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 $L_{max}$) connected in series with a light bulb $B$ as shown. The mains electrical supply is $220\, V$ at $50 \,Hz$, 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
$R = \frac{V^2}{P} = \frac{220^2}{1100} = 44 \,\Omega$
$220 = \frac{(200)^2 \times 44}{44^2 + (L\omega)^2} ; 44^2 + (L \omega)^2 = 220$
$(L \omega)^2 = \sqrt{220 \times 44 - 44^2}$
$= \sqrt{44(220 - 44)} = \sqrt{44 \times 176} = 88 \,\Omega$
$L = \frac{88}{2 \pi \times f} = \frac{88}{2\pi \times 50} $
$= \frac{88}{2 \times 22} \times \frac{7}{50} = 0.28\,H$