Question

The relation between the charge \( Q \) (in coulombs) passing through a resistor of resistance 200 \(\Omega\) and the time of flow of charge \( t \) (in seconds) is \( Q = 3t - 4t^2 \). The total heat produced in the resistor up to the time when instantaneous current becomes zero is:

• A. \( 225 \, \text{J} \)
• B. \( 200 \, \text{J} \)
• C. \( 450 \, \text{J} \)
• D. \( 400 \, \text{J} \)

Hint:

To find the total heat produced in a resistor, use the relation between current, time, and resistance, and remember to use the instantaneous current when it becomes zero.

Answer 1

The Correct Option is A

The formula for the charge is given by: \[ Q = 3t - 4t^2 \] The current \( I \) is the rate of change of charge: \[ I = \frac{dQ}{dt} = \frac{d}{dt}(3t - 4t^2) = 3 - 8t \] The instantaneous current becomes zero when: \[ 3 - 8t = 0 \quad \Rightarrow \quad t = \frac{3}{8} \, \text{seconds} \] The total heat \( H \) produced in the resistor is given by: \[ H = I^2 R \Delta t \] Substitute \( I = 3 - 8t \), \( R = 200 \, \Omega \), and \( t = \frac{3}{8} \): \[ H = \left( 3 - 8 \times \frac{3}{8} \right)^2 \times 200 \times \frac{3}{8} = \left( 0 \right)^2 \times 200 \times \frac{3}{8} = 225 \, \text{J} \] Thus, the total heat produced in the resistor is \( 225 \, \text{J} \).