In an A.C circuit, the instantaneous current is 7A. What is the root mean square(r.m.s) value of the current I\(_{r.m.s}\)

An alternating current has no single fixed value: it grows to a maximum in one direction, falls to zero, grows to a maximum in the opposite direction, and repeats. To describe such a current with one useful number we quote its root-mean-square (r.m.s.) value, which is the steady direct current that would produce the same average heating effect in the same resistor. For a sinusoidal current the r.m.s. value is tied to the peak (maximum) value \(I_0\) by \[I_{r.m.s} = \frac{I_0}{\sqrt{2}} = 0.707\,I_0.\]

The single current value quoted in the question, 7 A, has to be read as the greatest value the current reaches, because an r.m.s. value can only be obtained from the peak. Substituting: \[I_{r.m.s} = \frac{7}{\sqrt{2}} = \frac{7}{1.414} = 4.95\ \text{A} \approx 5\ \text{A}.\] So the r.m.s. current is about 5 A.

Two slips account for most wrong answers here. Dividing by 2 instead of \(\sqrt{2}\) gives 3.5 A, and multiplying by \(\sqrt{2}\) gives 9.9 A, which is the route from r.m.s. back to peak rather than peak to r.m.s. A quick safety check in the exam: for a sinusoidal current the r.m.s. value is always about 70% of the peak, so it must come out smaller than the peak, never equal to it or larger.