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Concepts Of Physics MCQ Edition [Volume 2]
Exam
Concepts Of Physics MCQ Edition [Volume 2]
Subject
Physics
Chapter
Electric Current in Conductors

Concepts Of Physics MCQ Edition [Volume 2] · Physics · Electric Current in Conductors

In a capacitor-charging circuit, let Q_1 denote the charge...

In a capacitor-charging circuit, let Q1Q_1 denote the charge delivered to the capacitor during an initial time interval of 10 ms10 \text{ ms}, and let Q2Q_2 denote the charge delivered during the subsequent 10 ms10 \text{ ms} interval. Additionally, assume that an initial 10μC10 \mu\text{C} of charge is deposited over a time interval t1t_1, while the next 10μC10 \mu\text{C} of charge is deposited over the following time interval t2t_2.

Options

  1. AQ1<Q2Q_1 < Q_2, t1>t2t_1 > t_2
  2. BQ1<Q2Q_1 < Q_2, t1<t2t_1 < t_2
  3. CQ1>Q2Q_1 > Q_2, t1>t2t_1 > t_2
  4. DQ1>Q2Q_1 > Q_2, t1<t2t_1 < t_2

Correct answer

D. Q1>Q2Q_1 > Q_2, t1<t2t_1 < t_2

Step-by-step solution

The charge on a capacitor during charging is given by q(t)=Q0(1−e−t/RC)q(t) = Q_0(1 - e^{-t/RC}).

The charging current is the rate of flow of charge, given by i(t)=dqdt=Q0RCe−t/RCi(t) = \dfrac{dq}{dt} = \dfrac{Q_0}{RC} e^{-t/RC}.

Since the current i(t)i(t) decreases exponentially with time, the rate at which charge is delivered to the capacitor continuously decreases.

For a fixed time interval, the charge delivered is the area under the i(t)i(t) curve. Because i(t)i(t) is a decreasing function, the charge delivered in the first 10 ms10 \text{ ms} interval is greater than the charge delivered in the subsequent 10 ms10 \text{ ms} interval. Thus, Q1>Q2Q_1 > Q_2.

Similarly, to deposit a fixed amount of charge (10μC10 \mu\text{C}), it takes less time initially when the current is high, and more time later when the current is lower. Therefore, the time t1t_1 required to deposit the first 10μC10 \mu\text{C} is less than the time t2t_2 required to deposit the next 10μC10 \mu\text{C}. Thus, t1<t2t_1 < t_2.
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