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To understand the origin of the torque on a current loop due to the magnetic forces on the current-carrying wires.

This problem will show you how to calculate the torque on a magnetic dipole in a uniform magnetic field. We start with a rectangular current loop, the shape of which allows us to calculate the Lorentz forces explicitly. Then we generalize our result. Even if you already know the general formula to solve this problem, you might find it instructive to discover where it comes from.

a) Give a more general expression for the magnitude of the torque τ. Rewrite the answer found in Part A in terms of the magnitude of the magnetic dipole moment of the current loop m. Define the angle between the vector perpendicular to the plane of the coil and the magnetic field to be ϕ, noting that this angle is the complement of angle θ in Part A.

Give your answer in terms of the magnetic moment m, magnetic field B, and ϕ.

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Answer:

The torque is  [tex]\tau = B * m sin(\phi )[/tex]

Explanation:

Generally when current flow through a rectangular loop the magnetic dipole generated we can denote this as  [tex]\vec m[/tex]

Here k is the length of the side of one side of the rectangular loop

Now when this loop is place in a magnetic field, the torque it experience is mathematically represented as

      [tex]\vec \tau = \vec B \ X \ \vec m[/tex]

Here  X stands for cross - product

From the question we are told that the angle between the vector ( [tex]\vec m[/tex])perpendicular to the plane of the coil and magnetic field(B) as [tex]\phi[/tex]

So

=>     [tex]\tau = B * m sin(\phi )[/tex]

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