Question

A $\dfrac{9}{{100}}\pi $H inductor and a $12\omega $ resistance are connected in series to a 225 V, 50 Hz AC source. Calculate the current (in A) in the circuit.

Answer 1

- Hint: Here we have to find current and for which we apply the formulae ${i_{rms}} = \dfrac{{{V_{rms}}}}{Z}$ and here 225V is given so we have to find Z through formulae $Z = \sqrt {{X_L}^2 + {R^2}} $ and where \[{X_{_L}} = L.\omega \]. Now we have to just put all the values according to the question and get an answer.

Formula used:
V=I.Z
$
  Z = \sqrt {{X_L}^2 + {R^2}} \\
  {X_{_L}} = L.\omega \\
 $

Complete step-by-step answer:
Impedance- It is the effective resistance of the circuit which results from the combined effects of the resistor and reactance.
Two types of reactance are found in any circuit, the capacitive and the inductive reactance. As is clear from their names, the former is the impedance of the capacitor and the latter is the impedance of an inductor.
Resistance of the circuit = 12 ohms.
Inductance of the circuit = $\dfrac{9}{{100}}\pi $ Henry
The inductive reactance ${X_L}$ = $\omega .L = 2\pi \nu .L$
$
  {X_L} = 2\pi (50) \times \dfrac{9}{{100\pi }} \\
  {X_L} = 9{\text{ ohms}} \\
 $
The impedance of the circuit:
$
  Z = \sqrt {{{12}^2} + {9^2}} \\
  Z = \sqrt {225} \\
  Z = 15{\text{ ohms}} \\
$
The RMS current flowing in the circuit can now be calculated using V=I.Z . So we have, I= V/Z.
 $
  {i_{rms}} = \dfrac{{{V_{rms}}}}{Z} \\
   = \dfrac{{225}}{{15}} = 15{\text{ Amps}} \\
 $
The current value is 15 A.

Note: Whenever we get this type of question the key concept of solving is first we have to remember all the formulae used and also understand the meaning of r.m.s value and terms like impedance and reactance then this type of questions will be easily solved.