First slide

Power in a.c. circuit

Question

In an LR-circuit, the inductive reactance is equal to the resistance R of the circuit. An e.m.f. $E = E_{0} \cos (ωt)$ applied to the circuit. The power consumed in the circuit is

Moderate

A

$\frac{E_{0}^{2}}{R}$
B

$\frac{E_{0}^{2}}{2 R}$
C

$\frac{E_{0}^{2}}{4 R}$
D

$\frac{E_{0}^{2}}{8 R}$

Solution

$P = E_{rms} i_{rms} cosϕ = \frac{E_{0}}{\sqrt{2}} \times \frac{i_{0}}{\sqrt{2}} \times \frac{R}{Z}$
$\Rightarrow \frac{E_{0}}{\sqrt{2}} \times \frac{E_{0}}{Z \sqrt{2}} \times \frac{R}{Z} \Rightarrow P = \frac{E_{0}^{2} R}{2 Z^{2}}$
Given: $X_{L} = R, so Z = \sqrt{2} R \Rightarrow P = \frac{E_{0}^{2}}{4 R}$

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