Question

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

Answer 1

Hint: In order to answer this question, we need to understand that first write the given data and then find the angular frequency by using the suitable formula. Secondly, find the inductive reactance. Then use the formula for impedance for the LR circuit. Now, using this information we can answer the first and second part of the question.

Formula Used:
Angular frequency,
\[\omega = 2\pi f\]
Inductive reactance,
\[{{\text{X}}_L} = \omega L\]
Impedance for LR circuit,
\[z = \sqrt {X_L^2 + {R^2}} \]

Complete step by step solution:
Inductance, L= 0.5H
Resistance, R= \[100{\text{ }}\Omega \]
RMS voltage, \[{V_{rms}}\]= 240 V
Frequency, f= 50 Hz
Now, angular frequency,
\[\omega = 2\pi f\]
Substituting the values we get,
\[ \Rightarrow \omega = 2\pi \times 50\]
\[ \Rightarrow \omega = 100\pi {\text{ rad/sec}}\]
Inductive reactance,
\[{{\text{X}}_L} = \omega L\]
Substituting the values we get,
\[ \Rightarrow {{\text{X}}_L} = 100\pi \times 0.5\]
\[ \Rightarrow {{\text{X}}_L} = 50\pi \]
(a)
Formula for impedance for LR circuit,
\[z = \sqrt {X_L^2 + {R^2}} \]
Substituting the values we get,
\[ \Rightarrow z = \sqrt {{{\left( {50\pi } \right)}^2} + {{\left( {100} \right)}^2}} \]
\[ \Rightarrow z = 186.209{\text{ }}\Omega \]
We know that;
\[{V_{rms}} = z \times {I_{rms}}\]
\[ \Rightarrow 240 = 186.209 \times {I_{rms}}\]
\[ \Rightarrow {I_{rms}} = \left( {\dfrac{{240}}{{186.209}}} \right)\]
\[ \Rightarrow {I_{rms}} = 1.2888{\text{ }}A\]
\[{I_{rms}} = \left( {\dfrac{{{I_{\max }}}}{{\sqrt 2 }}} \right)\]
\[ \Rightarrow {I_{\max }} = \sqrt 2 {I_{rms}}\]
\[ \Rightarrow {I_{\max }} = \sqrt 2 \times 1.2888\]
\[ \Rightarrow {I_{\max }} = 1.8226{\text{ A}}\]
Maximum current in the circuit is \[{I_{\max }} = 1.8226{\text{ A}}\]
(b)
The power factor of an alternating current is defined as the ratio of the true power flowing through the circuit to the apparent power present in the circuit.
\[\cos \phi = \dfrac{R}{z}\]
R- resistance in the circuit
Z- impedance of the circuit.
\[ \Rightarrow \cos \phi = \dfrac{{100}}{{186.209}}\]
\[ \Rightarrow \phi = {\cos ^{ - 1}}\left( {\dfrac{{100}}{{186.209}}} \right)\]
\[ \Rightarrow \phi = 1{\text{ randian}}\]
\[t = \dfrac{\phi }{w}\]
\[ \Rightarrow t = \left( {\dfrac{1}{{100\pi }}} \right)\]
\[\therefore t = \dfrac{1}{{100\pi }}\sec .\]

Note:
It should be remembered that Ohm’s law for the RMS value of an alternating current is calculated by dividing the RMS voltage by the impedance. The average power delivered to a LCR circuit varies with the phase angle. The power factor of an alternating current is defined as the ratio of the true power flowing through the circuit to the apparent power present in the circuit. It is usually in the interval of -1 to 1 and is dimensionless.