What is meant by root mean square value of alternating current? Derive an expression for r.m.s. value of alternating current.

Root mean square (r.m.s.) or virtual value of a.c.: It is that steady current, which when passed through a resistance for a given time will produce the same amount of heat as the alternating current does in the same resistance and in the same time. It is denoted
I_rms or I_v.

Derivation of r.m.s. value of current:

The instantaneous value of a.c. passing through a resistance R is given by
                  I = I₀ sin ωt
The alternating current changes continuously with time.

Suppose, that the current through the resistance remains constant for an infinitesimally small time dt. 

Then, small amount of heat produced the resistance R in time dt is given by
dH = I² R dt
     = (1₀ sin ωt)² R dt
     = I₀² R sin² ωt dt. 

The amount of heat produced in the resistance in time T/2 is
                          $$\begin{gathered} \mathrm{H} = \underset{0}{\overset{\mathrm{T}/2}{\int }} {\mathrm{I}}_0^2 \mathrm{R} {\sin }^2 \mathrm{\omega t} \mathrm{dt} \\ = {\mathrm{I}}_0^2 \mathrm{R} \underset{0}{\overset{\mathrm{T}/2}{\int }}\frac{1-\cos 2\mathrm{\omega t}}{2}\mathrm{dt} \end{gathered}$$

                          $\mathrm{H} = \frac{{\mathrm{I}}_0^2\mathrm{R}}{2}{\left[\mathrm{t}-\frac{\sin 2\mathrm{\omega t}}{2\mathrm{\omega }}\right]}_0^{\mathrm{T}/2}$

                         $\mathrm{H} = \frac{{\mathrm{I}}_0^2\mathrm{R}}{2}\left[\frac{\mathrm{T}}{2}-\frac{\sin 2\mathrm{\omega t}.{\displaystyle \frac{\mathrm{T}}{2}}}{2\mathrm{\omega }}-0\right]$

                         $\mathrm{H} = \frac{{\mathrm{I}}_0^2\mathrm{R}}{2}\left[\frac{\mathrm{T}}{2}-\frac{\sin 2. {\displaystyle \frac{2\mathrm{\pi }}{\mathrm{T}}}.{\displaystyle \frac{\mathrm{T}}{2}}}{2\mathrm{\omega }}\right]$

                         $\mathrm{H} = \frac{{\mathrm{I}}_0^2\mathrm{R}}{2}\left[\frac{\mathrm{T}}{2}-\frac{\sin 2\mathrm{\pi }}{2\mathrm{\omega }}\right]$

                         $\mathrm{H} = \frac{{\mathrm{I}}_0^2\mathrm{R}}{2}\left[\frac{\mathrm{T}}{2}-\frac{\sin 2\mathrm{\pi }}{2\mathrm{\omega }}\right]$ 

                         $\mathrm{H} = \frac{{\mathrm{I}}_0^2\mathrm{R}}{2}.\frac{\mathrm{T}}{2}$                 ...(i)      $\left[\because \sin 2\mathrm{\pi } = 0\right]$
             
If I_rms be the r.m.s. value of a.c., then by definition,
                         $\mathrm{H} = {{\mathrm{I}}^2}_{\mathrm{rms}} \mathrm{R} \frac{\mathrm{T}}{2}$                                 ...(ii) 

From equtions (i) and (ii), we have
                      ${{\mathrm{I}}^2}_{\mathrm{rms}}\mathrm{R}\frac{\mathrm{T}}{2} = \frac{{\mathrm{I}}_0^2\mathrm{R}}{2}.\frac{\mathrm{T}}{2}$ 

                            ${{\mathrm{I}}^2}_{\mathrm{rms}} = \frac{{\mathrm{I}}_0^2}{2}$ 

                             ${\mathrm{I}}_{\mathrm{rms}} = \frac{{\mathrm{I}}_0}{\sqrt{2}} = 0.707{\mathrm{I}}_0.$