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

Calculation of Charge from Current-Time Graph

Given Information:

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

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

In this interval, the constant current forms a rectangular area:

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

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

For this interval, the current-time graph depicts a triangle. The area of a triangle is calculated as:

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

Using the given values (base = \( 6 - 4 = 2 \, \text{s} \), height = 2 A):

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

Comparison of Charges:

Based on the calculations:

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

Therefore, the relationship between \( Q_1 \) and \( Q_2 \) is: \[ Q_1 > Q_2 \]