A coil of 16 H is coupled with a coil of 4 H in such a way that coefficient of coupling is unity. Find the mutual inductance. 

Explanation:

To determine the mutual inductance (M) between two coils, we can use the formula:

Formula: \( M = k \sqrt{L_1 \times L_2} \)

Where:

• M is the mutual inductance.
• k is the coefficient of coupling.
• L₁ is the inductance of the first coil.
• L₂ is the inductance of the second coil.

Given the data:

• Inductance of the first coil, L₁ = 16 H
• Inductance of the second coil, L₂ = 4 H
• Coefficient of coupling, k = 1

Let's calculate the mutual inductance using the given values:

Step-by-Step Solution:

1. First, we identify the given values in the problem:

• Inductance of the first coil, L₁ = 16 H
• Inductance of the second coil, L₂ = 4 H
• Coefficient of coupling, k = 1

2. Using the formula for mutual inductance, \( M = k \sqrt{L_1 \times L_2} \), we substitute the given values:

• \( M = 1 \times \sqrt{16 \times 4} \)

3. Calculate the product inside the square root:

• \( 16 \times 4 = 64 \)

4. Calculate the square root of the product:

• \( \sqrt{64} = 8 \)

5. Finally, multiply by the coefficient of coupling (which is 1 in this case):

• \( M = 1 \times 8 \)

So, the mutual inductance M is 8 H.

Conclusion:

The mutual inductance between the two coils is 8 H, which matches Option 2.

Correct Option Analysis:

The correct option is:

Option 2: 8 H

This option correctly represents the calculated mutual inductance based on the provided inductances and the coefficient of coupling.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: 10 H

This option is incorrect. By following the same calculation steps, it is clear that the mutual inductance should be calculated as \( M = 1 \times \sqrt{16 \times 4} = 8 H \). Therefore, the value 10 H is not supported by the given data and formula.

Option 3: 12 H

This option is also incorrect. The mutual inductance calculation yields 8 H, not 12 H. The value 12 H does not align with the correct calculation using \( M = k \sqrt{L_1 \times L_2} \) with the given parameters.

Option 4: 16 H

This option is incorrect. The mutual inductance is calculated to be 8 H. The value 16 H does not match the correct calculation. It appears to be a misunderstanding or misapplication of the given formula.

Conclusion:

Understanding the calculation of mutual inductance is essential for accurately determining the interaction between two coupled coils. The mutual inductance is influenced by the inductance of each coil and the coefficient of coupling. By correctly applying the formula \( M = k \sqrt{L_1 \times L_2} \), we can ensure accurate results and avoid common pitfalls in the calculation process.