The magnetic moments associated with two closely wound circular coils `A` and `B` of radius `r_(A)=10cm` and `r_(B)=20cm` respectively are equal if: (Where `N_(A),I_(A)` and `N_(B),I_(B)`.Is are number of turn and current of `A` and `B` respectively)

Options

A. `2N_(A)I_A=N_BI_(B)`
B. `N_(A)=2N_(B)`
C. `N_(A)I_(A)=4N_BI_B `
D. `4N_(A)I_A=N_BI_B`

Answer

To solve the problem, we need to determine the relationship between the magnetic moments of two circular coils, A and B, given their respective radii and the conditions that their magnetic moments are equal.

### Step-by-Step Solution:

1. **Understand the Magnetic Moment Formula**:
The magnetic moment \( m \) of a circular coil is given by the formula:
\[ m = N \cdot I \cdot A \]
where \( N \) is the number of turns, \( I \) is the current, and \( A \) is the area of the coil.

2. **Calculate the Area of Each Coil**:
The area \( A \) of a circular coil is given by:
\[ A = \pi r^2 \]
For coil A:
\[ A_A = \pi r_A^2 = \pi (0.1)^2 = \pi \times 0.01 \, \text{m}^2 \]
For coil B:
\[ A_B = \pi r_B^2 = \pi (0.2)^2 = \pi \times 0.04 \, \text{m}^2 \]

3. **Write the Magnetic Moment Equations**:
The magnetic moments for coils A and B can be expressed as:
\[ m_A = N_A \cdot I_A \cdot A_A = N_A \cdot I_A \cdot \pi \cdot (0.1)^2 \]
\[ m_B = N_B \cdot I_B \cdot A_B = N_B \cdot I_B \cdot \pi \cdot (0.2)^2 \]

4. **Set the Magnetic Moments Equal**:
Since the magnetic moments are equal, we have:
\[ m_A = m_B \]
Substituting the expressions for \( m_A \) and \( m_B \):
\[ N_A \cdot I_A \cdot \pi \cdot (0.1)^2 = N_B \cdot I_B \cdot \pi \cdot (0.2)^2 \]

5. **Cancel Out Common Terms**:
We can cancel \( \pi \) from both sides:
\[ N_A \cdot I_A \cdot (0.1)^2 = N_B \cdot I_B \cdot (0.2)^2 \]
Simplifying further gives:
\[ N_A \cdot I_A \cdot 0.01 = N_B \cdot I_B \cdot 0.04 \]

6. **Rearranging the Equation**:
Dividing both sides by 0.01:
\[ N_A \cdot I_A = 4 \cdot N_B \cdot I_B \]

7. **Conclusion**:
The relationship between the number of turns and current for the two coils is:
\[ N_A \cdot I_A = 4 \cdot N_B \cdot I_B \]

### Final Answer:
The magnetic moments associated with the two coils A and B are equal if:
\[ N_A \cdot I_A = 4 \cdot N_B \cdot I_B \]