A square coil of side 10 cm is rotating about its vertical axis in a region of uniform magnetic field of 0.2 T. The angular speed of coil is 35 rad `s^(-1)`. Calculate the rms value of emf induced in the coil. Also find the power dissipated if resistance of coil is 6 `omega` . Number of turns in coil is 10.

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the area of the square coil
The side length of the square coil is given as 10 cm. We need to convert this to meters for our calculations.

\[ \text{Side length} = 10 \, \text{cm} = 0.1 \, \text{m} \]

The area \( A \) of the square coil is given by:

\[ A = \text{side}^2 = (0.1 \, \text{m})^2 = 0.01 \, \text{m}^2 \]

### Step 2: Use the formula for induced EMF
The induced EMF (\( E \)) in a rotating coil in a magnetic field can be calculated using the formula:

\[ E = N \cdot B \cdot A \cdot \omega \cdot \sin(\theta) \]

Where:
- \( N \) = number of turns = 10
- \( B \) = magnetic field strength = 0.2 T
- \( A \) = area of the coil = 0.01 m²
- \( \omega \) = angular speed = 35 rad/s
- \( \theta \) = angle between the magnetic field and the area vector (maximum value is when \( \sin(\theta) = 1 \))

### Step 3: Calculate the peak EMF
Substituting the values into the formula for peak EMF:

\[ E_0 = N \cdot B \cdot A \cdot \omega \]

\[ E_0 = 10 \cdot 0.2 \cdot 0.01 \cdot 35 \]

Calculating this gives:

\[ E_0 = 10 \cdot 0.2 \cdot 0.01 \cdot 35 = 0.07 \, \text{V} \]

### Step 4: Calculate the RMS value of the induced EMF
The RMS value of the induced EMF is given by:

\[ E_{\text{RMS}} = \frac{E_0}{\sqrt{2}} \]

Substituting the peak EMF into this formula:

\[ E_{\text{RMS}} = \frac{0.07}{\sqrt{2}} \approx 0.0495 \, \text{V} \approx 0.05 \, \text{V} \]

### Step 5: Calculate the power dissipated in the coil
The power \( P \) dissipated in the coil can be calculated using:

\[ P = \frac{E_{\text{RMS}}^2}{R} \]

Where \( R \) is the resistance of the coil, given as 6 ohms.

Substituting the values:

\[ P = \frac{(0.05)^2}{6} = \frac{0.0025}{6} \approx 0.00041667 \, \text{W} \approx 0.00042 \, \text{W} \]

### Final Answers
- The RMS value of the induced EMF is approximately **0.05 V**.
- The power dissipated in the coil is approximately **0.00042 W**.