Let's break down the situation step by step.
Given:
There is an infinitely long current-carrying wire and a small current-carrying loop in the plane of the paper.
The radius of the loop is "a," and the distance of the loop's center from the wire is "d" (with the condition d >> a).
The loop applies a force F on the wire.
Step 1: Magnetic Field of the Wire
The magnetic field at a distance "r" from a long straight current-carrying wire is given by Ampère's law:
B = (μ₀ * I) / (2π * r)
Where:
μ₀ is the permeability of free space.
I is the current in the wire.
r is the distance from the wire.
In this case, the distance between the wire and the center of the loop is "d." Thus, the magnetic field at the center of the loop is:
B = (μ₀ * I) / (2π * d)
Step 2: Force on the Loop
Now, we consider the force on the current-carrying loop due to the magnetic field created by the wire. The force on a small current-carrying element (dl) in the loop due to a magnetic field (B) is given by:
dF = I * dl × B
Since the magnetic field is radially outward from the wire and the current in the loop is circular, the force will be directed radially towards or away from the wire, depending on the direction of the current.
Step 3: Net Force on the Loop
The net force on the loop can be derived by integrating the forces on all current elements of the loop. The result of this integration gives us the total force F that the loop applies on the wire.
From the expression for the magnetic field, we can derive the force F to be proportional to:
F ∝ (μ₀ * I² * a²) / d³
Where:
"a" is the radius of the loop.
"d" is the distance of the loop's center from the wire.
Step 4: Conclusion
Since we are asked to determine the proportionality of the force F, we focus on the relationship:
F ∝ (a²) / (d³)
This matches the option A:
A. F ∝ (a² / d³)
So, the correct answer is option A.
