Question

A coil has resistance 20 Ω and inductance 0.35 H. Compute its impedance to an alternating current of 25 cycles/s.

• A. 50.5 Ω
• B. 48.5 Ω
• C. 58.5 Ω
• D. 68.5 Ω

Hint:

The impedance of an inductive coil depends on both the resistance and the inductive reactance. The total impedance is the square root of the sum of the squares of these values.

Answer 1

The Correct Option is C

The impedance (Z) of a coil in an alternating current (AC) circuit is determined by its resistance (R) and its inductive reactance (X_L). This can be calculated using the formula:

Z = √(R² + X_L²) 

Given:

• Resistance, R = 20 Ω
• Inductance, L = 0.35 H
• Frequency, f = 25 cycles/s

The inductive reactance is given by:

X_L = 2πfL

Substituting the given values:

X_L = 2 × π × 25 × 0.35

X_L = 2 × 3.1416 × 25 × 0.35

X_L ≈ 55 Ω

Now, calculate the impedance:

Z = √(20² + 55²)

Z = √(400 + 3025)

Z = √3425

Z ≈ 58.5 Ω

Therefore, the impedance of the coil to an alternating current of 25 cycles/s is approximately 58.5 Ω, confirming the correct answer.

Answer 2

The impedance \( Z \) of a coil is given by: \[ Z = \sqrt{R^2 + (X_L)^2} \] Where:
- \( R = 20 \, \Omega \),
- \( X_L = 2\pi f L \) is the inductive reactance, with:
- \( f = 25 \, \text{Hz} \),
- \( L = 0.35 \, \text{H} \). First, calculate \( X_L \): \[ X_L = 2 \pi \times 25 \times 0.35 = 54.98 \, \Omega \] Then, the impedance is: \[ Z = \sqrt{20^2 + 54.98^2} = 58.5 \, \Omega \]

Answer 3

This question asks for the total impedance of a coil that has both resistance and inductance when driven by an alternating current of 25 Hz. Let's check the four options directly using the impedance relation \( Z = \sqrt{R^2 + X_L^2} \), where \( X_L = 2\pi f L \) is the inductive reactance.

First find the reactance: \( X_L = 2\pi \times 25 \times 0.35 \approx 54.98 \, \Omega \). So we need \( Z = \sqrt{20^2 + 54.98^2} = \sqrt{400 + 3022.8} = \sqrt{3422.8} \).

1. 50.5 Ω: Squaring this gives \( 50.5^2 = 2550.25 \), well below 3422.8, so this value is too small to be the impedance.
2. 48.5 Ω: Squaring this gives \( 48.5^2 = 2352.25 \), even further from 3422.8, so this option can be ruled out too.
3. 58.5 Ω: Squaring this gives \( 58.5^2 = 3422.25 \), which matches \( 3422.8 \) almost exactly, well within rounding. This is the value that satisfies the impedance relation.
4. 68.5 Ω: Squaring this gives \( 68.5^2 = 4692.25 \), far higher than 3422.8, so this option overshoots the true impedance.

Only the value 58.5 Ω squares back to the resistance-plus-reactance sum calculated from the given data, so it is the impedance that fits the circuit.

Therefore, the correct answer is 58.5 Ω.