Question

Question: What multiple of the time constant $\tau$gives the time taken by an initially uncharged capacitor in an RC series circuit to be charged to99,0%of its final charge?

Short answer

Answer:

The multiple of the time constant $\tau$ is 4.61 .

Step-by-step solution

The given data

The final charge of the capacitor, $q=0.99q_0$

Understanding the concept of time constant

Using the charging and discharging concept of the capacitor, we can get the required charge relation using the time constant of the RC circuit. Thus, the condition will give the multiplication value of the time constant.

Formulae:

The charge equation of an RC circuit, $q=q_0{e}^{-t/RC}$ (1)

The time constant of the RC circuit, $\tau =RC$ (2)

Calculation of the multiple of the time constant

Now, the multiple of the time constant can be given using equation (2) in equation (1) as follows:

$$\begin{gathered} 0.99q_0=q_0{e}^{-t/\tau } \\ 0.99=e^{-t/\tau } \\ \frac{t}{\tau }=\ln \left(\frac{100}{99}\right) \\ =4.61 \\ t=4.61\tau \end{gathered}$$

Hence, the value of the multiple of the time is 4.61