In an ideal sinusoidal alternating current (AC), which statement is NOT true?

• A. The root-mean-square current satisfies $I_{\text{rms}}=\frac{I_0}{\sqrt{2}}$.
• B. The average value of the current over one complete cycle is $0$.
• C. The current has a nonzero net (DC) component over each full cycle, so $\frac{1}{T}{\int }_0^{T}i\left(t\right)dt\ne 0$.
• D. The instantaneous current can be written as $i\left(t\right)=I_0\sin (\omega t+\phi )$.

Answer: (C)

Verified step by step guidance

1. Recall that for an ideal sinusoidal alternating current (AC), the instantaneous current is given by the equation \(i(t) = I_0 \sin(\omega t + \phi)\), where \(I_0\) is the peak current, \(\omega\) is the angular frequency, and \(\phi\) is the phase angle.
2. Understand that the root-mean-square (rms) current \(I_{rms}\) is related to the peak current \(I_0\) by the formula \(I_{rms} = \frac{I_0}{\sqrt{2}}\). This relationship comes from the definition of rms as the square root of the average of the square of the instantaneous current over one full cycle.
3. Recognize that the average value of the current over one complete cycle of a pure sinusoidal AC is zero, because the positive half-cycle cancels out the negative half-cycle when integrated over the period \(T\).
4. Analyze the statement about the net (DC) component: for a pure sinusoidal AC, the net or average current over one full cycle is zero, meaning there is no DC component. Mathematically, this is expressed as \(\frac{1}{T} \int_0^T i(t) \, dt = 0\).
5. Conclude that the statement claiming a nonzero net (DC) component over each full cycle is NOT true for an ideal sinusoidal AC, since the integral of \(i(t)\) over one period must be zero.