JEE Main & Advanced Physics Alternating Current / प्रत्यावर्ती धारा Important Values of Alternating Quantities

(1) Peak value (\[{{i}_{0}}\] or \[{{V}_{0}}\]) : The maximum value of alternating quantity (i or V) is defined as peak value or amplitude.

(2) Mean square value \[\mathbf{(}\,\overline{{{V}^{\mathbf{2}}}}\,or \overline{\,{{i}^{2}}}\,\mathbf{)}\] : The average of square of instantaneous values in one cycle is called mean square value. It is always positive for one complete cycle.  e.g.  \[\overline{{{V}^{2}}}=\frac{1}{T}\int_{0}^{T}{{{V}^{2}}dt}=\frac{V_{0}^{2}}{2}\] or \[\overline{\,{{i}^{2}}}=\frac{i_{0}^{2}}{2}\]

(3) Root mean square (r.m.s.) value : Root of mean of square of voltage or current in an ac circuit for one complete cycle is called r.m.s. value. It is denoted by \[{{V}_{rms}}\] or \[{{i}_{rms}}\]

\[{{i}_{rms}}=\sqrt{\frac{i_{1}^{2}+i_{2}^{2}+...}{n}}=\sqrt{\overline{{{i}^{2}}}}=\sqrt{\frac{\int_{\,0}^{\,T}{{{i}^{2}}dt}}{\int_{\,0}^{\,T}{dt}}}=\frac{{{i}_{0}}}{\sqrt{2}}\]\[=0.707\,{{i}_{0}}=70.7%\,\,{{i}_{0}}\]

Similarly \[{{V}_{rms}}=\frac{{{V}_{0}}}{\sqrt{2}}=0.707\,{{V}_{0}}=70.7%\,\]of \[{{V}_{0}}\]

\[\left[ \langle {{\sin }^{2}}(\omega \,t)\rangle =\langle {{\cos }^{2}}(\omega \,t)\rangle =\frac{1}{2} \right]\]

(i) The r.m.s. value of alternating current is also called virtual value or effective value.

(ii) In general when values of voltage or current for alternating circuits are given, these are r.m.s. value.

(iii) ac ammeter and voltmeter are always measure r.m.s. value. Values printed on ac circuits are r.m.s. values.

(iv) In our houses ac is supplied at 220 V, which is the r.m.s. value of voltage. It's peak value is \[\sqrt{2}\times 200=311V.\]

(v) r.m.s. value of ac is equal to that value of dc, which when passed through a resistance for a given time will produce the same amount of heat as produced by the alternating  current when passed through the same resistance for same time.

(4) Mean or Average value (\[{{i}_{av}}\] or \[{{V}_{av}}\]) : The average value of alternating quantity for one complete cycle is zero.

The average value of ac over half cycle (\[t=0\] to \[T/2\])

\[{{i}_{av}}=\frac{\int_{\,0}^{\,T/2}{\,i\,dt}}{\int_{\,0}^{\,T/2}{dt}}=\frac{2{{i}_{0}}}{\pi }=0.637{{i}_{0}}=63.7%\] of \[{{i}_{0}}\],

Similarly \[{{V}_{av}}=\frac{2{{V}_{0}}}{\pi }=0.637{{V}_{0}}=63.7%\] of \[{{V}_{0}}\].

(5) Peak to peak value : It is equal to the sum of the magnitudes of positive and negative peak values

\[\therefore \] Peak to peak value \[={{V}_{0}}+{{V}_{0}}=2{{V}_{0}}\]

\[=2\sqrt{2}\,{{V}_{rms}}=2.828\,{{V}_{rms}}\]

(6) Form factor and peak factor : The ratio of r.m.s. value of ac to it's average during half cycle is defined as form factor. The ratio of peak value and r.m.s. value is called peak factor