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. `5xx10^(-7)T`
B. `sqrt5xx10^(-7)T`
C. `10^(-7)T`
D. `2xx10^(-7)T`

Answer

To solve the problem of finding the resultant magnetic field at a point midway 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 \( m = 1 \, \text{Am}^2 \), placed 2 meters apart. The point of interest is the midpoint between the two dipoles, which is 1 meter away from each dipole.

### Step 2: Identify the Magnetic Field Due to Each Dipole
The magnetic field \( B \) at a point due to a magnetic dipole is given by the formulas for axial and equatorial points:

1. **Axial Point (along the axis of the dipole)**:
\[ B_{\text{axial}} = \frac{\mu_0}{4\pi} \cdot \frac{2m}{d^3} \]
where \( d \) is the distance from the dipole to the point.

2. **Equatorial Point (perpendicular to the axis of the dipole)**:
\[ B_{\text{equatorial}} = \frac{\mu_0}{4\pi} \cdot \frac{m}{d^3} \]

### Step 3: Calculate the Magnetic Field from Each Dipole
- **For Dipole 1 (Axial Point)**:
- Distance \( d = 1 \, \text{m} \)
- Magnetic field:
\[ B_1 = \frac{\mu_0}{4\pi} \cdot \frac{2m}{d^3} = \frac{10^{-7}}{4\pi} \cdot \frac{2 \cdot 1}{1^3} = \frac{10^{-7}}{2} = 2 \times 10^{-7} \, \text{T} \]
This field is directed along the axis of Dipole 1.

- **For Dipole 2 (Equatorial Point)**:
- Distance \( d = 1 \, \text{m} \)
- Magnetic field:
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{m}{d^3} = \frac{10^{-7}}{4\pi} \cdot \frac{1}{1^3} = \frac{10^{-7}}{4\pi} \approx 7.96 \times 10^{-8} \, \text{T} \]
This field is directed perpendicular to the axis of Dipole 2.

### Step 4: Determine the Resultant Magnetic Field
Since \( B_1 \) and \( B_2 \) are perpendicular to each other, we can use the Pythagorean theorem to find the resultant magnetic field \( B \):
\[ B = \sqrt{B_1^2 + B_2^2} \]
Substituting the values:
\[ B = \sqrt{(2 \times 10^{-7})^2 + (7.96 \times 10^{-8})^2} \]
Calculating:
\[ B = \sqrt{(4 \times 10^{-14}) + (6.33 \times 10^{-15})} = \sqrt{4.633 \times 10^{-14}} \approx 6.81 \times 10^{-7} \, \text{T} \]

### Final Result
The resultant magnetic field at the midpoint between the two dipoles is approximately:
\[ B \approx 6.81 \times 10^{-7} \, \text{T} \]