The current in a circuit is given by \( I=5\cos(10\pi t) \) A. What is the average value of the current over one complete cycle?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → zero
A sinusoidal current has equal positive and negative values in each cycle, so over one complete cycle the instantaneous values add up to zero. Here \( I=5\cos(10\pi t)\,\mathrm{A} \), so \( I_0=5\,\mathrm{A} \) and \( \omega=10\pi\,\mathrm{rad\,s^{-1}} \).
The time period of the current is:
\( \begin{aligned} T &= \frac{2\pi}{\omega} \\ &= \frac{2\pi}{10\pi\,\mathrm{rad\,s^{-1}}} \\ &= 0.2\,\mathrm{s} \end{aligned} \)
The average over one period is:
\( \begin{aligned} I_{\mathrm{av}} &= \frac{1}{T}\int_0^{T}5\cos(\omega t)\,dt \\ &= \frac{5}{\omega T}\Big[\sin(\omega t)\Big]_0^{T} \\ &= \frac{5}{2\pi}\,\big(\sin2\pi-\sin0\big) \\ &= 0 \end{aligned} \)
Therefore, the average value of the current over one complete cycle is zero.
NCERT (2026-27):
AC Voltage Applied to a Resistor – “We see that, like the applied voltage, the current varies sinusoidally and has corresponding positive and negative values during each cycle. Thus, the sum of the instantaneous current values over one complete cycle is zero, and the average current is zero.”