A coil of resistance R connected to an external battery is...

A coil of resistance $R$ connected to an external battery is placed inside an adiabatic cylinder fitted with a frictionless piston and containing an ideal gas. A current $I=a_{0} t$ flows through the coil $\left(a_{0}\right.$ is a teve constant). For time interval $t=0$ to $t=t_{0}$, the piston goes up to a height of $($ Assume $\Delta \mathrm{U}=0)$ (A) $\frac{a_{0}^{2} R^{2} t_{0}^{2}}{2 m g}$ (B) $\frac{a_{0}^{2} R t_{0}^{3}}{2 m g}$ (C) $\frac{a_{0}^{2} R t_{0}^{3}}{3 m g}$ (D) $\frac{a_{0}^{2} R t_{0}^{2}}{3 m g}$

A coil of resistance $R$ connected to an external battery is placed inside an adiabatic cylinder fitted with a frictionless piston and containing an ideal gas. A current $I=a_{0} t$ flows through the coil $\left(a_{0}\right.$ is a teve constant). For time interval $t=0$ to $t=t_{0}$, the piston goes up to a height of $($ Assume $\Delta \mathrm{U}=0)$
(A) $\frac{a_{0}^{2} R^{2} t_{0}^{2}}{2 m g}$
(B) $\frac{a_{0}^{2} R t_{0}^{3}}{2 m g}$
(C) $\frac{a_{0}^{2} R t_{0}^{3}}{3 m g}$
(D) $\frac{a_{0}^{2} R t_{0}^{2}}{3 m g}$

Key Concepts

Joule Heating

Joule heating describes the conversion of electrical energy into thermal energy when current flows through a resistor. The power dissipated in this way is proportional to the square of the instantaneous current multiplied by the resistance, and it represents a fundamental mechanism by which energy is transferred in electrical circuits.

Energy Conservation and the First Law of Thermodynamics

The first law of thermodynamics establishes that the total energy of an isolated system is conserved, relating changes in internal energy to the net heat added to the system and the work done by the system. In processes where the internal energy change is zero, any energy introduced into the system is entirely converted into work.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged between the system and its surroundings. In such processes, any work done by or on the system must come solely from energy conversions within the system, making the analysis of energy transfers, such as work done on a piston, straightforward.

Mechanical Work Against Gravity

Mechanical work against gravity involves lifting a mass, where the work done is given by the product of the gravitational force (or weight) and the vertical displacement. This concept is essential when energy from another source, like electrical heating, is converted into gravitational potential energy through the upward movement of a piston.

Integration of Time-Dependent Electrical Power

When the electrical current varies with time, the total energy dissipated by a resistor is obtained by integrating the time-dependent power (which is proportional to the square of the instantaneous current) over the duration of interest. This method allows for the calculation of cumulative energy inputs in systems with non-constant electrical conditions.

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