Question

A 10Vbattery is connected to a series ofcapacitors, each of capacitance$\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\mu F}$. If the total stored energy is$\mathbf{25}\mathbf{}\mathbf{\mu F}$, what is n?

Short answer

The number of capacitors connected in series is 4.

Step-by-step solution

The given data

• Potential of a battery is 10 V
• Capacitance of each capacitor is$2.0\mathrm{\mu F}$
• Total store energy$25\mathrm{\mu F}$
• There are n capacitors in the series.

Understanding the concept of the capacitance

The capacitors are the devices that store the energy. The capacitance of the capacitor is their ability to store energy. The energy of the capacitor can be written in terms of capacitance and voltage across the capacitor.

All capacitors are identical. So, the voltage will get divided between all capacitors, then the voltage drop across each capacitor is the same and it is given as V/N. As the capacitors are identical, we have to sum up the energy stored in each capacitor, which is the same for all.

Formulae:

The total stored energy due to N number of capacitors,$U_{total}=\sum _{i=1}^N{U}_i$ …(i)

The voltage of each identical capacitor for N capacitors connected in series,

$V\text{'}=\frac{V}{N}$ …(ii)

Here, V is the voltage across all the capacitors, V' is the voltage across each capacitor, N is the number of capacitors.

The energy stored within the capacitor plates,$U=\frac{1}{2}{C}_i{(V\text{'})}^2$ …(iii)

Here, U is the energy across each capacitor, C is the capacitance of each capacitor, and , V' is the voltage across each capacitor.

Calculation of the value of n

Since all the capacitors are identical, we can say

Thus, the total energy stored can be given using equation (iii) in equation (i) as follows:

Using the above expression, we get the number of capacitors for the given value of energy as:

Hence, there are four capacitors connected in series.