Question

A typical light dimmer, used to dim the stage lights in a theatre consists of a variable induction for $L$ (where inductance is adjustable between zero and $\left.L_{\max }\right)$, is connected in series with a light bulb $B$ as shown. The mains electrical supply is $220 \,V$ at $50\, Hz$ and the light bulb is rated at $220\, V , 1100\, W$. What $L_{\max }$ is required if the rate of energy dissipation in the light bulb is to be varied by a factor of 5 from its upper limit of $1100 \,W$ ?

• A. 0.69 H
• B. 0.28 H
• C. 0.38 H
• D. 0.56 H

Answer 1

Answer: (B)

For power to be consumed at the rate of
$\frac{1100}{5}=220\, W$,
we have $P=E_{v} I_{v} \cos \theta$
$220=\frac{220 \times 220}{\sqrt{R^{2}+L^{2} \omega^{2}}} \times \frac{R}{\sqrt{R^{2}+L^{2} \omega^{2}}}$
where $R=\frac{V^{2}}{P}=\frac{220^{2}}{1100}=44\, \Omega$
$220=\frac{(220)^{2} \times 44}{44^{2}+(L \omega)^{2}} $
$\Rightarrow 44^{2}+(L \omega)^{2}=220 \times 44$
$(L \omega)^{2}=\sqrt{220 \times 44-44^{2}} $
$=\sqrt{44(220)-44)}$
$=\sqrt{44 \times 176}=88\, \Omega$
$L=\frac{88}{2 \pi \times f}=\frac{88}{2 \pi \times 50}$
$=\frac{88}{2 \times 22} \times \frac{7}{50}=0.28 \,H$