Question

The figure shows the plot of current through a cross-section of wire over two different time intervals. Compare the charges Q1 and Q2 that pass through the cross-section during these time intervals.

Answer 1

Charge Calculation from Current-Time Graph

Given:

• Charge \( Q \) is the area under the current-time (\( I \)-\( t \)) graph.
• Formula for charge: \[ Q = \int I(t) \, dt \]

Interval for \( Q_1 \): From \( t = 1 \, \text{s} \) to \( t = 3 \, \text{s} \), current is constant at 2 A

For this interval, the current is constant, so the area under the curve is simply a rectangle:

\[ Q_1 = I \times \Delta t = 2 \times (3 - 1) = 4 \, \text{C} \]

Interval for \( Q_2 \): From \( t = 4 \, \text{s} \) to \( t = 6 \, \text{s} \), current increases linearly from 0 to 2 A

For this interval, the current-time graph forms a triangle. The area of a triangle is given by:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

Substituting the values (base = 2 s, height = 2 A):

\[ Q_2 = \frac{1}{2} \times (6 - 4) \times 2 = 2 \, \text{C} \]

Comparison:

From the calculations above:

• Charge \( Q_1 \): \( 4 \, \text{C} \)
• Charge \( Q_2 \): \( 2 \, \text{C} \)

Thus, we have: \[ Q_1 > Q_2 \]