Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • NCERT
  • In the LCR circuit shown in Fig 7.7, the ac driving voltage is v=v_{m} sin omega t (i) Write down the equation of motion for q (t). (ii) At t=t_{0}, the voltage source stops and R is short circuited.

In the LCR circuit shown in Fig 7.7, the AC driving voltage is v=v_{m}\sin \omega t

(i) Write down the equation of motion for q (t).
(ii) At t=t_{0}, the voltage source stops, and R is short-circuited. Now write down how much energy is stored in each of L and C.
(iii) Describe the subsequent motion of charges.

 

Answers (1)

Explanation:-

a) In an LCR circuit (comprising an inductor L, capacitor C, and resistor R), when an AC voltage source drives the circuit, the equation governing the charge q(t) on the capacitor is derived from Kirchhoff's Voltage Law (KVL).

V=V_{m}\sin \omega t

L\frac{di}{dt}+iR+\frac{q}{C}=V_{m}\sin \omega t

L\frac{d^{2}q}{dt^{2}}+\frac{dq}{dt}R+\frac{q}{C}=V_{m}\sin \omega t

b) Once the driving voltage is turned off and the resistor is short-circuited (essentially removing the resistor from the circuit), the circuit becomes an LC circuit, which oscillates without any energy loss due to resistance.

Let q=q_{m}\sin \left ( \omega t+\phi \right )

\frac{dq}{dt}=\omega q_{m}\cos (\omega t+\phi )

i_{m}=\frac{V_{m}}{Z}=\frac{V_{m}}{\sqrt{(R)^{2}+(X_{c}-X_{L})^{2}}}

\phi =\tan ^{-1}\left ( \frac{X_{c}-X_{L}}{R} \right )

At t=t_{0}, Resistance is short-circuited and the inductor and capacitor store energy

U_{L}=\frac{1}{2}Li^{2}=\frac{1}{2}L\left [ \frac{V_{m}}{\sqrt{(R)^{2}+(X_{c}-X_{L})^{2}}} \right ]^{2}\sin ^{2} (\omega t_{0}+\phi )

U_{c}=\frac{q^{2}}{2C}=\frac{1}{2C\omega ^{2}}\left [ \frac{V_{m}}{\sqrt{(R)^{2}+(X_{C}-X_{L})^{2}}} \right ]^{2}\; \cos ^{2}(\omega t_{0}+\phi )

(c) When R is short-circuited, the circuit becomes an L-C oscillator. When the capacitor discharges, all its energy goes from the capacitor to the inductor. Energy oscillates from electrostatic to magnetic and from magnetic to electrostatic.

 

Posted by

infoexpert23

View full answer

Similar Questions

Latest Question