Two identical magnetic dipoles of magnetic moments `2Am^2` are placed at a separation of `2m` with their axes perpendicular to each other in air. The resultant magnetic field at a mid point between the dipole is

Options

A. `4sqrt(5)xx10^(-5)` T
B. `2sqrt(5)xx10^(-5)` T
C. `4sqrt(5)xx10^(-7)` T
D. `2sqrt(5)xx10^(-7)` T

Explanation

To find the resultant magnetic field at the midpoint between two identical magnetic dipoles with their axes perpendicular to each other, we can follow these steps:

### Step 1: Understand the Configuration
We have two identical magnetic dipoles, each with a magnetic moment \( M = 2 \, \text{Am}^2 \). They are separated by a distance of \( 2 \, \text{m} \), and we need to find the magnetic field at the midpoint between them.

### Step 2: Determine the Distance to the Midpoint
Since the dipoles are \( 2 \, \text{m} \) apart, the distance from each dipole to the midpoint \( P \) is:
\[ R = \frac{2}{2} = 1 \, \text{m} \]

### Step 3: Calculate the Magnetic Field Due to Each Dipole
The magnetic field \( B \) due to a magnetic dipole at a distance \( R \) is given by the formula:
\[ B = \frac{\mu_0 M}{4 \pi R^3} \]
where \( \mu_0 = 4 \pi \times 10^{-7} \, \text{T m/A} \).

For each dipole, substituting \( M = 2 \, \text{Am}^2 \) and \( R = 1 \, \text{m} \):
\[ B = \frac{(4 \pi \times 10^{-7}) \times 2}{4 \pi \times (1)^3} = \frac{2 \times 10^{-7}}{1} = 2 \times 10^{-7} \, \text{T} \]

### Step 4: Determine the Direction of the Magnetic Fields
Since the axes of the dipoles are perpendicular, the magnetic field \( B_1 \) due to dipole 1 will be in one direction (say along the x-axis), and the magnetic field \( B_2 \) due to dipole 2 will be in the perpendicular direction (along the y-axis).

### Step 5: Calculate the Resultant Magnetic Field
The resultant magnetic field \( B_P \) at point \( P \) can be calculated using the Pythagorean theorem:
\[ B_P = \sqrt{B_1^2 + B_2^2} \]
Substituting \( B_1 = 2 \times 10^{-7} \, \text{T} \) and \( B_2 = 2 \times 10^{-7} \, \text{T} \):
\[ B_P = \sqrt{(2 \times 10^{-7})^2 + (2 \times 10^{-7})^2} = \sqrt{2 \times (2 \times 10^{-7})^2} = 2 \times 10^{-7} \sqrt{2} \]

### Step 6: Final Calculation
Calculating the resultant:
\[ B_P = 2 \times 10^{-7} \sqrt{2} \approx 2 \times 10^{-7} \times 1.414 = 2.828 \times 10^{-7} \, \text{T} \]

### Conclusion
The resultant magnetic field at the midpoint between the two dipoles is approximately:
\[ B_P \approx 2.828 \times 10^{-7} \, \text{T} \]