Question

The circuit shown below contains an AC generator which provides a source of sinusoidally varying emf 8 sin wt, a resistor with resistance R, and a black box. which contains either an inductor or a capacitor, but not both. The amplitude of the driving emf, Eo, is 100 v2 V, and the angular frequency wis 10 rad/s. We measure the current in the circuit and find that it is given as a function of time by the expression 1(1) = (10A) sin (at + π/4). Note: π/4rad = 45° and tan (π/4) = +1. 1E(1) o) (a) Does the current lead or lag the emf? (b) What is the unknown circuit element in the black box-an inductor or a capacitor? (c) What is the numerical value of the resistance R? d) What is the numerical value of the capacitance or the inductance in the black box?

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Answer #1

a)The current leads the driving emf by pi/4 deg, because the driving frequency is less than the resonance frequency.

b)The circuit is purely capacitive, hence the device connected is a capacitor.

c)The impedence Z of the circuit will be:

Z = e0/I ; from current eqn given, I = 10 A and e = 100 sqrt(2)

Z = 100 x sqrt(2)/10 = 100 x 1.414/10 = 14.14 Ohm

We know that,

cos(theta) = R/Z

R = Z cos(theta) = 14.14 x cos(pi/4) = 10 Ohm

Hence, R = 10 Ohm

d)we know that

Z = sqrt (Xc^2 + R^2)

Xc = sqrt (Z^2 - R^2 ) = sqrt (14.14^2 - 10^2) = 10

1/w C = 10

C = 1/10 w = 1/10 x 10 = 1/100 = 10^-2 F

Hence, C = 10^-2 F

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