A typical ”light dimmer” used to dim the stage lights in a theater consists of a

Question

A typical ”light dimmer” used to dim the stage lights in a theater consists of a variable inductor L (whose inductance is adjustable between zero and Lmax) connected in series with a lightbulb B as shown in the figure. The electrical supply provides an emf E (rms) at a frequency f; the lightbulb is rated for power P at this emf. (a) What Lmax is required if the rate of energy dissipation in the lightbulb is to be varied by a factor of 4 from its upper limit? Assume that the resistance of the lightbulb is independent of its temperature. (b) What would be the maximum resistance of a variable resistor (adjustable between zero and Rmax) which can be used in place of an inductor? State your answers in terms of the given variables.

Explanation / Answer

Power = Vrms^2/R

R = Vrms^2/P

if we put the inductor then total current will decrease and the power dissipated across resistor will also decrease.

We want to know what Lmax will give rate of energy dissipation equals to

P' = P/4

Now

P' = (Vrms/Z)^2*R

P' =(Vrms/sqrt(w^2*Lmax^2 + R^2))*R

P' = Vrms^2*R/(w^2*Lmax^2 + R^2)

(w^2*Lmax^2 + R^2) = Vrms^2*R/P'

Lmax^2 = (Vrms^2*R/P' - R^2)/w^2

Lmax = sqrt [(Vrms^2*R/P' - R^2)/w^2]

B.

for max reistance

[(Rmax + Rbulb)/Rbulb]^2 = 4

Rmax = [sqrt (4) - 1]*Rbulb

Rmax = 2*Rbulb

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