Question

In the phase difference between Alternating Voltage and Alternating Current is $\dfrac{\pi }{6}$ and the resistance in the circuit is $\sqrt {300} \Omega $, then the impedance of the circuit will be
A. $25\Omega $
B. $50\Omega $
C. $20\Omega $
D. $100\Omega $

Answer 1

Hint: Impedance is defined as the opposition to alternating and/or direct electric current that an electronic component, circuit, or system offers. Also, Impedance is a vector quantity consisting of two independent scalar quantities: resistance and reactance. Here, we are given the value of resistance and phase angle. Therefore, we need to use the relation between impedance, resistance and phase angle to solve this problem.

Formula used:
$R = Z\cos \theta $,
where, $R$ is the resistance, $Z$ is the impedance and $\theta$ is the phase difference between Alternating Voltage and Alternating Current.

Complete step by step solution:
In the question, we are given two values: First, the phase difference between Alternating Voltage and Alternating Current is $\dfrac{\pi }{6}$ and second, the resistance in the circuit is $\sqrt {300} \Omega $.
Thus, $\theta = \dfrac{\pi }{6}$ and $R = \sqrt {300} \Omega $
We need to find the value of impedance $Z$.
We know that $R = Z\cos \theta $
$
Z = \dfrac{R}{{\cos \theta }} \\
\Rightarrow Z = \dfrac{{\sqrt {300} }}{{\cos \dfrac{\pi }{6}}} \\
\Rightarrow Z = \dfrac{{10\sqrt 3 }}{{\dfrac{{\sqrt 3 }}{2}}} \\
\therefore Z = 20\Omega $
Thus, the impedance of the circuit will be $20\Omega $.

Hence, option C is the right answer.

Note: Here, we have seen the relation between resistance and impedance. But we also need to understand the difference between resistance and impedance. A resistance is defined as the opposition to steady electric current and it does not change with frequency when connected with DC. Whereas, an impedance is defined as the opposition to the AC electricity which is created due to inductance and capacitance and impedance varies with the frequency.