Question

The electric current flowing through a given conductor varies with time as shown in the graph below. The number of free electrons which flow through a given cross-section of the conductor in the time interval 0 ≤ t ≤ 20s is

• A. 3.125×10¹⁹
• B. 1.6×10¹⁹
• C. 6.25×10¹⁸
• D. 1.625×10¹⁸

Answer 1

The Correct Option is A

Given:

• Electric current varies with time as per the graph.
• Time interval: \( 0 \leq t \leq 20 \, \text{s} \)
• Charge of one electron: \( e = 1.6 \times 10^{-19} \, \text{C} \)

Step 1: Analyze the Graph

From the graph:

• From \( 0 \) to \( 10 \, \text{s} \): Current increases linearly from \( 100 \, \text{mA} \) to \( 300 \, \text{mA} \).
• From \( 10 \) to \( 20 \, \text{s} \): Current remains constant at \( 300 \, \text{mA} \).

Step 2: Calculate Total Charge

The total charge is equal to the area under the current-time graph:

From \( 0 \) to \( 10 \, \text{s} \): It's a trapezium (since current varies linearly):

\[ Q_1 = \frac{1}{2} \times (I_1 + I_2) \times t = \frac{1}{2} \times (100 + 300) \times 10 = 2000 \, \text{mC} \]

From \( 10 \) to \( 20 \, \text{s} \): It's a rectangle:

\[ Q_2 = I \times t = 300 \times 10 = 3000 \, \text{mC} \]

Total charge:

\[ Q = Q_1 + Q_2 = 2000 + 3000 = 5000 \, \text{mC} = 5 \, \text{C} \]

Step 3: Calculate Number of Free Electrons

\[ n = \frac{Q}{e} = \frac{5}{1.6 \times 10^{-19}} = 3.125 \times 10^{19} \]

The number of free electrons is \( {3.125 \times 10^{19}} \), so the correct answer is (A).