A rigid circular loop has a radius of 0.20 m and is in the xy-plane. A clockwise current I is carried
by the loop, as shown. The magnitude of the magnetic moment of the loop is 0.75 A · m2. A
uniform external magnetic field, B = 0.20 T in the positive x-direction, is present. An external
torque changes the orientation of the loop from one of lowest potential energy to one of highest
potential energy. The work done by this external torque is closest to
To find the work done by the external torque in changing the orientation of the loop from one of lowest potential energy to one of highest potential energy, we can use the formula:
Work = Change in Potential Energy.
The potential energy of a magnetic dipole in a magnetic field is given by:
Potential Energy = -magnetic moment (μ) • magnetic field (B) • cos(θ)
Where:
- magnetic moment (μ) = magnitude of the magnetic moment of the loop = 0.75 A · m²
- magnetic field (B) = 0.20 T
- θ = angle between the magnetic moment (loop) and the magnetic field.
When the loop is in the orientation of lowest potential energy, the magnetic moment is aligned parallel to the magnetic field. Therefore, θ = 0°.
Substituting these values into the potential energy formula:
Potential Energy (lowest) = -0.75 A · m² • 0.20 T • cos(0°) = -0.75 A · m² • 0.20 T • 1 = -0.15 J
When the loop is in the orientation of highest potential energy, the magnetic moment is aligned anti-parallel to the magnetic field. Therefore, θ = 180°.
Substituting these values into the potential energy formula:
Potential Energy (highest) = -0.75 A · m² • 0.20 T • cos(180°) = -0.75 A · m² • 0.20 T • -1 = 0.15 J
The change in potential energy is given by:
Change in Potential Energy = Potential Energy (highest) - Potential Energy (lowest)
= 0.15 J - (-0.15 J)
= 0.30 J
Therefore, the work done by the external torque in changing the orientation of the loop is 0.30 Joules.
To find the work done by the external torque, we need to consider the change in potential energy of the loop when it rotates from the lowest potential energy orientation to the highest potential energy orientation.
The potential energy of a magnetic moment in a magnetic field is given by the equation U = -m · B, where U is the potential energy, m is the magnetic moment, and B is the magnetic field.
In this case, the loop initially has the magnetic moment m = 0.75 A · m^2 and is in a magnetic field B = 0.20 T. The work done by the external torque is equal to the change in potential energy, which can be calculated as:
ΔU = -Δm · B
Since the orientation of the loop is changing from the lowest potential energy to the highest potential energy, the change in magnetic moment, Δm, is equal to the total magnetic moment:
Δm = m
Substituting the given values:
ΔU = -0.75 A · m^2 · 0.20 T
ΔU = -0.15 J
The work done by the external torque is closest to -0.15 J.