Question

The charge flowing through a resistance \( R \) varies with time as \( q = at - bt^2 \), where \( a, b \) are positive constants. The total heat produced in \( R \) is:

• A. \( \frac{a^3 R}{6b} \)
• B. \( \frac{a^3 R}{2b} \)
• C. \( \frac{a^3 R}{3b} \)
• D. \( \frac{a^3 R}{b} \)

Hint:

To calculate the total heat produced in a resistor, use the formula \( H = \int I^2 R \, dt \), where \( I = \frac{dq}{dt} \).

Answer 1

The Correct Option is A

The current is: \[ I = \frac{dq}{dt} = a - 2bt \] The power dissipated is: \[ P = I^2 R = (a - 2bt)^2 R \] The total heat produced is: \[ H = \int P \, dt = \int (a - 2bt)^2 R \, dt = \frac{a^3 R}{6b} \] Thus, the total heat produced is \( \frac{a^3 R}{6b} \).