Question

The root mean square (r.m.s) value of alternating current equation I = 60 sin 100 πt is

• A. 60√2
• B. 30
• C. 100
• D. Zero
• E. None of these

Hint:

The RMS value for a sinusoidal waveform is always the peak value divided by \(\sqrt{2}\) (approximately 0.707 times the peak value). Conversely, the peak value is \(\sqrt{2}\) times the RMS value. This is a crucial relationship in AC circuit analysis. When your answer doesn't match, double-check your calculation and then look for plausible typos in the question.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The root mean square (r.m.s) value of an alternating current (AC) represents the effective DC value that would deliver the same average power to a resistor. For a sinusoidal current, it is directly related to its peak (maximum) value.
Step 2: Key Formula or Approach:
The general equation for a sinusoidal AC is given by: \[ I(t) = I_0 \sin(\omega t) \] where \(I_0\) is the peak amplitude or maximum current.
The r.m.s value (\(I_{\text{rms}}\)) is related to the peak current by the formula: \[ I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \] Step 3: Detailed Explanation:
The given equation for the current is \(I = 60 \sin(100 \pi t)\).
By comparing this with the standard form \(I = I_0 \sin(\omega t)\), we can identify the peak current: \[ I_0 = 60 \, \text{A} \] Now, we apply the formula for the r.m.s value: \[ I_{\text{rms}} = \frac{I_0}{\sqrt{2}} = \frac{60}{\sqrt{2}} \] To simplify this expression, we rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{2}\): \[ I_{\text{rms}} = \frac{60}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{60\sqrt{2}}{2} = 30\sqrt{2} \, \text{A} \] Step 4: Final Answer:
The calculated r.m.s value is \(30\sqrt{2}\) A, which is approximately \(42.4\) A. This value does not match any of the given options. There appears to be an error in the question or the options. In such cases in an exam, it is common that there was a typo in the question. For instance, if the equation had been \(I = 30\sqrt{2} \sin(100 \pi t)\), then \(I_{\text{rms}}\) would be \(\frac{30\sqrt{2}}{\sqrt{2}} = 30\) A, matching option (B). Given the choices, it's highly probable this was the intended question. However, based strictly on the question provided, none of the options is correct.