Question

$220\;V$, $50\;Hz$, AC source is connected to an inductance of $0.2\;H$ and a resistance of $20\Omega $in series. What is the current in the circuit?

Answer 1

Hint: To find the current in the given AC circuit, we divide the voltage by the total impedance of the circuit. The impedance can be calculated by taking the root of squared and added values of resistance and the inductive reactance of the coil.
Formula used:
$I = \dfrac{V}{Z}$ and
$Z = \sqrt {{R^2} + {X_L}^2} $

Complete step by step solution:
The current in a circuit can be calculated by using Ohm’s Law. But in an A.C. circuit, apart from resistance, the current flow is also obstructed by the capacitive and inductive properties of the circuit. This phenomenon is known as the impedance of a circuit.
Therefore, the current in the circuit is given by-
$I = \dfrac{V}{Z}$
where $I$ is the current, $V$ is the voltage across the circuit, and $Z$ is the impedance of the circuit.
In the question, we are provided with-
The voltage across the circuit, that is $V = 220$ V
The circuit has inductive as well as resistive properties whose-
Inductance $L = 0.2\;H$
And Resistance $R = 20\Omega $
The frequency of the A.C. source $f = 50\;Hz$
The impedance of an RL circuit can be given by-
$Z = \sqrt {{R^2} + {X_L}^2} $
where ${X_L}$is the inductive reactance of the circuit which is given by the formula-
${X_L} = \omega L$
Here $\omega $ stands for the angular frequency of the current,
That is,
$\omega = 2\pi f$
$ \Rightarrow \omega = 2\pi \times 50$
$ \Rightarrow \omega = 100\pi = 314.15$
Thus the inductive reactance is,
${X_L} = 314.15 \times 0.2 = 62.83$
The impedance of the circuit is-
$Z = \sqrt {{{20}^2} + {{62.83}^2}} $
$ \Rightarrow Z = \sqrt {4347} $
$ \Rightarrow Z = 65.93\Omega $
Now that the impedance of the circuit is known, we can calculate the current in the circuit by using,
$I = \dfrac{V}{Z}$
$ \Rightarrow I = \dfrac{{220}}{{65.93}}$
$ \Rightarrow I = 3.34A$

The current flowing in the given RL circuit is equal to $3.34$ Amperes.

Note: The impedance in the circuit combines the real and imaginary parts of the overall resistance to current flow. While the resistance causes heat loss and physical stoppage to the current, the inductive reactance tries to stop the current because of electromagnetic induction in the inductor.