A circular wire loop has radius R = 0.327 m and carries current i = 0.291 A. The loop is placed in the xy-plane in a uniform magnetic field pointing in the negative z direction with magnitude 3.83 T, as shown in the figure.

a) Determine the magnitude of the loop’s magnetic moment.

b) Calculate the potential energy of the loop in the position shown

To determine the magnitude of the loop's magnetic moment (part a), we can use the equation:

μ = iA

where μ is the magnetic moment, i is the current flowing through the loop, and A is the area of the loop.

To find the area of the loop, we can use the equation:

A = πR^2

where R is the radius of the loop.

So the magnetic moment can be calculated as:

μ = iπR^2

Substituting the given values, we have:

μ = (0.291 A)(π)(0.327 m)^2

Calculating this expression will give us the magnitude of the loop's magnetic moment.

To calculate the potential energy of the loop in the given position (part b), we can use the equation:

U = -μ • B

where U is the potential energy, μ is the magnetic moment, and B is the magnetic field.

Since the magnetic field is pointing in the negative z direction, we can write it as:

B = -Bz k

where Bz is the magnitude of the magnetic field.

Substituting these values into the equation, we have:

U = -μ • (-Bz k)

Simplifying further, we get:

U = μ • Bz k

Now, we just need to plug in the values we have:

U = (μ)(3.83 T)(k)

Calculating this expression will give us the potential energy of the loop in the given position.