Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

I. E. Irodov Solutions for Physics Class 11 & Class 12

JEE : Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

The document Irodov Solutions: Electric Oscillations- 2 Notes | EduRev is a part of the JEE Course I. E. Irodov Solutions for Physics Class 11 & Class 12.
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Q.111. An oscillating circuit incorporates a leaking capacitor. Its capacitance is equal to C and active resistance to R. The coil inductance is L. The resistance of the coil and the wires is negligible. Find:
 (a) the damped oscillation frequency of such a circuit;
 (b) its quality factor. 

 Ans. In a leaky condenser

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.112. Find the quality factor of a circuit with capacitance C = = 2.0 μF and inductance L = 5.0 mH if the maintenance of undamped oscillations in the circuit with the voltage amplitude across the capacitor being equal to Vm  = 1.0 V requires a power (P) 0.10 mW. The damping of oscillations is sufficiently low. 

 Ans. 

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.113. What mean power should be fed to an oscillating circuit with active resistance R = 0.45Ω to maintain undamped harmonic oscillations with current amplitude Im = 30 mA? 

 Ans. Energy is lost across the resistance and the mean power lass is

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

This power should be fed to the circuit to maintain undamped oscillations.

 

Q.114. An oscillating circuit consists of a capacitor with capacitance C = 1.2 nF and a coil with inductance L = 6.0 iLtli and active resistance R = 0.50Ω. What mean power should be fed to the circuit to maintain undamped harmonic oscillations with voltage amplitude across the capacitor being equal to Vm  = 10 V? 

 Ans. Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.115. Find the damped oscillation frequency of the circuit shown in Fig. 4.30. The capacitance C, inductance L, and active resistance R are supposed to be known. Find how must C, L, and R be interrelated to make oscillations possible. 

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

  Ans.

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

A solution exists only if 

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Thus   Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

ωis the oscillation frequency. Oscillations are possible only if Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

i.e.   Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.116. There are two oscillating circuits (Fig. 4.31) with capacitors of equal capacitances. How must inductances and active resistances of the coils be interrelated for the frequencies and damping of free oscillations in both circuits to be equal? The mutual inductance of coils in the left circuit is negligible. 

 Ans. We have

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Then differentiating we have the equations

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Look for a solution

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

This set of simultaneous equations has a nontrivial solution only if

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

This cubic equation has one real root which we ignore and two complex conjugate roots. We require the condition that this pair of complex conjugate roots is identical with the roots of the equation

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

The general solution of this problem is not easy.We look for special cases. If R= R2 = 0, thei 

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRevThese are the quoted solution but they are misleading.
We shall give the solution for small R1, R2 . Then we put Irodov Solutions: Electric Oscillations- 2 Notes | EduRev is small

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.117. A circuit consists of a capacitor with capacitance C and a coil of inductance L connected in series, as well as a switch and a resistance equal to the critical value for this circuit. With the switch 

disconnected, the capacitor was charged to a voltage Vo, and at the moment t = 0 the switch was closed. Find the current I in the circuit as a function of time t. What is Imax  equal to? 

 Ans.

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

An independent solution is Irodov Solutions: Electric Oscillations- 2 Notes | EduRev Thus

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Thus filially

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

The current has been defined to increase the charge. Hence the minus sign.

The current is maximum when

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 This gives Irodov Solutions: Electric Oscillations- 2 Notes | EduRev and the magnitude of the maximum current is

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.118. A coil with active resistance R and inductance L was connected at the moment t = 0 to a source of voltage V = Vm  cos ωt. Find the current in the coil as a function of time t. 

 Ans. The equation o f the circuit is ( I is th e current)

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

From the theory of differential equations

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

where IP is a particular integral and IC is the complementary function (Solution of the differential equation with the RHS = 0 ). Now

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Substituting we get

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Thus Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Now in an inductive circuit I = 0 at t = 0

because a current cannot change suddenly.

Thus Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 and so

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.119. A circuit consisting of a capacitor with capacitance C and a resistance R connected in series was connected at the moment t = 0 to a source of ac voltage V = Vm cos ωt. Find the current in the circuit as a function of time t. 

 Ans. Here the equation is (Q is charge on the capacitor)

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

A solution subject to Q = 0 at t = 0 is of the form (as in the previous problem)

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Substituting back

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

so 

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 This leads to

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Hence

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.120. A long one-layer solenoid tightly wound of wire with resistivity p has n turns per unit length. The thickness of the wire insulation is negligible. The cross-sectional radius of the solenoid is equal to a. Find the phase difference between current and alternating voltage fed to the solenoid with frequency v. 

