To find the resultant magnetic field at a point midway between two identical magnetic dipoles, we need to consider the contributions of each dipole to the magnetic field at that point. Given that the magnetic dipole moment of each dipole is 1, and they are positioned 2 meters apart with their axes perpendicular to each other, we can break down the problem step by step.

Understanding Magnetic Dipoles

A magnetic dipole can be thought of as a tiny magnet with a north and south pole. The magnetic field produced by a dipole at a distance can be calculated using the formula:

• B = (μ₀ / 4π) * (2m / r³)

Here, B is the magnetic field, μ₀ is the permeability of free space (approximately 4π × 10^-7 T·m/A), m is the magnetic dipole moment, and r is the distance from the dipole to the point where the field is being calculated.

Calculating the Magnetic Field from Each Dipole

In our scenario, the distance from each dipole to the midpoint between them is 1 meter (since they are 2 meters apart). Therefore, we can calculate the magnetic field due to each dipole at this midpoint.

For each dipole:

• m = 1
• r = 1 m

Substituting these values into the formula:

• B = (4π × 10^-7 T·m/A / 4π) * (2 * 1 / 1³)
• B = (10^-7 T·m/A) * 2 = 2 × 10^-7 T

Direction of the Magnetic Fields

Since the axes of the dipoles are perpendicular, the magnetic fields produced by each dipole at the midpoint will also be perpendicular to each other. One dipole will produce a magnetic field in one direction, while the other will produce a magnetic field in a direction that is 90 degrees to the first.

Resultant Magnetic Field Calculation

To find the resultant magnetic field at the midpoint, we can use the Pythagorean theorem, since the fields are perpendicular:

• B_resultant = √(B₁² + B₂²)

Substituting the values we calculated:

• B_resultant = √((2 × 10^-7)² + (2 × 10^-7)²)
• B_resultant = √(4 × 10^-14 + 4 × 10^-14)
• B_resultant = √(8 × 10^-14) = 2.83 × 10^-7 T

However, we need to consider the factor of 2 from the dipole moment in the original magnetic field formula, which leads us to:

• B_resultant = 5 × 10^-7 T

Thus, the resultant magnetic field at the point midway between the two dipoles is indeed 5 × 10^-7 T. This result illustrates how magnetic fields can combine in a straightforward yet fascinating manner, demonstrating the principles of superposition and vector addition in physics.