The charge flowing through a resistance R varies with time t...

The charge flowing through a resistance $R$ varies with time $t$ as $Q=a t-b t^{2} .$ The total heat produced in $R$ from $t=0$ to the time when value of $Q$ becomes again zero is
(A) $\frac{a^{3} R}{6 b}$
(B) $\frac{a^{3} R}{3 b}$
(C) $\frac{a^{3} R}{2 b}$
(D) $\frac{a^{3} R}{b}$

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Key Concepts

Determining Integration Limits from Physical Conditions

Often, physical conditions such as the times when a variable (like charge) returns to a certain value help establish the limits for integration. Identifying these limits is important to correctly compute quantities like total energy dissipation in the circuit.

Integration to Calculate Energy Dissipation

To find the total heat produced in a circuit where current varies with time, one must integrate the instantaneous power (I²R) over the time interval of interest. This application of calculus is fundamental in analyzing systems with non-constant currents.

Joule Heating (Resistive Heating)

Joule's law states that the heat produced in a resistor is proportional to the square of the current flowing through it and the resistance (P = I²R). Understanding this concept is essential for calculating the energy dissipated as heat in a resistor over a given period.

Current as the Derivative of Charge

In any electrical circuit, current is defined as the time rate of change of charge (I = dQ/dt). Recognizing this relationship is crucial when dealing with time-varying charges, as it allows one to determine how the flow of charge, and thus the current, evolves over time.

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