Question

An inductance of $1mH$, a condenser of $10\mu F$ and resistance of $50\Omega $ are connected in series. The reactance of inductors and condensers are the same. The reactance of either will be-
a. $100\Omega $
b. $30\Omega $
c. $3.2\Omega $
d. $10\Omega $

Answer 1

Hint: Reactance is the opposition of a circuit element to the flow of current due to that element's inductance or capacitance. Greater reactance leads to smaller currents for the same voltage applied.

Complete step-by-step solution:
The circuit resistance opposites the flow of the current if the alternating current is flowing around a loop. Other components like the inductor or the condenser can be used. These elements are therefore opposed to the movement of current. The inductor and condenser property to oppose current flow is understood to be reactance. The reactance of inductor is called Inductive Reactance and d is denoted by ${{X}_{L}}$ and the reactance of capacitor is called capacitive reactance and is denoted by ${{X}_{C}}$.
It is the magnitude of the imaginary part of the acoustic or mechanical impedance. The current across an inductive reactant varies when there is a potential difference in it. The potential difference and rate of current change is relative. Reactance does not lead to power dissipation. The following equation gives the inductive reaction for an inductor connected to the circuit along with the AC power supply.
${{X}_{L}}=\omega L$
Where, ${{X}_{L}}$ is the reactance, $\omega $is the angular frequency and $L$ is the inductance
Now, we know that the reactance of inductor and the condenser are same, as given in the question,
Therefore,
$\omega L=\dfrac{1}{\omega C}$
Where $C$ is the capacitance
Now,
${{\omega }^{2}}=\dfrac{1}{LC}$
Now we have,
$L=1mH$ OR \[{{10}^{-3}}H\]
$C=10\mu F$ OR ${{10}^{-5}}F$
Now,
$\Rightarrow$${{\omega }^{{}}}=\dfrac{1}{\sqrt{{{10}^{-3}}\times {{10}^{-5}}}}$
$\Rightarrow$$\omega =\dfrac{1}{\sqrt{{{10}^{-8}}}}$
$\Rightarrow$$\omega ={{10}^{4}}$
Now,
Calculating Reactance using the formula
${{X}_{L}}=\omega L$
$\Rightarrow$${{X}_{L}}={{10}^{4}}\times {{10}^{-3}}$
$\Rightarrow$${{X}_{L}}=10\Omega $
Therefore, the reactance of inductor and condenser is $10\Omega $.

Option D is the correct answer.

Note: You can also use the formula for inductive reactance as ${{X}_{L}}=2\pi fL$ where $f$ is the frequency and $L$ is inductance.