Concept:

The power dissipated through resistor, in a circuit consist of voltage source and a resistor.

P = V² / R = I²R

Where P = Power

V = Source voltage

I = Current through resistance

R = Resistor

Calculation:

Given, V = 105 sin 377t V

R = 10 Ω 

P = V2 / R = (105 sin 377t)² / 15 = 735 sin2 377t W

The instantaneous power in ac circuits can be obtained by taking the product of the instantaneous values of current and voltage.

The current waveform in this case can be represented as

i(t) = I sin (ωt - ϕ)

Now, the expression of the instantaneous power,

p (t) = u(t).i(t)

⇒ p (t) = V sin ωt I sin (ωt - ϕ)

⇒ p (t) = VI sin ωt sin (ωt - ϕ)

⇒ p (t) = VI sin ωt (sin ωt cos ϕ – cos ωt sin ϕ)

⇒ p (t) = VI (sin2ωt cos ϕ – sin ωt cos ωt sin ϕ)

\(\Rightarrow p\left( t \right) = \frac{V}{{\sqrt 2 }}.\frac{I}{{\sqrt 2 }}\left( {2{{\sin }^2}\omega t\cos \phi - 2\sin \omega t\cos \omega t\sin \phi } \right)\)

\(\Rightarrow p\left( t \right) = {V_{rms}}{I_{rms}}\cos \phi \left( {1 - \cos 2\omega t} \right) - {V_{rms}}{I_{rms}}\sin \phi \sin 2\omega t]\)

[Where Vrms and Irms is the root mean square value of voltage and current waveform]

= P (1 – cos 2ωt) – Q sin 2ωt