a cicular loop of radius R carries a current I , another circular loop of radius r<

To find the torque acting on the smaller loop due to the larger loop, we need to consider the magnetic field generated by the larger loop and how it interacts with the smaller loop. Let's break this down step by step.

Understanding the Magnetic Field

The larger loop of radius R carrying a current I generates a magnetic field in its center. The formula for the magnetic field (B) at the center of a circular loop is given by:

B = (μ₀ * I) / (2 * R)

where μ₀ is the permeability of free space, approximately equal to 4π × 10⁻⁷ T·m/A.

Magnetic Field Direction

Using the right-hand rule, we can determine the direction of the magnetic field. If you curl the fingers of your right hand in the direction of the current I in the larger loop, your thumb will point in the direction of the magnetic field B, which is perpendicular to the plane of the loop.

Torque on the Smaller Loop

The smaller loop, with radius r (where r << R), carries a current i and is placed at the center of the larger loop. The torque (τ) acting on a current-carrying loop in a magnetic field can be calculated using the formula:

τ = μ × B

Here, μ is the magnetic moment of the smaller loop, which is given by:

μ = i * A

where A is the area of the smaller loop. For a circular loop, the area A can be calculated as:

A = π * r²

Thus, the magnetic moment becomes:

μ = i * π * r²

Calculating the Torque

Now, substituting the expressions for μ and B into the torque formula:

τ = (i * π * r²) * (μ₀ * I) / (2 * R)

This simplifies to:

τ = (μ₀ * I * i * π * r²) / (2 * R)

Direction of the Torque

The direction of the torque will be such that it tends to align the magnetic moment of the smaller loop with the magnetic field produced by the larger loop. If the currents in the loops are in the same direction, the torque will try to rotate the smaller loop to align with the magnetic field. If the currents are in opposite directions, the torque will act to rotate the smaller loop in the opposite direction.

Considering Both Loops

In this scenario, the torque acting on the smaller loop is primarily due to the magnetic field created by the larger loop. The smaller loop does not create a significant magnetic field that would affect the larger loop because its radius is much smaller (r << R). Therefore, the torque is predominantly influenced by the larger loop's magnetic field.

Final Thoughts

In summary, the torque acting on the smaller loop due to the larger loop can be expressed as:

τ = (μ₀ * I * i * π * r²) / (2 * R)

This equation captures the relationship between the currents, the sizes of the loops, and the magnetic field, allowing us to understand how the smaller loop behaves in the presence of the larger loop's magnetic field.