Q. A source of emf E = 10V and having negligible internal resistance is connected to a variable resistance. The resistance varies as shown in figure. The total charge that has passed through the resistor R during the time interval from t1 to t2 is
(a) 40 loge 4
(b) 30 loge 3
(c) 20 loge 2
(d) 10 loge 2
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Ans: (d)
Sol: Equation of straight line passing through the points (10 , 20) & (30 , 40)
$ \displaystyle R – 20 = \frac{40-20}{30-10} (t-10) $
R – 20 = (t-10)
R = t + 10
Since , dq = I dt
$ \displaystyle q = \int_{t_1}^{t_2} I dt $
$ \displaystyle q = \int_{10}^{20} \frac{10}{t+10} dt $ (since , I = E/R)
$ \displaystyle q = [10 log(t+10)]_{10}^{30} $
$ \displaystyle = 10 log(\frac{40}{20}) $
= 10 loge 2