Question

An alternating current is represented by the equation, $\mathrm{i}=100 \sqrt{2} \sin (100 \pi \mathrm{t})$ ampere. The RMS value of current and the frequency of the given alternating current are

• A. $100 \sqrt{2} \mathrm{~A}, 100 \mathrm{~Hz}$
• B. $\frac{100}{\sqrt{2}} \mathrm{~A}, 100 \mathrm{~Hz}$
• C. $100 \mathrm{~A}, 50 \mathrm{~Hz}$
• D. $50 \sqrt{2} \mathrm{~A}, 50 \mathrm{~Hz}$

Hint:

The RMS value of an alternating current is given by the peak value divided by $\sqrt{2}$.

Answer 1

The Correct Option is C

1. RMS value of current: \[ i_{\text{rms}} = \frac{i_0}{\sqrt{2}} = 100 \mathrm{~A} \]
2. Frequency of the current: \[ f = \frac{\omega}{2\pi} = \frac{100\pi}{2\pi} = 50 \mathrm{~Hz} \] Therefore, the correct answer is (3) $100 \mathrm{~A}, 50 \mathrm{~Hz}$.

Answer 2

We are given the alternating current equation:

\[ i = 100\sqrt{2} \sin(100\pi t) \]

We need to find the RMS value of the current and the frequency of the alternating current.

Concept Used:

The general equation of an alternating current is given by:

\[ i = I_0 \sin(\omega t) \]

where \( I_0 \) is the peak current (maximum current) and \( \omega = 2\pi f \) is the angular frequency. The RMS value of the current is:

\[ I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \]

Step-by-Step Solution:

Step 1: Identify the peak current \( I_0 \) from the given equation.

\[ I_0 = 100\sqrt{2} \, \text{A} \]

Step 2: Calculate the RMS value of the current using \( I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \).

\[ I_{\text{rms}} = \frac{100\sqrt{2}}{\sqrt{2}} = 100 \, \text{A} \]

Step 3: Determine the angular frequency \( \omega \) from the given equation.

\[ \omega = 100\pi \]

Step 4: Relate angular frequency to linear frequency \( f \) using \( \omega = 2\pi f \).

\[ 100\pi = 2\pi f \] \[ f = 50 \, \text{Hz} \]

Final Computation & Result:

The RMS current is \( 100 \, \text{A} \) and the frequency is \( 50 \, \text{Hz} \).

Final Answer: \( I_{\text{rms}} = 100 \, \text{A}, \, f = 50 \, \text{Hz} \)

Correct Option: (3) 100 A, 50 Hz