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 t_{1} to t_{2} is

(a) 40 log* _{e}* 4

(b) 30 log* _{e}* 3

(c) 20 log* _{e}* 2

(d) 10 log* _{e}* 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 log* _{e}* 2