Question

A current i₁ = i₀ sin ωt passes through a resistor of resistance R. How much thermal energy is produced in one time period? A current i₂ = −i₀ sin ωt passes through the resistor. How much thermal energy is produced in one time period? If i₁ and i₂ both pass through the resistor simultaneously, how much thermal energy is produced? Is the principle of superposition obeyed in this case?

Answer 1

The thermal energy produced for an AC circuit in one time period is given by,
$H={I_{rms}}^2\times R\times \frac{2\mathrm{\pi }}{\omega }$
For current, i₁ = i₀ sin ωt,
$I_{rms}=\frac{i_0}{\sqrt{2}}$
$\Rightarrow H=\frac{{i_0}^{2}R}{2}\times \frac{2\mathrm{\pi }}{\omega }=\frac{\mathrm{\pi }{i_0}^{2}R}{\omega }$

For current, i₂ = −i₀ sin ωt,
$I_{rms}=\frac{i_0}{\sqrt{2}}$
Hence, the same thermal energy will be produced due to this current.

Since, the direction of i₁ and i₂ are opposite and their magnitude is same, the net current through the resistor will become zero when both are passed together. Yes, the principle of superposition is obeyed in this case.