Question

A series R-C combination is connected to an AC voltage of angular frequency $(\omega =500rad/\sec )$. If the impedance of the R-C circuit is $R\sqrt{1.25}$, the time constant (in millisecond) of the circuit is:
A. 1
B. 2
C. 3
D. 4

Answer 1

Hint: The RC time Constant is known as tau denoted by ($\tau $). It is calculated in seconds. The value of the time constant of the RC circuit is equivalent to the product of the resistance in ohms and the capacitance in farads of the circuit. It is the time required by the source to charge the capacitor through the used resistor in the circuit.

Complete answer:
As per the given data,
Frequency ($\omega $) = $500 rad/s$
Impedance (Z) = $R\sqrt{1.25}$

As per the question, it is mentioned that two passive components are resistor and capacitor are connected in series with an AC voltage source.
In an RC circuit, the total impedance is the vector sum of the resistance of the resistor and the reactance of the capacitor.
Mathematically,
${{Z}^{2}}={{R}^{2}}+{{X}_{c}}^{2}$
Which implies,
$\begin{align}
  & Z=\sqrt{{{R}^{2}}+{{X}_{c}}^{2}} \\
 & \Rightarrow Z=\sqrt{{{R}^{2}}+{{\dfrac{1}{\omega C}}^{2}}} \\
\end{align}$
By putting the values as per the given data,
$\begin{align}
  & R\sqrt{1.25}=\sqrt{{{R}^{2}}+\dfrac{1}{500{{C}^{2}}}} \\
 & \Rightarrow 1.25{{R}^{2}}={{R}^{2}}+\dfrac{1}{500{{C}^{2}}} \\
 & \Rightarrow 0.25{{R}^{2}}=\dfrac{1}{500{{C}^{2}}} \\
 & \Rightarrow {{(RC)}^{2}}=\dfrac{{{10}^{-6}}}{{{0.25}^{2}}} \\
\end{align}$
The Time Constant of an RC circuit is the product of the resistance and the capacitance of the circuit. So the time constant of the circuit will be given as,
$\begin{align}
  & T=RC \\
 & \Rightarrow \sqrt{{{(RC)}^{2}}}=\sqrt{\dfrac{{{10}^{-6}}}{{{0.25}^{2}}}} \\
 & \Rightarrow RC=0.004s \\
 & T=4ms \\
\end{align}$

So, the correct answer is “Option D”.

Note:
There are three passive components (resistor, inductor, and capacitor) that are used to form a circuit. These components oppose the flow of current in the circuit. The resistor is a linear component whereas the inductor and capacitor have storage capacity. In-circuit which has both linear and storage components the total sum of property that opposes the flow is current is known as impedance.