Question

A coil has resistance 30 ohm and inductive reactance 20 ohm at 50Hz frequency. If an AC source of 200 volts, 100Hz is connected across the coil, the current in the coil
A. 2.0A
B. 4.0A
C. 8.0A
D. 20A

Answer 1

Inductive reactance is directly proportional to frequency of AC source. As frequency increases inductive reactance also increases.

Formula used:
Impedance of the coil,$Z = \sqrt {\left( {{R^2} + {X^2}_L} \right)}$
Where, r is the resistance of the coil (ohm), ${X_L}$ is the inductive reactance (ohm)
Current in the coil,$I = \dfrac{{{V_{rms}}}}{Z}$
${V_{rms}}$ Is the RMS value of voltage (V)

Complete step by step answer:
Given, resistance of the coil r=30ohm
The inductive reactance of the coil ${X_L}$=20 ohm
Frequency, f=50hz, ${V_{rms}}$=200v
We know that ${X_L} = \omega L$
$= 2\pi fL$
Then,$20 = \left( {2\pi } \right)50 \times L$ ……………………. (1)
When frequency of ac source is changed to 100hz, then
${X^I}_L = {\omega ^I}L$
$= 2\pi {f^I}L$
Then,${X^I}_L = \left( {2\pi } \right)100L$
Above equation can be rearranged as,
${X^I}_L = \left( {2\pi } \right)50 \times L \times 2$
Now substitute equation (1) in the above equation we get,
${X^I}_L = 20 \times 2 = 40\Omega$
Therefore,
Impedance of the coil,$Z = \sqrt {\left( {{R^2} + {X^I}{{^2}_L}} \right)}$
$Z = \sqrt {\left( {{{30}^2} + {{40}^2}} \right)} = 50\Omega$
Then, current in the coil,$I = \dfrac{{{V_{rms}}}}{Z}$
$I = \dfrac{{200}}{{50}}$
$= 4A$

So, the correct answer is “Option B”.

Additional Information:
When l or c is present in an ac circuit, energy is required to build up the magnetic field around l or electric field in c. This energy comes from the source. However, power consumed in l or c is zero because all the power received from the source in a quarter cycle is returned to the source in the next quarter cycle. This power oscillates back and forth and is called reactive power. The total opposition to alternating current in an AC circuit is called impedance of the circuit Z.

Note:
1. Opposition offered by an inductor for the flow of ac is called ‘inductive reactance’.
2. An alternating voltage is one whose magnitude changes with time and direction changes periodically.
3. Frequency of direct current is zero. That is direct current is independent of frequency.