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कर्नाटक बोर्ड पी.यू.सी.पीयूसी विज्ञान 2nd PUC Class 12

In the LCR circuit shown in figure, the ac driving voltage is v = vm sin ωt. Write down the equation of motion for q (t). At t = t0, the voltage source stops and R is short circuited - Physics

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प्रश्न

In the LCR circuit shown in figure, the ac driving voltage is v = vm sin ωt.

  1. Write down the equation of motion for q (t).
  2. At t = t0, the voltage source stops and R is short circuited. Now write down how much energy is stored in each of L and C.
  3. Describe subsequent motion of charges.

दीर्घउत्तर
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उत्तर

i. Consider series LCR circuit and tapping key K to short circuit R. Let i be the current in circuit. Then by Kirchhoff's voltage law, when key K is open.

VR+ VL + VC = Vm sin ωt

`iR + L (di(t))/(dt) + (q(t))/C - V_m` sin ωt = 0  .....[∵ i(t) = i = lm sin[ωt + `phi`]

⇒ As charge q(t) changes in circuit with time in AC,

Then `i = (dq(t))/(dt)`

`(di)/(dt) = (d^2q(t))/(dt^2)`    .....(Differentiating again)

`R (dq(t))/(dt) + L (d^2q(t))/(dt^2) + (q(t))/C = V_m` sin ωt

`L (d^2q(t))/(dt) + R (dq(t))/(dt) + (q(t))/C = V_m` sin ωt

This is the equation for variation of motion of charge with respect to time.

ii. Let time-dependent charge in circuit is at phase angle with voltage then q = qm cos (ωt + `phi`)

i = `(dq)/(dt) = ω q_m sin (ωt + phi)`  ......(I)

`i_m = V_m/Z = V_m/sqrt(R^2 + (X_C - X_L)^2)`  ......(II)

`tan phi = (X_C - X_L)/R`

At t = t0, R is short-circuited, then energy stored in L and C, when K is closed will be, `U_L = 1/2 Li`  ......(III)

At t = t0

i = im sin (ωt0 + `phi`)  ......(IV)

From (II)

`i = V_m/sqrt(R^2 + (X_C - X_L)^2) sin (ωt_0 + phi)`  ......(V)

∴ UL = `1/2[V_m/sqrt(R^2 + (X_C - X_L)^2]]^2 sin^2(ωt_0 + phi)`

UC = `q^2/(2C) = 1/(2C) [q_m^2  cos^2 (ωt_0 + pji)]`

Comparing (IV) and (I) Im = qmω

∴ `q_m = i_m/ω`

∴ UC = `1/(2C) * (i_m^2)/ω^2 cos^2 (ωt_0 + phi) = (i_m^2)/(2Cω^2) cos^2 (ωt_0 + phi)`

Using equation (II)

UC = `1/(2Cω^2) [(V_m^2)/(R^2 + (X_C - X_L)^2)]^2 cos^2 (ωt_0 + phi)`

iii. When R is short-circuited, the circuit becomes L-C oscillator. The capacitor will go discharging and all energy will transfer to L and back and forth. Hence there is oscillation of energy from electrostatic to magnetic and vice versa.

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अध्याय 7: Alternating Current - MCQ I [पृष्ठ ४६]

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एनसीईआरटी एक्झांप्लर Physics [English] Class 12
अध्याय 7 Alternating Current
MCQ I | Q 7.31 | पृष्ठ ४६

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