Physics

Physics

Problem

An alternating emf applied across a purely resistive load is given by v=V0sin⁡ωtv = V_0 \sin \omega t, and the resulting current is i=I0sin⁡ωti = I_0 \sin \omega t. Attempt all parts: a. (a) Write the expression for the instantaneous power p(t)p(t) supplied to the resistor in terms of V0V_0, I0I_0 and ωt\omega t. b. (b) Show, by integrating p(t)p(t) over one complete cycle, that the average power ⟨P⟩\langle P \rangle supplied to the resistor is V0I02\dfrac{V_0 I_0}{2}. c. (c) Given V0=311 VV_0 = 311\,\text{V} (which corresponds to an rms value of 220 V) and I0=5.0 AI_0 = 5.0\,\text{A}, calculate the average power delivered to the resistor. d. (d) If the amplitude of the applied voltage is doubled while the resistance of the load remains unchanged, by what factor will the average power change? Justify your answer.
Intuitive explanation

In a purely resistive circuit, voltage and current oscillate in perfect sync—they reach their peaks and zeros at exactly the same time. This is the key insight: resistance creates no phase shift. Use Ohm's law applied to the instantaneous values to relate the amplitudes, and remember that average power depends on the product of voltage and current averaged over a complete cycle, not just their peak values multiplied together.

Intuitive explanation

In a purely resistive circuit, voltage and current oscillate in perfect sync—they reach their peaks and zeros at exactly the same time. This is the key insight: resistance creates no phase shift. Use Ohm's law applied to the instantaneous values to relate the amplitudes, and remember that average power depends on the product of voltage and current averaged over a complete cycle, not just their peak values multiplied together.

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