Alternating Current

JEE Main level, single correct, medium

4 of 190

Question 4 of 190

The electric current in the circuit is given as i=io(t/T)i=i_{\mathrm{o}}(t / T). The r.m.s current for the period t=0t=0 to t=Tt=T is ____\_\_\_\_.

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Answer: (C) io3\frac{i_{o}}{\sqrt{3}}

The RMS current is calculated using Irms=1T∫0Ti2dtI_{rms} = \sqrt{\frac{1}{T}\int_0^T i^2 dt}.
Given i=i0(t/T)i = i_0(t/T):
Irms=1T∫0Ti02(tT)2dt=i02T3∫0Tt2dt=i02T3⋅T33I_{rms} = \sqrt{\frac{1}{T}\int_0^T i_0^2\left(\frac{t}{T}\right)^2 dt} = \sqrt{\frac{i_0^2}{T^3}\int_0^T t^2 dt} = \sqrt{\frac{i_0^2}{T^3} \cdot \frac{T^3}{3}}
=i023=i03= \sqrt{\frac{i_0^2}{3}} = \frac{i_0}{\sqrt{3}}.

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