 Ans. The current lags behind the voltage by the phase angle

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.121. A circuit consisting of a capacitor and an active resistance R = 110Ω connected in series is fed an alternating voltage with amplitude Vm = 110 V. In this case the amplitude of steady-state current is equal to I= 0.50 A. Find the phase difference between the current and the voltage fed.

 Ans.

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 where

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Now

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Thus the current is ahead of the voltage by 

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.122. Fig. 4.32 illustrates the simplest ripple filter. A voltage V = V0  (1. + cos ωt) is fed to the left input. Find: (a) the output voltage V' (t); (b) the magnitude of the product RC at which the output amplitude of alternating voltage component is η = 7.0 times less than the direct voltage component, if ω = 314 s-1

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

  Ans.

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Ignoring transients, a solution has the form

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Then Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

It satisfies Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.123. Draw the approximate voltage vector diagrams in the electric circuits shown in Fig. 4.33 a, b. The external voltage V is assumed to be alternating harmonically with frequency ω. 

 Ans.
Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

as  Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.124. A series circuit consisting of a capacitor with capacitance C = 22 μF and a coil with active resistance R = 20Ω and inductance L = 0.35 H is connected to a source of alternating voltage with amplitude Vm = 180 V and frequency ω = 314 s-1. Find:
 (a) the current amplitude in the circuit;
 (b) the phase difference between the current and the external voltage;
 (c) the amplitudes of voltage across the capacitor and the coil. 

 Ans. (a)

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

(b)Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 Current lags behind the voltage  Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.125. A series circuit consisting of a capacitor with capacitance C, a resistance R, and a coil with inductance L and negligible active resistance is connected to an oscillator whose frequency can be varied without changing the voltage amplitude. Find the frequency at which the voltage amplitude is maximum
 (a) across the capacitor;
 (b) across the coil.

 Ans. (a)

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

This is maximum whenIrodov Solutions: Electric Oscillations- 2 Notes | EduRev

(b)

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 This is maximum when

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

or

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.126. An alternating voltage with frequency ω = 314s-1 and amplitude Vm = 180 V is fed to a series circuit consisting of a capacitor and a coil with active resistance R = 40Ω and inductance L = 0.36 H. At what value of the capacitor's capacitance will the voltage amplitude across the coil be maximum? What is this amplitude equal to? What is the corresponding voltage amplitude across the condenser? 

 Ans.

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

for a given ω , L , R , this is maximum when

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 At this C,  Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.127. A capacitor with capacitance C whose interelectrode space is filled up with poorly conducting medium with active resistance R is connected to a source of alternating voltage V = Vm  cos ωt. Find the time dependence of the steady-state current flowing in lead wires. The resistance of the wires is to be neglected. 

 Ans.

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

We use the complex voltage Irodov Solutions: Electric Oscillations- 2 Notes | EduRev Then the voltage across the capacitor is

 Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

and that across the resistance R I ' and both equal V . Thus

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 Hence

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

The actual voltage is obtained by taking the real part Then

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

WhereIrodov Solutions: Electric Oscillations- 2 Notes | EduRev

Note → A condenser with poorly conducting material (dielectric of high resistance) be the plates is equvalent to an an ideal condenser with a high resistance joined in p between its plates.

 

Q.128. An oscillating circuit consists of a capacitor of capacitance C and a solenoid with inductance L1. The solenoid is inductively connected with a short-circuited coil having an inductance L2  and a negligible active resistance. Their mutual inductance coefficient is equal to L12. Find the natural frequency of the given oscillating circuit. 

 Ans.

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Then

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Thus the current oscillates with frequency

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.129. Find the quality factor of an oscillating circuit connected in series to a source of alternating emf if at resonance the voltage across the capacitor is n times that of the source. 

 Ans.

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

where

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Then, Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 As resonance the voltage amplitude across the capacitor

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

So             Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Now       Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

 

Q.130. An oscillating circuit consisting of a coil and a capacitor connected in series is fed an alternating emf, with coil inductance being chosen to provide the maximum current in the circuit. Find the quality factor of the system, provided an n-fold increase of inductance results in an ii-fold decrease of the current in the circuit. 

 Ans. For maximum current amplitude

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Now  
Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

So

Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

Now  Irodov Solutions: Electric Oscillations- 2 Notes | EduRev

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