Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

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

JEE : Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

The document Irodov Solutions: Electric Oscillations- 3 Notes | EduRev is a part of the JEE Course I. E. Irodov Solutions for Physics Class 11 & Class 12.
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Q.131. A series circuit consisting of a capacitor and a coil with active resistance is connected to a source of harmonic voltage whose frequency can be varied, keeping the voltage amplitude constant. At frequencies ω1 and ω2  the current amplitudes are n times less than the resonance amplitude. Find:
 (a) the resonance frequency;
 (b) the quality factor of the circuit.

Ans. At resonance

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 and 

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.132. Demonstrate that at low damping the quality factor Q of a circuit maintaining forced oscillations is approximately equal to ωo/Δ ω, where ω is the natural oscillation frequency, Δω is the width of the resonance curve I (ω) at the "height" which is Irodov Solutions: Electric Oscillations- 3 Notes | EduRev times less than the resonance current amplitude. 

Ans.

Irodov Solutions: Electric Oscillations- 3 Notes | EduRevI= current amplitude at resouance.

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.133. A circuit consisting of a capacitor and a coil connected in series is fed two alternating voltages of equal amplitudes but different frequencies. The frequency of one voltage is equal to the natural oscillation frequency (ωo) of the circuit, the frequency of the other voltage is η times higher. Find the ratio of the current amplitudes (l0/l) generated by the two voltages if the quality factor of the system is equal to Q. Calculate this ratio for Q = 10 and 100, if η = 1.10. 

Ans. At resonance

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.134. It takes t hours for a direct current l0 to charge a storage battery. How long will it take to charge such a battery from the mains using a half-wave rectifier, if the effective current value is also equal to l0

Ans. The a.c. current must be 

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Then D.C. component of the rectified current is

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 Since the charge deposited must be the same

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

The answer is incorrect.

 

Q.135. Find the effective value of current if its mean value is l0 and its time dependence is
 (a) shown in Fig. 4.34; 
Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Ans. (a)

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Now mean current

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

ThenIrodov Solutions: Electric Oscillations- 3 Notes | EduRev

 Now mean square current 

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

so effective currentIrodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

so Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Then, mean square current  Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

so effective current Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.136. A solenoid with inductance L = 7 mH and active resistance R = 44Ω is first connected to a source of direct voltage Vo  and then to a source of sinusoidal voltage with effective value V = V0. At what frequency of the oscillator will the power consumed by the solenoid be η = 5.0 times less than in the former case? 

Ans.

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Thus Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

or  Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.137. A coil with inductive resistance XL = 30Ω and impedance Z = 50Ω is connected to the mains with effective voltage value V = 100 V. Find the phase difference between the current and the voltage, as well as the heat power generated in the coil. 

Ans.

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 The current lags by φ behind the voltage.

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

also Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.138. A coil with inductance L =- 0.70 H and active resistance r = 20Ω is connected in series with an inductance-free resistance R. An alternating voltage with effective value V = 220 V and frequency w = 314 s-1 i s applied across the terminals of this circuit. At what value of the resistance R will the maximum heat power be generated in the circuit? What is it equal to? 

Ans.

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

This is maximum when R + r = ωL for

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Thus R = ωL - r  for maximum power and Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Substituting the values, we get Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.139. A circuit consisting of a capacitor and a coil in series is connected to the mains. Varying the capacitance of the capacitor, the heat power generated in the coil was increased n = 1.7 times. How much (in per cent) was the value of cos 11) changed in the process? 

Ans.

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Varying the capacitor does not change R so if P increases n times

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 Thus

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.140. A source of sinusoidal emf with constant voltage is connected in series with an oscillating circuit with quality factor Q = 100. At a certain frequency of the external voltage the heat power generated in the circuit reaches the maximum value. How much (in per cent) should this frequency be shifted to decrease the power generated n = 2.0 times? 

Ans.

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Power generated will decrease n times when

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

(taking only the positive sign in the first term to ensure positive value for Irodov Solutions: Electric Oscillations- 3 Notes | EduRev)

Now

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Thus

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 For large Q

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.141. A series circuit consisting of an inductance-free resistance R = 0.16 kΩ and a coil with active resistance is connected to the mains with effective voltage V = 220 V, Find the heat power generated in the coil if the effective voltage values across the resistance R and the coil are equal to V1 =  80 V and V2  = 180 V respectively. 

Ans. We have 

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Heat generated in the coil Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.142. A coil and an inductance-free resistance R = 25Ω are connected in parallel to the ac mains. Find the heat power generated in the coil provided a current I = 0.90 A is drawn from the mains. The coil and the resistance R carry currents I1 = 0.50 A and l2  = = 0.60 A respectively. 

Ans. Here

Irodov Solutions: Electric Oscillations- 3 Notes | EduReveffective voltage

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 Now mean power consumed in the coil

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.143. An alternating current of frequency ω = 314 s- 1  is fed to a circuit consisting of a capacitor of capacitance C = 73 ηF and an active resistance R = 100Ω connected in parallel. Find the impedance of the circuit. 

