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In the circuit shown in figure, The batt...

In the circuit shown in figure, The battery is an ideal one. With emf V. The capacitor is initially uncharged. The switch S is closed at time `t=0`.
(i) If `Q(t)=(CV)/(2)(1-e^(-(xt)/(3RC)))`, find x.
If the liming value of current in AB at `t to oo` is `(V)/(xR)`, then find x.

Text Solution

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The correct Answer is:
(i) 2
(ii) 2
Let at any time t charge on capacitor C be Q and currents are if as shown, since,charge Q will increase with time t Therefore,
(i)Applying Kinchhoff's second law in loop MNABM
`V=(i-i_1)R+iR or V=2iR-i_1R`….(i)

Similarly,applying Kishhoff's second in loop MNSTM ,we have
`V=i_1R+Q/C+iR` .....(ii)
Eliminating i from Eqs (i) and (ii), we get
`V=3i_1R+(2Q)/c or 3i_1R=V-(2Q)/c`
or `i_1=1/(3R)(V-(2Q)/C) or (dQ)/(dt)=1/(3R)(V-(2Q)/c)`
or `(dQ)/(V-(2Q)/C)=(dt)/(3R) or int_0^(Q) (dQ)/(V-(2Q)/C)=int_0^(t) (dt)/(3R)`
This equation given `Q=(CV)/2(1-e^(-2t//3RC))` (ii)`i_1=(dQ)/(dt)=(V)/(3R)e^(-2t//3RC)`
From Eq (i) `i=(V+i_1R)/(2R)=(V+V/3e^(-2t//3RC))/(2R)therefore ` Current through AB
`i_2=i-i_1=(V+V/3e^(-2t//3RC))/(2R)-v/(3R)e^(-2t//3RC, i_2=V/(2R)-V/(6R)e^(-2t//3RC)`
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