Chapter 28, Problem 1Q

To determine

The value of magnitude of the charge on the capacitor while charging is $Q=Q_0\left(1-e^{\frac{-t}{\tau }}\right)$ and while the discharging is $Q=Q_0{e}^{\frac{-t}{\tau }}$.

Answer to Problem 1Q

The value of magnitude of the charge on the capacitor while charging and discharging is $Q=Q_0\left(1-e^{\frac{-t}{\tau }}\right)$.and $Q=Q_0{e}^{\frac{-t}{\tau }}$ respectively.

Explanation of Solution

Write the expression for the realtion between the voltage and the capacitor.

    $V\left(t\right)=V_0\left(1-e^{\frac{-t}{RC}}\right)$                                                                                 

Here,  $V\left(t\right)$ is the volatge as the function of time, $V_0$ is the initial voltage, $t$ is the time, $R$ is the resistance and $C$ is the capacitance.

Write the expression for the charge.

    $Q=CV$

Here, $Q$ is the charge, $C$ is the capacitance and $V$ is the voltage.

Substitute $V_0\left(1-e^{\frac{-t}{RC}}\right)$ for  $V$ in the above equation.

    $Q=C{V}_0\left(1-e^{\frac{-t}{RC}}\right)$

Substitute $Q_0$ for $C{V}_0$ in the above equation.

    $Q=Q_0\left(1-e^{\frac{-t}{RC}}\right)$        (I)

Here, $Q_0$ is the initial charge.

Write the expression for the time constant in the $RC$ circuit.

    $\tau =RC$

Here, $\tau$ is the the time consatnt in the $RC$ circuit, $R$ is the resistance and $C$ is the capacitance.

Substitute $\tau$ for $RC$ in equation (I).

    $Q=Q_0\left(1-e^{\frac{-t}{\tau }}\right)$

Write the expression for the voltage that is time dependent for the discharging voltage.

  $V\left(t\right)=V_0{e}^{\frac{-t}{RC}}$                                                                                         

Here,  $V\left(t\right)$ is the volatge as the function of time, $V_0$ is the initial voltage, $t$ is the time and $\tau$ is the time constant.

Write the expression for the charge.

    $Q=CV$

Substitute $V_0{e}^{\frac{-t}{RC}}$ for $V$ in the above equation.

    $Q=C{V}_0{e}^{\frac{-t}{RC}}$

Substitute $Q_0$ for $C{V}_0$ and $\tau$ for $RC$ in the above equation.

    $Q=Q_0{e}^{\frac{-t}{\tau }}$                                                                                     

Here, $Q_0$ is the initial charge.

Conclusion:

Thus, the value of magnitude of the charge on the capacitor while charging and discharging is $Q=Q_0\left(1-e^{\frac{-t}{\tau }}\right)$.and $Q=Q_0{e}^{\frac{-t}{\tau }}$ respectively.