Expression for the RMS value of an alternating EMF

In this short piece of the article, we are going to derive an expression for the RMS value of an alternating voltage, so let’s get started…

Root mean square value of an alternating emf

It is defined as that value of a steady voltage that produces the same amount of heat in a given resistance as is produced by the given alternating emf when applied to the same resistance for the same time. It is also called the virtual or effective value of the alternating emf. It is denoted by $\varepsilon_{r m s}$ or $\varepsilon_{e f f}$ or $\varepsilon_{v}$.

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Alternating current class 12 – amplitude, time period, frequency

Relation between the RMS value and the peak value of an alternating emf

RMS Voltage of a Sinusoidal AC Waveform
Relation between the RMS value and the peak value of an alternating emf

Suppose an alternating emf $\varepsilon$ applied to a resistance $R$ is given by $$ \varepsilon=\varepsilon_{0} \sin \omega t $$ Heat produced in a small time $d t$ will be $$ d H=\frac{\varepsilon^{2}}{R} d t=\frac{\varepsilon_{0}^{2}}{R} \sin ^{2} \omega t d t $$ Let $T$ be the time period of the alternating emf. Then heat produced in time $T$ will be

$$ \begin{aligned} H &=\int d H=\int_{0}^{T} \frac{\varepsilon_{0}^{2}}{R} \sin ^{2} \omega t d t \\ &=\frac{\varepsilon_{0}^{2}}{R} \int_{0}^{T} \frac{(1-\cos 2 \omega t)}{2} d t=\frac{E_{0}^{2}}{2 R}\left[t-\frac{\sin 2 \omega t}{2 \omega}\right]_{0}^{T} \\ &=\frac{\varepsilon_{0}^{2}}{2 R}\left[(T-0)-\frac{1}{2 \omega}\left|\sin \frac{4 \pi}{T} t\right|_{0}^{T}\right] \\ &=\frac{\varepsilon_{0}^{2}}{2 R}\left[T-\frac{1}{2 \omega} \sin (4 \pi-\sin 0)\right] \\ \text { or } \quad H &=\frac{\varepsilon_{0}^{2}}{2 R}[T-0]=\frac{\varepsilon_{0}^{2} T}{2 R} \end{aligned} $$

If $\varepsilon_{rms}$ is the root mean square value of the alternating emf, then the amount of heat produced by it in the same resistance $R$ in the time $T$ will be $$ H=\frac{\varepsilon_{rms}^{2} T}{R} $$

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From the above two equations, we get $$\frac{\varepsilon_{rms}^2 T}{R}=\frac{\varepsilon_{0}^2 T}{2R}$$ $$\text{Or}\qquad \varepsilon_{rms}=\frac{\varepsilon_0}{\sqrt{2}}=0.707\varepsilon_0$$

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