Question

A coil has an inductance of 1 Henry.
1. At what frequency (in Hz) will it have a reactance of 3142 ohm?
2. What should be the capacity of the condenser ($in \ \mu F$) which has the same reactance at that frequency?
A. 600, 0.11
B. 500, 0.11
C. 700, 0.12
D. 800, 0.13

Answer 1

Hint: In DC circuit, the total opposition offered by a load is called resistance and in AC circuit, the total opposition offered by a load is called its reactance. In DC, the resistance of the load is fixed whereas in AC it is the function of time and it depends upon the frequency of the source.

Formula used:
$\chi_{L} = \omega L, \ \chi_{C} = \dfrac{1}{\omega C}$

Complete step by step answer:
Since the relation between angular frequency ($\omega$) inductive reactance ($\chi_L$) and inductance of the coil (L) is$\chi_L = \omega L$, hence:
 Given, $\chi_L = 3142 \ ohm$and L=1H
Hence, $\chi_L = \omega L$
$\omega = \dfrac{3142}{1} = 3142 rads^{-1}$
But since, we are asked about frequency, which is $\nu = \dfrac{\omega}{2\pi}$
Hence $\nu = \dfrac{3142}{2\pi} = 500Hz$
Hence, at a frequency of 500 Hz, the reactance will be 3142 ohm.

Now, for the relation between angular frequency ($\omega$), capacitive reactance ($\chi_C$) and capacitance (C), we use:
$\chi_C = \dfrac1{\omega C}$
Hence, $\nu = 500 Hz$
And $\omega = 2\pi\nu = 2 \times \pi \times 500 = 3142 rad s^{-1}$
Hence using $\chi_C = \dfrac1{\omega C}$
$C = \dfrac1{\chi_C \omega} = \dfrac 1{3142 \times 3142} = 0.11\mu F$
Hence at the same inductance and frequency, the capacitance will be $0.11 \mu F$.
Hence option B. is correct.

Additional information:
Resonating frequency – It is a special type of frequency at which, the inductive reactance and capacitive reactance of the circuit becomes numerically equal. At this frequency, the graph of both reactance cuts each other and maximum current is flown in the circuit.

Note:
What type of frequency is this? This is a special type of angular frequency which is called ‘Resonating frequency’. At this frequency, the current in the circuit achieves its maximum value in the circuit. Mathematically it is equal to $\omega = \dfrac{1}{\sqrt{LC}}$.