Question

A resistance $R$ draws a power $P$ when connected to an AC source. If an inductance is now placed in series with the resistance, such that the impedance of the circuit becomes $Z$, then the power drawn will be:
$\begin{aligned} & A.P\left( \dfrac{R}{Z} \right) \\\ & B.P \\\ & C.P{{\left( \dfrac{R}{Z} \right)}^{2}} \\\ & D.P\sqrt{\dfrac{R}{Z}} \\\ \end{aligned}$

Answer 1

We know that the power in the circuit is the rate at which electrical energy is transferred with respect to time. In a LCR circuit is where the inductor $L$ , capacitance $C$ and resistance $R$ are connected to an AC source, thus the current in the circuit varies.
Formula: $P=I^{2}_{rms}R=\dfrac({V_{rms}}{R})^{2}R$

Complete answer:
We know that the source of an AC circuit is sinusoidal. Then there is a phase difference between the voltage and current. If the phase difference between the current and voltage is zero, then both are said to be in phase, and if the phase difference is not equal to zero, then both are said to be out of phase.
Given that $R$ is the resistance when connected to say $V$ source gives $I$ current and absorbs $P$ power.
$P=I^{2}_{rms}R=\dfrac({V_{rms}}{R})^{2}R$
If $L$ inductance is added to the circuit, then the $Z$ is the impedance of the circuit or the total resistance offered by the circuit, it is given as $Z=\sqrt{R^{2}+(X_{L}-X_{C})^{2}}$. Where, $X_{L},X_{C}$ are the inductive reactance and the capacitive reactance.
Here,$Z=\sqrt{R^{2}+(X_{L})^{2}}$
We know that the new power $P\prime=I^{2}_{rms}R$
$\implies P\prime=\dfrac({V_{rms}}{Z})^{2}R$
$\implies\dfrac{ P\prime}{P}=\dfrac{\dfrac({V_{rms}}{R})^{2}R}{\dfrac({V_{rms}}{Z})^{2}R}$
$\implies \dfrac{P\prime}{P}=\left(\dfrac{R}{Z}\right)^{2}$
$\therefore P\prime=R\left(\dfrac{R}{Z}\right)^{2}$

Thus, the correct answer is option $C.P{{\left( \dfrac{R}{Z} \right)}^{2}}$ .

Additional information:
We know that, in mechanics, One watt , is named after the scientist Watt, is the energy consumed by doing one joule for one second. i.e. $1W=\dfrac{1J}{1s}$. And in electrical terms, it can be defined as the current flow of $1\;A$ of $1\;V$. i.e. $1W=1A\times 1V$.The unit watt,is named after the scientist Watt, who discovered the steam engine. Power is used to measure work in two different contexts, one in terms of mechanical work and other in terms of electrical work.

Note:
Since here, we need to calculate the ratio of the powers, we can find the individual powers using the formula and then take the ratio between them. Also note that in both cases the same formula is used, but instead of resistance, we are taking the impedance in the second case.