r.m.s. value of current i=3+4sin `(omegat+pi//3) is

Options

A. 5A
B. `sqrt17A`
C. `(5)/(sqrt2A)`
D. `(7)/(sqrt2)A`

Answer

To find the root mean square (RMS) value of the current \( i = 3 + 4 \sin(\omega t + \frac{\pi}{3}) \), we will follow these steps:

### Step 1: Square the current expression
We start by squaring the current:
\[ i^2 = (3 + 4 \sin(\omega t + \frac{\pi}{3}))^2 \]
Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ i^2 = 3^2 + 2 \cdot 3 \cdot 4 \sin(\omega t + \frac{\pi}{3}) + (4 \sin(\omega t + \frac{\pi}{3}))^2 \]
Calculating each term:
\[ = 9 + 24 \sin(\omega t + \frac{\pi}{3}) + 16 \sin^2(\omega t + \frac{\pi}{3}) \]

### Step 2: Find the mean of \( i^2 \)
Next, we need to find the mean value of \( i^2 \). The mean of a constant is the constant itself, and the mean of \( \sin^2 \) over one complete cycle (period) is \( \frac{1}{2} \):
\[ \text{Mean of } i^2 = 9 + 24 \cdot 0 + 16 \cdot \frac{1}{2} \]
Calculating this:
\[ = 9 + 0 + 8 = 17 \]

### Step 3: Take the square root to find the RMS value
Finally, we take the square root of the mean value we just calculated:
\[ I_{\text{RMS}} = \sqrt{17} \]

### Final Answer
Thus, the RMS value of the current \( i \) is:
\[ I_{\text{RMS}} = \sqrt{17} \text{ Amperes} \]