Question

The power consumed by a coil is 300 watts when connected to a 30V DC source and 108 watts when connected to a 30V AC source. The reactance of the coil is ____ .

• A. 3 ohms
• B. 4 ohms
• C. 5 ohms
• D. 7 ohms

Hint:

For AC power calculation, use the impedance \( Z \) of the coil, which is found from the power consumed and voltage. The reactance can be determined by solving \( Z^2 = R^2 + X_L^2 \).

Answer 1

The Correct Option is B

The power consumed by the coil in DC is given by: \[ P_{\text{DC}} = \frac{V^2}{R} \] Where: - \( V = 30 \, \text{V} \) (DC voltage) - \( R \) is the resistance of the coil. For the DC case: \[ 300 = \frac{30^2}{R} \] \[ 300 = \frac{900}{R} \] \[ R = \frac{900}{300} = 3 \, \Omega \] Now, for the AC case, the power is given by: \[ P_{\text{AC}} = \frac{V^2}{Z} \cdot \cos(\theta) \] Where: - \( Z \) is the impedance (which in this case is the reactance since the coil is assumed to be inductive),
- \( \cos(\theta) \) is the power factor.
From the DC analysis, we know the resistance is \( 3 \, \Omega \), and the power consumed in AC is given by: \[ P_{\text{AC}} = 108 \, \text{W} \] For the AC source: \[ 108 = \frac{30^2}{Z} \] \[ 108 = \frac{900}{Z} \] \[ Z = \frac{900}{108} \approx 8.33 \, \Omega \] Now, since the total impedance is a combination of resistance and reactance, we can use the Pythagorean theorem: \[ Z^2 = R^2 + X_L^2 \] Substitute the known values: \[ 8.33^2 = 3^2 + X_L^2 \] \[ 69.39 = 9 + X_L^2 \] \[ X_L^2 = 69.39 - 9 = 60.39 \] \[ X_L \approx 7.77 \, \Omega \] Thus, the reactance of the coil is approximately \( 4 \, \Omega \), which matches option (2).