Question

In the circuit shown in Fig. E26.47 each capacitor initially has a charge of magnitude 3.50 nC on its plates. After the switch S is closed, what will be the current in the circuit at the instant that the capacitors have lost 80.0% of their initial stored energy?

Short answer

If each capacitor initially has a charge of magnitude 3.50 nC on the plates and after the switch, S is closed then the current in the circuit at the instant that the capacitors lost 80.0% of their initial stored energy is 13.5 A

Step-by-step solution

Capacitors in parallel

It is known that the capacitors are connected in parallel, therefore the equivalent capacitance is given by:

$\frac{\mathbf{1}}{\mathbf{C}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{C}}_{\mathbf{1}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{C}}_{\mathbf{2}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{C}}_{\mathbf{3}}}$

Calculating the stored energy

Given data,

C_1 =15.0pF, C_2=20.0pF, C_3=10.0pF and resistor R=25ohm. Each of the capacitor has a charge of magnitude Q=3.50nC.

Since the capacitors are connected in parallel therefore the charge over the plates also has the same magnitude, given by:

$\mathrm{Q}={\mathrm{Q}}_0{\mathrm{e}}^{\frac{-\mathrm{t}}{\mathrm{RC}}}$

And the stored energy is given by:

$\mathrm{U}=\frac{{\mathrm{Q}}^2}{2\mathrm{C}}$

Now, from both the equation we get:

$$\begin{gathered} \mathrm{U}=\frac{{\left({\mathrm{Q}}_0{\mathrm{e}}^{\frac{-\mathrm{t}}{\mathrm{RC}}}\right)}^2}{2\mathrm{C}} \\ =\left(\frac{{\mathrm{Q}}_0^2}{2\mathrm{C}}\right){\mathrm{e}}^{\frac{-21}{\mathrm{RC}}} \end{gathered}$$

But,${\mathrm{U}}_0=\frac{{\mathrm{Q}}_0^2}{2\mathrm{C}}$

Thus:$\mathrm{U}={\mathrm{U}}_0{\mathrm{e}}^{\frac{-21}{\mathrm{RC}}}$

Solving for t

Now, if the capacitor has lost 80% of its stored energy, then the stored energy is 20% of the total energy. Or it means that

$$\begin{gathered} \mathrm{U}=20\%{\mathrm{U}}_0 \\ =\frac{{\mathrm{U}}_0}{5} \end{gathered}$$

Or it can also be written as:

$\frac{1}{5}={\mathrm{e}}^{\frac{-21}{\mathrm{RC}}}$

Solving for t we get:

$\mathrm{t}=-\frac{\mathrm{RC}}{2}\mathrm{In}\left(\frac{1}{5}\right)$

Calculating the current

Since ${\mathrm{C}}_1=\frac{3}{2}{\mathrm{C}}_3$and${\mathrm{C}}_2=2{\mathrm{C}}_3$ therefore we can also write:

$$\begin{gathered} \frac{1}{\mathrm{C}}=\frac{1}{3{\mathrm{C}}_3}+\frac{1}{2{\mathrm{C}}_3}+\frac{1}{{\mathrm{C}}_3} \\ =\frac{13}{6{\mathrm{C}}_3} \end{gathered}$$

Substituting the values for t we get:

$$\begin{gathered} \mathrm{t}=-\frac{6{\mathrm{RC}}_3}{26}\mathrm{In}\left(\frac{1}{5}\right) \\ =-\frac{3{\mathrm{RC}}_3}{13}\mathrm{In}\left(\frac{1}{5}\right) \end{gathered}$$

Putting the values, we get:

$$\begin{gathered} \mathrm{t}=-\frac{3\left(25.0\mathrm{\Omega }\right)\left(10.0\times {10}^{-12}\mathrm{F}\right)}{13}\mathrm{In}\left(\frac{1}{5}\right) \\ =9.29\times {10}^{-12}\mathrm{S} \end{gathered}$$

Current in a circuit is given by:

$$\begin{gathered} \mathrm{I}=\frac{\mathrm{dQ}}{\mathrm{dt}} \\ =\left(\frac{{\mathrm{Q}}_0}{\mathrm{RC}}\right){\mathrm{e}}^{\frac{-\mathrm{t}}{\mathrm{RC}}} \end{gathered}$$

Putting the values, we get:

$$\begin{gathered} \mathrm{I}=\left(\frac{3.50\times {10}^{-9}\mathrm{C}}{\left(25.0\mathrm{\Omega }\right)\left(10.0\times {10}^{-12}\mathrm{F}\right)}\right){\mathrm{e}}^{-\left(\frac{9.29\times {10}^{-12}\mathrm{S}}{\left(25.0\mathrm{\Omega }\right)\left(10.0\times {10}^{-12}\mathrm{F}\right)}\right)} \\ =13.5\mathrm{A} \end{gathered}$$

Therefore, the current is:13.5 A