The rms value of the emf given by ` E = 8 sin omegat + 6 sin 2omegat` .

Options

A. `5sqrt2 V`
B. `7sqrt2 V`
C. `10V`
D. `10sqrt2V`

Explanation

To find the RMS (Root Mean Square) value of the given EMF \( E = 8 \sin(\omega t) + 6 \sin(2\omega t) \), we will follow these steps:

### Step 1: Understanding the RMS Value
The RMS value of a function over one complete cycle is given by the formula:
\[ E_{\text{rms}} = \sqrt{\frac{1}{T} \int_0^T E^2 \, dt} \]
where \( T \) is the period of the function.

### Step 2: Calculate \( E^2 \)
First, we need to square the expression for \( E \):
\[ E^2 = (8 \sin(\omega t) + 6 \sin(2\omega t))^2 \]
Expanding this:
\[ E^2 = 64 \sin^2(\omega t) + 36 \sin^2(2\omega t) + 96 \sin(\omega t) \sin(2\omega t) \]

### Step 3: Simplifying the Terms
Now, we will evaluate the average of each term over one complete cycle:
1. The average of \( \sin^2(\omega t) \) over one complete cycle is \( \frac{1}{2} \).
2. The average of \( \sin^2(2\omega t) \) over one complete cycle is also \( \frac{1}{2} \).
3. The average of \( \sin(\omega t) \sin(2\omega t) \) over one complete cycle is \( 0 \) (since it is a product of sine functions with different frequencies).

### Step 4: Calculate the Mean Square Value
Using the averages calculated:
\[ \langle E^2 \rangle = 64 \cdot \frac{1}{2} + 36 \cdot \frac{1}{2} + 96 \cdot 0 \]
\[ \langle E^2 \rangle = 32 + 18 + 0 = 50 \]

### Step 5: Calculate the RMS Value
Now, we can find the RMS value:
\[ E_{\text{rms}} = \sqrt{\langle E^2 \rangle} = \sqrt{50} = 5\sqrt{2} \]

### Final Answer
Thus, the RMS value of the EMF is:
\[ E_{\text{rms}} = 5\sqrt{2} \text{ volts} \]

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