Ans.

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.144. Draw the approximate vector diagrams of currents in the circuits shown in Fig. 4.35. The voltage applied across the points A and B is assumed to be sinusoidal; the parameters of each circuit are so chosen that the total current l0  lags in phase behind the external voltage by an angle φ. 

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Ans. (a) For the resistance, the voltage and the current are in phase. For the coil the voltage is ahead of the current by less than 90°. The current is obtained by addition because the elements are in parallel

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev   Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

(b) Ic is ahead of the voltage by 90°.

(c) The coil has no resistance so Ih is 90° behind the voltage.

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.145. A capacitor with capacitance C = 1.0 μF and a coil with active resistance R = 0.10Ω and inductance L = 1.0 mH are connected in parallel to a source of sinusoidal voltage V = 31 V. Find:
 (a) the frequency ω at which the resonance sets in;
 (b) the effective value of the fed current in resonance, as well as the corresponding currents flowing through the coil and through the capacitor. 

Ans. When the coil and the condenser are in parallel, the equation is 

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Thus, taking real parts  Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

whereIrodov Solutions: Electric Oscillations- 3 Notes | EduRev

and Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

(a) To get the frequency of resonance we must define what we mean by resouance. One definition requires the extremum (maximum or minimum) of current amplitude. The other definition requires rapid change of phase with φ passing through zero at resonance. For the series circuit.

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

both definitions give Irodov Solutions: Electric Oscillations- 3 Notes | EduRev at resonance. In the present case the two definitions do not agree (except when R = 0 ). The definition that has been adopted in the answer given in the book is the vanishing of phase. This requires

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Note that for small R , φ rapidly changes fromIrodov Solutions: Electric Oscillations- 3 Notes | EduRev  passes throughIrodov Solutions: Electric Oscillations- 3 Notes | EduRev

 (b) At resonance

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

so l = effective value of total currentIrodov Solutions: Electric Oscillations- 3 Notes | EduRev

similarly    Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 Note :- The vanishing of phase (its passing through zero) is considered a more basic definition of resonance.

 

Q.146. A capacitor with capacitance C and a coil with active resistance R and inductance L are connected in parallel to a source of sinusoidal voltage of frequency ω. Find the phase difference between the current fed to the circuit and the source voltage. 

Ans. We use the method of complex voltage 

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Then taking the real part

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

where Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.147. A circuit consists of a capacitor with capacitance C and a coil with active resistance R and inductance L connected in parallel. Find the impedance of the circuit at frequency ω of alternating voltage. 

Ans. From the previous problem   

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 

Q.148. A ring of thin wire with active resistance R and inductance L rotates with constant angular velocity ω in the external uniform magnetic field perpendicular to the rotation axis. In the process, the flux of magnetic induction of external field across the ring varies with time as Irodov Solutions: Electric Oscillations- 3 Notes | EduRev Demonstrate that
 (a) the inductive current in the ring varies with time as I =

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

(b) the mean mechanical power developed by external forces to maintain rotation is defined by the formula 

Irodov Solutions: Electric Oscillations- 3 Notes | EduRevIrodov Solutions: Electric Oscillations- 3 Notes | EduRev

Ans. (a) We have 

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 (b) Mean mechanical power required to maintain rotation = energy loss per unit time

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

  

Q.149. A wooden core (Fig. 4.36) supports two coils: coil 1 with inductance L1 a nd short-circuited coil 2 with active resistance R and inductance L2. The mutual inductance of the coils depends on 

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

the distance x between them according to the law L12 (x). Find the mean (averaged over time) value of the interaction force between the coils when coil 1 carries an alternating current Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Ans. We consider the force Irodov Solutions: Electric Oscillations- 3 Notes | EduRev that a circuit 1 exerts on another closed circuit 2 :-

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 Here Irodov Solutions: Electric Oscillations- 3 Notes | EduRev magnetic field at the site of the current element Irodov Solutions: Electric Oscillations- 3 Notes | EduRev due to the current I1 flowing in 1.

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

where Irodov Solutions: Electric Oscillations- 3 Notes | EduRev vector, from current element Irodov Solutions: Electric Oscillations- 3 Notes | EduRev to the current element Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 The integral involved will depend on the vector Irodov Solutions: Electric Oscillations- 3 Notes | EduRev that defines the separation of the (suitably chosen )centre of the coils. Let Cand C2 be the centres of the two coil suitably defined.
Write

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

The bracket defines the mutual inductance L12. Thus noting the definition of x

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

 where < > denotes time average. Now

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

The current in the coil 2 satisfies Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

or  Irodov Solutions: Electric Oscillations- 3 Notes | EduRev ( in the com plex case )

 taking the real part

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

Where tanIrodov Solutions: Electric Oscillations- 3 Notes | EduRev. Taking time average, we get

Irodov Solutions: Electric Oscillations- 3 Notes | EduRev

The repulsive nature of the force is also consistent with Lenz’s law, assuming, of comse, that L12 decreases with x.

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