Figure shows a circular wire loop of radius r carrying current i placed in a perpendicular magnetic field B. Suppose that the radius of cross section of the wire is a. Find the decrease in radius of loop when magnetic field is switched off if Young’s modulus is Y.
To find the decrease in radius of the loop when the magnetic field is switched off, we need to consider the forces acting on the loop.
When the loop is placed in a perpendicular magnetic field B, a force is exerted on the wire due to the interaction between the magnetic field and the current flowing through the wire. This force can be given by the equation F = BIL, where F is the force, B is the magnetic field, I is the current, and L is the length of wire in the magnetic field.
Since we need to find the decrease in radius, let's consider a small element of wire with length dl at a distance x from the center of the loop. The force acting on this element is given by dF = BIdl. The total force acting on the entire loop can be found by integrating this force over the entire length of wire.
dF = BIdl
Integrating both sides from 0 to L (the total length of the wire):
∫dF = B ∫Idl from 0 to L
F = BIL
Now, let's consider the Young's modulus and the decrease in radius. The Young's modulus relates to the stress and strain in a material. In this case, when the magnetic field is switched off, the loop experiences a decrease in length, causing a compression strain. This strain gives rise to a stress in the wire.
The stress σ in the wire can be given by:
σ = Y * (ΔL / L₀)
Where Y is the Young's modulus, ΔL is the change in length, and L₀ is the original length of the wire. In this case, ΔL is the decrease in length of the loop and L₀ is the original circumference of the loop.
Now, let's relate the variables in terms of radius and circumference. The circumference of the wire is given by:
C₀ = 2πr₀
Where r₀ is the original radius of the loop.
The decrease in circumference can be given by:
ΔC = 2π(r₀ - r)
Now, we need to relate the change in length (ΔL) to the change in circumference (ΔC).
ΔL = ΔC / π
Substituting these values into the equation for stress, we have:
σ = Y * (ΔC / π) / (2πr₀)
Finally, the force acting on the loop can also be expressed in terms of the stress and the cross-sectional area of the wire:
F = σ * (πr₀²)
Since the force acting on the loop is caused by the interaction between the magnetic field and the current, we can equate the force due to the magnetic field (BIL) to the force due to the stress (σ * πr₀²):
BIL = σ * πr₀²
Now we can substitute the expression for σ in terms of ΔC and solve for ΔC:
BIL = [Y * (ΔC / π) / (2πr₀)] * πr₀²
BIL = Y * (ΔC / 2) * r₀
ΔC = (2BIL) / (Y * r₀)
Finally, we need to convert the change in circumference to the change in radius:
Δr = ΔC / (2π)
Δr = [(2BIL) / (Y * r₀)] / (2π)
Δr = (BIL) / (πYr₀)
Therefore, the decrease in radius of the loop when the magnetic field is switched off is given by Δr = (BIL) / (πYr₀).