A group of electric lamps with a total power of 1000 W is supplied by an ac voltage \( E=200\sin(310t+60^\circ)\,\mathrm{V} \). The rms value of the current is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 5\sqrt{2}\,\mathrm{A} \)
Electric lamps are resistive, so the current is in phase with the voltage. The average power is then \( P=VI \), where \( V \) and \( I \) are rms values. The given expression has peak voltage \( V_m=200\,\mathrm{V} \). The angular frequency and the phase \( 60^\circ \) do not affect the rms values.
The rms voltage is:
\( \begin{aligned} V &= \frac{V_m}{\sqrt{2}} \\ &= \frac{200\,\mathrm{V}}{\sqrt{2}} \\ &= 100\sqrt{2}\,\mathrm{V} \end{aligned} \)
The rms current is then:
\( \begin{aligned} I &= \frac{P}{V} \\ &= \frac{1000\,\mathrm{W}}{100\sqrt{2}\,\mathrm{V}} \\ &= \frac{10}{\sqrt{2}}\,\mathrm{A} \\ &= 5\sqrt{2}\,\mathrm{A} \end{aligned} \)