d
A series combination of \(n_{1}\) capacitors each of capacitance \(C_{1}\) are connected to \(4 V\) source as shown in the figure.

Total capacitance of the series combination of the capacitors is

\(\frac{1}{C_{s}}=\frac{1}{C_{1}}+\frac{1}{C_{1}}+\frac{1}{C_{1}}+\ldots \ldots \text { upto } n_{1} \text { terms }=\frac{n_{1}}{C_{1}}\)

or \(C_{s}=\frac{C_{1}}{n_{1}}.........(i)\)

Total energy stored in a series combination of the capacitors is

\(U_{s}=\frac{1}{2} C_{s}(4 V)^{2}=\frac{1}{2}\left(\frac{C_{1}}{n_{1}}\right)(4 V)^{2} \quad(\text { Using }(\mathrm{i})).........(ii)\)

A parallel combination of \(n_{2}\) capacitors each of capacitance \(C_{2}\) are connected to \(V\) source as shown in the figure.

Total capacitance of the parallel combination of capacitors is

\(C_{p}=C_{2}+C_{2}+\ldots \ldots \ldots+\text { upto } n_{2} \text { terms }=n_{2} C_{2}\)

or \(\quad C_{p}=n_{2} C_{2}.........(iii)\)

Total energy stored in a parallel combination of capacitors is

\(U_{p} =\frac{1}{2} C_{p} V^{2}\)

\(=\frac{1}{2}\left(n_{2} C_{2}\right)(V)^{2}.........\) (Using \((iii))...(iv)\)

According to the given problem,

\(U_{s}=U_{p}\)

Subst tuting the values of \(U_s,\) and \(U_p,\) from equations \((ii)\) and \((iv)\), we get

\(\frac{1}{2} \frac{C_{1}}{n_{1}}(4 V)^{2}=\frac{1}{2}\left(n_{2} C_{2}\right)(V)^{2}\)

or \(\quad \frac{C_{1} 16}{n_{1}}=n_{2} C_{2}\) or \(C_{2}=\frac{16 C_{1}}{n_{1} n_{2}}\)