One type of \"light dimmer\" consists of a variable capacitor C (whose capacitan

Question

One type of "light dimmer" consists of a variable capacitor C (whose capacitance is adjustable between zero and Cmax) connected in series with the lightbulb. The electrical supply is 130 V (rms) at 61.0 Hz; the lightbulb is rated as "130 V,1100 W."

(a) What value of Cmax is required if the rate of energy dissipation in the lightbulb is to be varied by a factor of 5 from its upper limit of 1100 W? Assume that the resistance of the lightbulb is independent of its temperature.
F
(b) Suppose a variable resistor (adjustable between zero and Rmax) were used instead of a capacitor. What value of Rmax would be required?

Explanation / Answer

here the formula

Pmax / Pmin = Irmsmax^2 * R / ( Irmsmin^2 * R)

Pmax / Pmin = (Vrms^2 / Zmin)^2 / (Vrms^2 / Zmax)^2 = 5

5 = R^2 + ( 1 / w * Cmax) / ( R^2 + (1 / w * Cmin)

5 = R^2 + ( 1 / w*Cmax)^2 / R^2

5 = 1 + ( 1 / w*Cmax)^2 / R^2

4 * R^2 = (1 / w*Cmax)^2

2 * R = 1 / w * Cmax

Cmax = 2 * R * w

Cmax =( 2 * 130^2 /1100) * ( 2 * 3.14 * 60 )

Cmax = 11.6 * 10^3 F

so the Cmax is 11.6 * 10^3 F

b)

here

1 / 5 = R^2 / ( R^2 + Rmax^2)

Rmax = 2 * R

Rmax = 2 * V^2 / P

Rmax = 2 * 130 * 130 / 1100

Rmax = 30.72 ohm

so the Rmax is 30.72 ohm

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