The electric current in the circuit is given as $i=i_0(t / T)$. The r.m.s current for the period $t=

Question

The electric current in the circuit is given as $i=i_0(t / T)$. The r.m.s current for the period $t=0$ to $t=T$ is $\_\_\_\_$ .

Professor PC Thomas Classes · Answer

The root mean square (RMS) current is defined by the relation $I_{rms}^2 = \frac{1}{T} \int_0^T i^2 dt$. Substituting the given function $i = i_0 \frac{t}{T}$, we get $I_{rms}^2 = \frac{1}{T} \int_0^T \left(i_0 \frac{t}{T}\right)^2 dt = \frac{i_0^2}{T^3} \int_0^T t^2 dt = \frac{i_0^2}{T^3} \left[ \frac{t^3}{3} \right]_0^T = \frac{i_0^2}{T^3} \cdot \frac{T^3}{3} = \frac{i_0^2}{3}$. Taking the square root gives $I_{rms} = \frac{i_0}{\sqrt{3}}$.