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Atypical 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 in figure. The mains electrical supply rs 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 dissipated in the light bulb is to be varied by a factor of 5 from its upper limit of 1100 W?

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0.69 H

0.28 H

0.38 H

0.56 H

answer is B.

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Detailed Solution

Resistance of bulb: $R = \frac{V_{0}^{2}}{P_{0}} = \frac{{(220)}^{2}}{1100} = 44 Ω$

when Z is maximum, power consumed will be minimum which is

$P = \frac{1100}{5} = 220 W$

$N o w P = I_{v}^{2} R \Rightarrow 220 = I_{v}^{2} \times 44 \Rightarrow I_{v} = \sqrt{5} A$

$I_{v} = \frac{E_{v}}{Z} \Rightarrow \sqrt{5} = \frac{220}{\sqrt{44^{2} + X_{L}^{2}}} \Rightarrow X_{L} = 88 Ω$

$\Rightarrow 2 π f L_{max} = 88 \Rightarrow 2 \times \frac{22}{7} \times 50 L_{max} = 88$

$\Rightarrow L_{max} = 0.28 H$

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