Question

A parallel plate capacitor is filled by a dielectric whose permittivity varies with applied voltage according to relation εr=αV, where α=1(volt)^-1. The same capacitor containing no dielectric, charged to a voltage of 72 volt is connected in parallel to the first non-linear uncharged capacitor. The final voltage across the capacitor is?

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{final}\mathrm{potential}\mathrm{difference}\mathrm{be}\mathrm{V}\mathrm{volt},\mathrm{then}\mathrm{the}\mathrm{capacity}\mathrm{of}\mathrm{other}\mathrm{capacitor}\mathrm{will}\mathrm{be}\mathrm{given}\mathrm{by}, \\ \mathrm{C}\times \mathrm{\alpha V} \\ \mathrm{V}=\frac{\mathrm{C}\times 72+\mathrm{\alpha CV}\times 0}{\mathrm{C}+\mathrm{C\alpha V}} \\ \mathrm{CV}[1+\mathrm{\alpha V}]=72\mathrm{C} \\ \mathrm{Or}{\mathrm{V}}^2+\mathrm{V}-72=0 \\ \mathrm{Or}{\mathrm{V}}^2+9\mathrm{V}-8\mathrm{V}-72=0 \\ \mathrm{Or}\mathrm{V}(\mathrm{V}+9)-8(\mathrm{V}+9)=0 \\ \mathrm{Or}\mathrm{V}=8\mathrm{V}\mathrm{and}\mathrm{V}=-9\mathrm{V} \\ \mathrm{Leaving}\mathrm{negative}\mathrm{root}, \\ \mathrm{we}\mathrm{get}\mathrm{V}=8\mathrm{Volt} \end{gathered}$$