A coil has a resistance of $$ 10\Omega $$ and an inductance of $$0.4$$ henry. It is connected to an AC source of $$6.5V,\dfrac{30}{\pi }$$Hz. Find the average power consumed in the circuit.
$$R=10\Omega $$ L= 0.4 Henny
E=6.5 V $$\displaystyle f=\frac{30}{\pi }Hz$$
$$\displaystyle Z=\sqrt{R^{2}+X_{L}^{2}}=\sqrt{R^{2}+\left ( 2\pi fL \right )^{2}}$$
$$ Power = V_{rms}I_{rms}\cos \phi $$
$$\displaystyle = 6.5\times \frac{6.5}{Z}\times \frac{R}{Z}=\frac{6.5\times 6.5\times 10}{\left [ \sqrt{R^{2}+\left ( 2\pi fL \right )^{2}} \right ]^{2}}=\frac{6.5\times 6.5\times 10}{10^{2}\left ( 2\pi \times \frac{30}{\pi \times 0.4} \right )^{2}}=\frac{6.5\times 6.5\times 10}{100+576}=0.625=\frac{5}{8\ }W$$
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