A circuit consisting of a capacitor C, a resistor of resistance R and an ideal battery of emf V, as shown in figure is known as RC series circuit. As…

A circuit consisting of a capacitor $\displaystyle \mathrm{C}$, a resistor of resistance $\displaystyle \mathrm{R}$ and an ideal battery of emf V, as shown in figure is known as RC series circuit. $$

As soon as the circuit is completed by closing key $\displaystyle \mathrm{S}_{1}$ (keeping $\displaystyle \mathrm{S}_{2}$ open) charges begin to flow between the capacitor plates and the battery terminals. The charge on the capacitor increases and consequently the potential difference $\displaystyle V_{c}(=\mathrm{q} / \mathrm{C})$ across the capacitor also increases with time. When this potential difference equals the potential difference across the battery, the capacitor is fully charged $\displaystyle (Q=V \mathrm{C})$. During this process of charging, the charge $\displaystyle \mathrm{q}$ on the capacitor changes with time $\displaystyle \mathrm{t}$ as $\displaystyle \mathrm{q}=Q\left[1-\mathrm{e}^{-\mathrm{t} / \mathrm{RC}}\right]$ The charging current can be obtained by differentiating it and using $\displaystyle \frac{d}{d x}\left(\mathrm{e}^{m x}\right)=m \mathrm{e}^{m x}$. Consider the case when $\displaystyle \mathrm{R}=20 \mathrm{k} \Omega, \mathrm{C}=500 \mu \mathrm{~F}$ and $\displaystyle V=10 \mathrm{~V}$.

The final charge on the capacitor, when key $\displaystyle \mathrm{S}_{1}$ is closed and $\displaystyle \mathrm{S}_{2}$ is open, is

$\displaystyle 5 \mu \mathrm{C}$

$\displaystyle 5$ mC

$\displaystyle 25$ mC

0.$\displaystyle 1$ C

For sufficient time the key $\displaystyle \mathrm{S}_{1}$ is closed and $\displaystyle \mathrm{S}_{2}$ is open. Now key $\displaystyle \mathrm{S}_{2}$ is closed and $\displaystyle \mathrm{S}_{1}$ is open. What is the final charge on the capacitor?

Zero

$\displaystyle 5$ mC

2.$\displaystyle 5$ mC

$\displaystyle 5 \mu \mathrm{C}$

The dimensional formula for RC is

$\displaystyle \left[\mathrm{M} \mathrm{L}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\right]$

$\displaystyle \left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{-1} \mathrm{~A}^{0}\right]$

$\displaystyle \left[\mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{~T}^{4} \mathrm{~A}^{2}\right]$

$\displaystyle \left[\mathrm{M}^{0} \mathrm{~L}^{0} T \mathrm{~A}^{0}\right]$

The key $\displaystyle \mathrm{S}_{1}$ is closed and $\displaystyle \mathrm{S}_{2}$ is open. The value of current in the resistor after $\displaystyle 5$ seconds, is

$\displaystyle \frac{1}{2 \sqrt{\mathrm{e}}} \mathrm{~mA}$

$\displaystyle \sqrt{\mathrm{e}} \mathrm{mA}$

$\displaystyle \frac{1}{\sqrt{\mathrm{e}}} \mathrm{~mA}$

$\displaystyle \frac{1}{2 \mathrm{e}} \mathrm{~mA}$

The key $\displaystyle \mathrm{S}_{1}$ is closed and $\displaystyle \mathrm{S}_{2}$ is open. The initial value of charging current in the resistor, is

$\displaystyle 5$ mA

0.$\displaystyle 5$ mA

$\displaystyle 2$ mA

$\displaystyle 1$ mA