Question

A coil has resistances 30$\Omega $ and inductive reactance $20\Omega $ at 50Hz frequency. If an ac source of 200V, 100Hz is connected across the coil, the current in the coil will be,
(A) 13A
(B)2.0A
(C )4.0A
(D)8.0A

Answer 1

Hint: First calculate the inductive reactance for 50Hz and 100Hz. And then comparing those two equations we get the inductive reactance for 100Hz. After that calculate the impedance of the circuit. Then finally calculate the value of current using the ac source voltage and impedance.

Formula used:
Inductive reactance, ${{X}_{L}}=\omega L=2\pi fL$ …………….(1)
Impedance, $z=\sqrt{\left( X_{L}^{'2} \right)+\left( {{R}^{2}} \right)}$ ………….(2)
Current, I = $\dfrac{V}{z}$ ……………(3)

Complete answer:
Given that,
Resistance, R=30$\Omega $
Inductive reactance, ${{X}_{L}}=20\Omega $
Frequency, f= 50Hz
Using equation (1),
${{X}_{L}}=\omega L=2\pi fL$
Substituting the values of inductive reactance and frequency we get,
 $20=2\pi \times 50\times L$ …………(4)
When the frequency of the ac source is changed to 100Hz,
New inductive reactance , $X_{L}^{'}=\omega 'L=2\pi \times 100\times L$
This can be rearranged as,
$X_{L}^{'}=2\pi \times (50\times 2)\times L$ ……….(5)
Substituting equation (4) in (5),
$X_{L}^{'}=2\times {{X}_{L}}=2\times 20=40\Omega $
Substituting $X_{L}^{'}\And R$ in equation (2),
$\Rightarrow $ Impedence, $z=\sqrt{{{40}^{2}}+{{30}^{2}}}=50\Omega $
$\Rightarrow $ Current, $I=\dfrac{V}{z}=\dfrac{200}{50}=4A$

Hence here option(C) is correct.

Additional information:
Inductive reactance is the name given to a changing current flow. The impedance is measured in ohms, just like resistance. That is, the inductive reactance has the same unit of resistance.
Capacitive reactance decreases with increasing AC frequency, while inductive reactance increases with increasing AC frequency.
When current enters a coil, then it will become electromagnetic. The current that flows through the coil will have an opposition upon its inductance and frequency waveform.

Note:
While calculating the impedance we should take the inductive reactance of the changed frequency of ac source not the first frequency. Also the inductive reactance has the same unit of resistance. That is, both are measured in ohms. The inductive reactance is always proportional to the angular frequency of AC voltage source.