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 Question:
We are asked for the mathematical expression of impedance ($Z$) for a basic series LR circuit driven by an AC source of frequency $f$.

Step 2: Key Formula or Approach:
Impedance in an AC circuit acts like the hypotenuse of a right-angled "impedance triangle", where the resistance $R$ is the base and the net reactance is the altitude.
$$Z = \sqrt{R^2 + X_L^2}$$
The inductive reactance $X_L$ depends on angular frequency $\omega$:
$$X_L = \omega L = 2\pi f L$$

Step 3: Detailed Explanation:
Substitute the expression for $X_L$ into the impedance formula:
$$Z = \sqrt{R^2 + (2\pi f L)^2}$$
Expand the squared term:
$$Z = \sqrt{R^2 + 4\pi^2 f^2 L^2}$$

Step 4: Final Answer:
The impedance expression perfectly matches option (d).