Question

What is${\mathit{C}}_{\mathbf{e}\mathbf{q}}$ of three capacitors, each of capacitance C, if they are connected to a battery (a) in series with one another and (b) in parallel? (c) In which arrangement is there more charge on the equivalent capacitance?

Short answer

a) The equivalent capacitance $C_{eq}$of three capacitors if they are in series with one another is$\frac{C}{3}$

b) The equivalent capacitance $C_{eq}$of three capacitors if they are in parallel with one another is 3C.

c) The parallel arrangement will end up having more charge on the equivalent capacitance $C_{eq}$.

Step-by-step solution

The given data

a) Capacitance of each capacitor is C.

b) There are three capacitors.

Understanding the concept of the equivalent capacitance

Using the formulae for the equivalent capacitance, for capacitors is in series and parallel, we can find the equivalent capacitance for series and parallel arrangement respectively. Using Eq.25-1, we can find out which arrangement has more charge on the equivalent capacitance.

Formulae:

If three capacitors are in series, the equivalent capacitanceis given by,

$\frac{1}{C_{eq}}=\sum _{}\frac{1}{C}$ …(i)

If three capacitors are in parallel, the equivalent capacitance$C_{eq}$is given by,

$C_{eq}=\sum _{}C$ …(ii)

The charge between the plates of the capacitor,$q=CV$ …(iii)

(a) Calculation of the equivalent capacitance with the capacitors in series

We are given that capacitance on each capacitor is given by,$C_1=C_2=C_3=C$

Thus, the equivalent capacitance$C_{eq}$ of these capacitors in series is given by:

$$\begin{gathered} \frac{1}{C_{eq}}=\frac{3}{C} \\ {C}_{eq}=\frac{C}{3} \end{gathered}$$

Therefore, the equivalent capacitance ${\mathrm{C}}_{\mathrm{eq}}\mathrm{is}\frac{\mathrm{C}}{3}$.

(b) Calculation of the equivalent capacitance with the capacitors in parallel

We are given that capacitance on each capacitor is given by,$C_1=C_2=C_3=C$

Thus,the equivalent capacitance$C_{eq}$of these capacitors in series is given by:

$$\begin{gathered} C_{eq}=C+C+C \\ =3C \end{gathered}$$

Therefore, the equivalent capacitance$C_{eq}$ is 3C.

(c) Calculation of the equivalent capacitance that has more charge

If capacitors are in series, then we get the charge using equivalent capacitance in series using equation (iii) as follows:

$q=\frac{1}{3}CV$

If capacitors are in parallel, then we get the charge using equivalent capacitance in series using equation (iii) as follows:

$q=3CV$

Therefore, we can say that the parallel will end up having more charge on the equivalent capacitance.