Two identical magnetic dipoles of magnetic moments `1*0Am^2` each are placed at a separation of `2m` with their axes perpendicular to each other. What is the resultant magnetic field at a point midway between the dipoles?

Options

A. `sqrt(5) xx 10^(-7)T`
B. `5 xx 10^(-7)T`
C. `sqrt(2) xx 10^(-7)T`
D. `10^(-7)T`

Answer

To solve the problem of finding the resultant magnetic field at a point midway between two identical magnetic dipoles, we can follow these steps:

### Step 1: Understand the Configuration
We have two magnetic dipoles, each with a magnetic moment \( m = 1 \, \text{A m}^2 \), placed at a separation of \( 2 \, \text{m} \). The axes of the dipoles are perpendicular to each other. We need to find the magnetic field at the midpoint between the two dipoles.

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

### Step 3: Calculate \( B_1 \) (Field from Dipole 1)
Since the midpoint is \( 1 \, \text{m} \) away from Dipole 1:
\[ B_1 = \frac{\mu_0}{4\pi} \cdot \frac{2m}{(1)^3} = \frac{10^{-7}}{1} \cdot 2 \cdot 1 = 2 \times 10^{-7} \, \text{T} \]
This field is directed away from Dipole 1.

### Step 4: Calculate \( B_2 \) (Field from Dipole 2)
For Dipole 2, which is also \( 1 \, \text{m} \) away from the midpoint:
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{m}{(1)^3} = \frac{10^{-7}}{1} \cdot 1 = 1 \times 10^{-7} \, \text{T} \]
This field is directed towards Dipole 2.

### Step 5: Determine the Direction of the Fields
Since the dipoles are perpendicular to each other:
- \( B_1 \) is directed along the axis of Dipole 1.
- \( B_2 \) is directed along the axis of Dipole 2.

### Step 6: Calculate the Resultant Magnetic Field
The resultant magnetic field \( B_{\text{net}} \) can be found using the Pythagorean theorem since the fields are perpendicular:
\[ B_{\text{net}} = \sqrt{B_1^2 + B_2^2} \]
Substituting the values:
\[ B_{\text{net}} = \sqrt{(2 \times 10^{-7})^2 + (1 \times 10^{-7})^2} \]
\[ B_{\text{net}} = \sqrt{4 \times 10^{-14} + 1 \times 10^{-14}} = \sqrt{5 \times 10^{-14}} = \sqrt{5} \times 10^{-7} \, \text{T} \]

### Step 7: Final Result
Thus, the resultant magnetic field at the midpoint between the dipoles is:
\[ B_{\text{net}} = \sqrt{5} \times 10^{-7} \, \text{T} \]