Question

An a.c. source of frequency 'f' is connected to a circuit containing an inductance 'L' and resistance 'R' in series. The impedance of this circuit is ______.

• A. $\sqrt{R^2 + 2\pi f L^2}$
• B. $\sqrt{R^2 + L^2}$
• C. $R + 2\pi f L$
• D. $\sqrt{R^2 + 4\pi^2 f^2 L^2}$

Hint:

Never simply add resistance and reactance directly ($Z \neq R + X_L$) because the voltage across the inductor leads the voltage across the resistor by exactly $90^\circ$. They must always be added using vector (phasor) addition.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In an L-R series circuit, the total opposition to the flow of alternating current is the impedance ($Z$). It is the phasor sum of resistance ($R$) and inductive reactance ($X_L$).

Step 2: Formula Application:
The inductive reactance is given by $X_L = \omega L = 2\pi f L$. The impedance is calculated using the formula $Z = \sqrt{R^2 + X_L^2}$.

Step 3: Explanation:
Substitute $X_L = 2\pi f L$ into the impedance formula: $Z = \sqrt{R^2 + (2\pi f L)^2} = \sqrt{R^2 + 4\pi^2 f^2 L^2}$.

Step 4: Final Answer:
The impedance of the circuit is $\sqrt{R^2 + 4\pi^2 f^2 L^2}$.