Question

The circuit shown below contains an AC generator which provides a source of sinusoidally varying emf epsilon (t) = epsilon _0 sin omega t, 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, epsilon _0, is 100 Squareroot 2 V, and the angular frequency omega is 10 rad/s. We measure the current in the circuit and find that it is given as a function of time by the expression I(t) = (10A) sin (omega t + pi/4). Does the current lead or lag the emf? What is the unknown circuit element in the black box-an inductor or a capacitor? What is the numerical value of the resistance R? What is the numerical value of the capacitance or the inductance in the black box?

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