Two identical magnetic dipoles of magnetic moment `2 Am^(2)` are placed at a separation of `2m` with their axes perpendicular to each other in air. The resultant magnetic field at a midpoint between the dipoles is `x sqrt(y) xx 10^(-7) T. Find `(x+y)`.

Answer

To solve the problem, we need to calculate the resultant magnetic field at the midpoint between two identical magnetic dipoles placed at a separation of 2 meters, with their axes perpendicular to each other. The magnetic moment of each dipole is given as \( 2 \, \text{Am}^2 \).

### Step 1: Understand the Configuration
We have two magnetic dipoles, \( m_1 \) and \( m_2 \), each with a magnetic moment \( m = 2 \, \text{Am}^2 \). They are placed at a distance of \( d = 2 \, \text{m} \) apart, and their axes are perpendicular to each other.

### Step 2: Determine the Midpoint
The midpoint between the two dipoles is at a distance of \( 1 \, \text{m} \) from each dipole.

### Step 3: Calculate the Magnetic Field Due to One Dipole
The magnetic field \( B \) at a point along the axis of a magnetic dipole is given by the formula:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2m}{r^3} \]
where:
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \) (permeability of free space),
- \( m \) is the magnetic moment,
- \( r \) is the distance from the dipole to the point of interest.

For our case, at the midpoint (1 m from each dipole):
\[ B_1 = \frac{\mu_0}{4\pi} \cdot \frac{2 \cdot 2}{1^3} = \frac{\mu_0}{4\pi} \cdot \frac{4}{1} = \frac{4\mu_0}{4\pi} = \frac{\mu_0}{\pi} \]

### Step 4: Calculate the Magnetic Field Due to the Second Dipole
Since the second dipole is perpendicular to the first, the magnetic field at the midpoint due to the second dipole will also be calculated using the same formula, but it will be perpendicular to the first dipole's field:
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{4}{1} = \frac{\mu_0}{\pi} \]

### Step 5: Calculate the Resultant Magnetic Field
Since \( B_1 \) and \( B_2 \) are perpendicular to each other, we can find the resultant magnetic field \( B_R \) using the Pythagorean theorem:
\[ B_R = \sqrt{B_1^2 + B_2^2} = \sqrt{\left(\frac{\mu_0}{\pi}\right)^2 + \left(\frac{\mu_0}{\pi}\right)^2} = \sqrt{2 \left(\frac{\mu_0}{\pi}\right)^2} = \frac{\mu_0}{\pi} \sqrt{2} \]

### Step 6: Substitute the Value of \( \mu_0 \)
Now, substituting \( \mu_0 = 4\pi \times 10^{-7} \):
\[ B_R = \frac{4\pi \times 10^{-7}}{\pi} \sqrt{2} = 4 \times 10^{-7} \sqrt{2} \, \text{T} \]

### Step 7: Express in the Required Form
The problem states that the resultant magnetic field can be expressed as \( x \sqrt{y} \times 10^{-7} \, \text{T} \). Here, we have:
\[ B_R = 4 \sqrt{2} \times 10^{-7} \, \text{T} \]
Thus, \( x = 4 \) and \( y = 2 \).

### Step 8: Find \( x + y \)
Finally, we calculate:
\[ x + y = 4 + 2 = 6 \]

### Final Answer
The answer is \( 6 \).