 Solution

ID: past-exam-of-the-physics-2-paper-of-the-university-of-oxford/2015/8/b/solution

Assume a passive linear electrical circuit in sinusoidal steady state, with transients absent and

V_{0}

the peak voltage. Write

Z = ∣ Z ∣ e^{i ϕ} = R + i X

I (t) = I_{0} cos (ω t - ϕ)

, with

I_{0} = V_{0} /∣ Z ∣

. Average the instantaneous electric power

V (t) I (t)

over a full period. The double-frequency term averages to zero, leaving

⟨ P ⟩ = \frac{V _{0} I _{0}}{2} cos ϕ, ∣ Z ∣^{2} = R^{2} + X^{2}, cos ϕ = \frac{R}{∣ Z ∣} .

Therefore

⟨ P ⟩ = \frac{V _{0}^{2} R}{2 ( R ^{2} + X ^{2} )} = \frac{1}{2} I_{0}^{2} R .

Equivalently, root mean square values give

⟨ P ⟩ = V_{rms} I_{rms} cos ϕ

. The reactance exchanges stored electric field or magnetic field energy with the source, whereas the real part produces Joule heating. A purely imaginary complex impedance dissipates no mean electric power under these ideal assumptions.

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