Question

The impedance of a circuit, when a resistance $R$ and an inductor of inductance $L$ are connected in series in an AC circuit of frequency$f$, is
(A)$\sqrt {R + 2{\pi ^2}{f^2}{L^2}}$
(B) $\sqrt {R + 4{\pi ^2}{f^2}{L^2}}$
(C) $\sqrt {{R^2} + 4{\pi ^2}{f^2}{L^2}}$
(D) $\sqrt {{R^2} + 2{\pi ^2}{f^2}{L^2}}$

Answer 1

The combined effect of the resistance and inductive resistance as whole is said to be known as Impedance of RL circuit. The expression for the impedance $Z$ can be written as,
$Z = \sqrt {({R^2}\; + {\text{ }}{X_L}^2)}$
Here, $R$ is resistance of the resistor and ${X_L}$ is inductive resistance.
The resistance caused by the inductor in the circuit is known as inductive resistance.

Formula used:
The expression of inductive resistance ${X_L}$ is written as,
${X_L} = 2\pi fL$
Here, $f$ is the frequency, and $L$ is inductance.
The expression for the impedance $Z$ of the RL circuit is written as,
$Z = \sqrt {({R^2}\; + {\text{ }}{X_L}^2)}$
Here, $R$ is resistance of the resistor

Complete step by step answer:
Write down the expression of inductive resistance ${X_L}$
${X_L} = 2\pi fL$
Here, $f$ is the frequency, and $L$ is inductance.
Write down the expression for the impedance $Z$ of the RL circuit,
$Z = \sqrt {({R^2}\; + {\text{ }}{X_L}^2)}$
Here, $R$ is resistance of the resistor
Substitute $2\pi fL$ for ${X_L}$

$$Z = \sqrt {{R^2}\; + {\text{ }}{{\left( {2\pi fL} \right)}^2}} \\\ \therefore Z = \sqrt {{R^2}\; + {\text{ 4}}{\pi ^2}{f^2}{L^2}} \\\$$

Therefore, option C is the correct choice.

Note: The expression of the inductive resistance is used in the expression of the impedance, to calculate the required impedance for the RL circuit, where resistor and inductor are connected in the series. The resistance $R$ is caused due to the presence of a resistor in the circuit. The combined effect of the resistance and inductive resistance as whole is said to be known as Impedance of RL circuit. The resistance caused by the inductor in the circuit is known as inductive resistance.