon _2}\right)\:\left[10\right]$$
$$\frac{Q}{V_T\epsilon _0A}=\frac{1}{\left(\frac{t_1}{\epsilon \:1}+\frac{t_2}{\epsilon \:_2}\right)}\:\left[11\right]$$
$$C=\frac{A\epsilon _{0\:}}{\left(\frac{t_1}{\epsilon \:1}+\frac{t_2}{\epsilon \:_2}\right)}\:\left[12\right]$$
where
$A$=$S$
As you can see I have an extra epsilon but I cant see how the epsilon has be canceled out.
