NCERT · Class 12 · Physics · Alternating CurrentExplain the concept of Root Mean Square (RMS) value of alternating current and derive its relationship with the peak value ($I0$).

The Root Mean Square (RMS) value of an alternating current is defined as that steady direct current (DC) which, when flowing through a given resistance for a given time period, produces the exact same amount of heat as is produced by the alternating current (AC) when flowing through the same resistance for the same duration of time. It is also referred to as the effective value or virtual value of alternating current ($I_{rms}$ or $I_{eff}$).

Derivation:\nLet the alternating current at any instant $t$ be given by: $$I = I_0 \sin(\omega t)$$ \nThe heat produced in a small time interval $dt$ across resistance $R$ is: $$dH = I^2 R dt = I_0^2 \sin^2(\omega t) R dt$$ \nThe total heat produced in one complete cycle of time period $T$ is: $$H = \int_0^T I_0^2 R \sin^2(\omega t) dt$$ $$H = I_0^2 R \int_0^T \frac{1 - \cos(2\omega t)}{2} dt$$\nSince the average of $\cos(2\omega t)$ over a full cycle is zero, we get: $$H = I_0^2 R \left( \frac{T}{2} \right) = \frac{I_0^2 R T}{2}$$ \nIf $I_{rms}$ is the steady current that produces the same heat $H$ in time $T$, then: $$H = I_{rms}^2 R T$$ \nEquating both equations of heat: $$I_{rms}^2 R T = \frac{I_0^2 R T}{2}$$ $$I_{rms}^2 = \frac{I_0^2}{2} \implies I_{rms} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0$$ \nHence, the RMS value of AC is equal to $0.707$ times or $70.7%$ of its peak value.