Question

Two square plates ($l\times l$) and dielectric $(\frac{l}{2}\times \frac{t}{2}\times l)$ are arranged as shown in figure. find the equivalent capacitance of the structure.

• A. $\frac{{\epsilon }_{0}A}{t}\left(\frac{3K+1}{2(K+1)}\right)$
• B. $\frac{{2\epsilon }_{0}A}{t}\left(\frac{K+1}{K+3}\right)$
• C. $\frac{{\epsilon }_{0}A}{t}\left(\frac{K+1}{K+3}\right)$
• D. $\frac{{\epsilon }_{0}A}{t}\left(\frac{2K+1}{2K+3}\right)$

Answer 1

The correct option is A $\frac{{\epsilon }_{0}A}{t}\left(\frac{3K+1}{2(K+1)}\right)$

The general expression of capacitance of parallel plate capacitor without dielectric or with dielectric is $C=\frac{A{ϵ}_0}{d}or\frac{AK{ϵ}_0}{d}$ where $d=$ separation of plates and $A=$ area of plate.
The equivalent network for the given structure is shown in figure below.
here, $C_1=\frac{A/2{ϵ}_0}{(t-t/2)}=\frac{A{ϵ}_0}{t}$;
$C_2=\frac{A/2K{ϵ}_0}{t/2}=\frac{AK{ϵ}_0}{t}$ and
$C_3=\frac{A/2{ϵ}_0}{t}=\frac{A{ϵ}_0}{2t}$
Now $C_{eq}=\frac{C_1{C}_2}{C_1+C_2}+C_3=\frac{\frac{A{ϵ}_0}{t}.\frac{AK{ϵ}_0}{t}}{\frac{A{ϵ}_0}{t}+\frac{AK{ϵ}_0}{t}}+\frac{A{ϵ}_0}{2t}=\frac{A{ϵ}_0}{t}\frac{K}{K+1}+\frac{A{ϵ}_0}{2t}=\frac{A{ϵ}_0}{t}[\frac{K}{K+1}+\frac{1}{2}]=\frac{A{ϵ}_0}{t}\left[\frac{3K+1}{2(K+1)}\right]$