Question

A $60\,Volt - 10\,Watt$ bulb is operated at $100\,Volt - 60\,Hz$ A.C. The inductance required is
A. $2.56\,Henry$
B. $0.32\,Henry$
C. $0.64\,Henry$
D. $1.28\,Henry$

Answer 1

Hint: In order to solve this question we need to understand faraday’s law of electromagnetic induction which states that if flux through any loop is changed then it induces emf in that loop and current flows in loop in direction such that it opposes the cause of its generation. When a current is flowing in loop and suppose it starts decreasing then back emf induced in loop so as to oppose this change, this phenomena is known as inductance and device that exhibit this behavior is known as inductor. The back emf induced is directly proportional to rate of decrease of current and proportionality constant is shown by $L$ and its SI unit is Henry.

Complete step by step answer:
Given, Power of bulb, $P = 10\,Watt$.
And voltage across the bulb is, $V = 60\,Volt$.
Let the current of the bulb be $I$.
So from the definition of power we get, $P = VI$.
$I = \dfrac{P}{V}$
Putting values we get,
$I = \dfrac{{10}}{{60}}A$
$\Rightarrow I = \dfrac{1}{6}A$
So the resistance develop across bulb is, $R = \dfrac{V}{I}$
Putting values we get,
$R = \dfrac{{60}}{{\dfrac{1}{6}}}\Omega $
$\Rightarrow R = 360\Omega $
Ac voltage is given, ${V_1} = 100\,Volt$
So impedance required in circuit is,
$Z = \dfrac{{{V_1}}}{I}$
Putting values we get,
$Z = \dfrac{{100}}{{\dfrac{1}{6}}}$
$\Rightarrow Z = 600\Omega $

Let inductor resistance be, ${X_L}$.
So using formula of impedance,
$Z = \sqrt {{R^2} + {X_L}^2} $
Squaring on both sides, ${Z^2} = {R^2} + {X_L}^2$
${X_L}^2 = {Z^2} - {R^2}$
$\Rightarrow {X_L} = \sqrt {{Z^2} - {R^2}} $
Putting values we get, ${X_L} = \sqrt {{{(600)}^2} - {{(360)}^2}} $
${X_L} = 480\Omega $
Frequency of inductor is given as, $f = 250Hz$
So, ${X_L} = 2\pi fL$
$L = \dfrac{{{X_L}}}{{2\pi f}}$
Putting values we get, $L = \dfrac{{480\Omega }}{{(2 \times \pi \times 60)}}$
$\therefore L = 1.2732\,Henry$

So the correct option is D.

Note: It should be remembered that flux across loops can be defined as a scalar product of magnetic field vector and area vector. Flux across the loop is changed by changing magnetic field, or by changing area under loop and by changing orientation of loop in magnetic field. Also impedance in circuit is a more general form of resistance as it may be complex.