Question 185: AC fundamentals

Question

An inductor stores 16 J of magnetic field energy and dissipates 32 W of thermal energy due to its resistance when an a.c. current of 2 A (rms) and frequency 50 Hz flows through it. The ratio of inductive reactance to its resistance is ______. ($\pi = 3.14$)

Accepted Answer

Energy stored ( U ) in inductor is $\mathrm{U}=16 \mathrm{~J}$
Power dissipated in resistance is $\mathrm{P}=32 \mathrm{~W}$
RMS current in the circuit is $\mathrm{I}_{\mathrm{rms}}=2 \mathrm{~A}$
Frequency (f) of the ac source is $f=50 \mathrm{~Hz}$
Power is dissipated only through the resistive component of the inductor.
$$ \mathrm{P}=\mathrm{I}_{\mathrm{rms}}^2 \cdot \mathrm{R} $$
$$\Rightarrow $$ $32=(2)^2 \cdot \mathrm{R} \Rightarrow 32=4 \mathrm{R} \Rightarrow \mathrm{R}=8 \Omega$
Magnetic energy is stored in the inductor.
$$ \mathrm{U}=\frac{1}{2} \mathrm{~L} \cdot i^2 $$
Where $L$ is the self inductance of the inductor and $i$ is the current through it. As the current source is AC, so energy stored is equivalent to current corresponding to the DC source, $i=i_{	ext {rms }}$
Substituting the values in energy formula :
$$ \mathrm{U}=\frac{1}{2} \mathrm{~L}\left(\mathrm{I}_{\mathrm{rms}}^2ight) $$
$$\Rightarrow $$ $16=\frac{L}{2} \cdot(2)^2 \Rightarrow 16=2 L $
$$\Rightarrow $$ $ L=8 H$
The inductive reactance $\left(X_Light)$ given as $X_L=\omega L=2 \pi f L$
$$ X_L=2 	imes 3.14 	imes 50 	imes 8 $$
$$\Rightarrow $$ $$X_L=2512 \Omega $$
So the ratio of $\left(\frac{X_L}{R}ight)$ is,
$$ \frac{X_L}{R}=\frac{2512}{8}=314 $$
Therefore, the ratio of the inductive reactance to the resistance of the inductor is 314.
Hence, the correct answer is $\mathbf{3 1 4}$.

AC fundamentals

An inductor stores 16 J of magnetic field energy and dissipates 32 W of thermal energy due to its resistance when an a.c. current of 2 A (rms) and frequency 50 Hz flows through it. The ratio of inductive reactance to its resistance is ______. (\pi = 3.14)

Integer / Numeric Answer Type