Question

A coil of inductance $0.5H$ and a resistor of resistance $100\Omega$ are connected in series to a $240V,50Hz$ supply.
(a) Find the maximum current in the circuit
(b) What is the time lag between voltage maximum and current maximum?

Answer 1

Given that:-

Inductance, $L=0.5$H

Resistance, $R=100\Omega$

RMS voltage, $V_{rms}=240$V

Frequency, $f=50$Hz

Now, angular frequency, $\omega =2\pi f$

$\omega =2\pi \times 50=100\pi$ rad/s

Inductive reactance, $X_L={\omega }_L$

$X_L=100\pi \times 0.5=50\pi omega$

Formula for impedance for LR circuit;

$z=\sqrt{X_L^2+R^2}$

$\therefore z=\sqrt{(50\pi {)}^2+(100{)}^2}=186.209\Omega$

(a) We know that;

$v_{rms}=z.I_{rms}$ $\Rightarrow$ Ohm's law for AC

$v_{rms}=z{I}_{rms}$

$\Rightarrow 240=186.209\times I_{rms}$

$\Rightarrow I_{rms}=\left(\frac{240}{186.209}\right)A=1.2888A$

As $I_{rms}=\left(\frac{I_{max}}{\sqrt{2}}\right)$

$\Rightarrow I_{max}=\sqrt{2}{I}_{rms}=(\sqrt{2}\times 1.2888)A=1.8226A$

Maximum current in the circuit, $I_{max}=1.8226A$

(b) $\cos \varphi =\frac{R}{z}=\left(\frac{100}{186.209}\right)$

$\Rightarrow \varphi ={\cos }^{-1}\left(\frac{100}{186.209}\right)=1$radian

Time lag, $t=\frac{\varphi }{w}=\left(\frac{1}{100\pi }\right)$sec.