Question

Two parallel-plate vacuum capacitors have areas $A_1$ and $A_2$ and equal plate spacing d. Show that when the capacitors are connected in parallel, the equivalent capacitance is the same as for a single capacitor with plate area $A_1+A_2$ and spacing d.

Answer 1

The capacitance of the capacitors are $C_1=\frac{A_1{ϵ}_0}{d}$ and $C_2=\frac{A_2{ϵ}_0}{d}$

When they are in parallel, the equivalent resistance $C=C_1+C_2=\frac{A_1{ϵ}_0}{d}+\frac{A_2{ϵ}_0}{d}=\frac{(A_1+A_2){ϵ}_0}{d}$, this is the capacitance of plate of area $A_1+A_2$.