First slide

AC Circuits

Question

A 750 Hz, 20 V (rms) source is connected to a resistance of $100 Ω$ , an inductance of 0.1803 H and a capacitance of $10 μ F$ all in series. The time in which the resistance (heat capacity 2 J/ºC) will get heated by 10ºC. (assume no loss of heat to the surroundings) is close to :

Moderate

A

418 s
B

348 s
C

365 s
D

245 s

Solution

$\begin{array}{l} ω = 2 π f \\ \Rightarrow ω = 2 π \times 750 = 1500 π \\ Impedance = Z = \sqrt{R^{2} + {(L ω - \frac{1}{C ω})}^{2}} \\ \Rightarrow Z = \sqrt{100^{2} + {((0.1803) (1500 π) - \frac{1}{(10 \times 10^{- 6}) (1500 π)})}^{2}} = 834 Ω \\ Power consumed by resistance = P = V_{r ms} i_{r ms} \cos θ = V_{r ms} . \frac{V_{rms}}{Z} . \frac{R}{Z} = \frac{20 \times 20 \times 100}{834 \times 834} = 0.058 W \\ H e a t d i s s i p a t e d = Pt = S θ \\ \Rightarrow t = \frac{S θ}{P} = \frac{2 \times 10}{0.058} = 348 s \end{array}$

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