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Dipole moment and torque on a rectangular wire

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The rectangular wire of the image is placed on yz plane and is immersed in a magnetic field which magnitude and direction is :
$$\vec B = \frac{0.05}{\sqrt{2}}(\hat i + \hat j)$$
a) Compute the magnetic dipole moment and torque over the wire when the current going through is 5A
b) Compute the magnetic dipole moment and torque over the wire when it turns counterclockwise 45 degrees respect to z axis and the current is also 5A

My try:

a) Pretty confident with it.

b) I have been thinking about 'counterclockwise 45 degrees with respect to z axis means that you look along the +z direction from the origin. The plane of the loop then lies along the $-(\hat{i}+\hat{j})$ direction so μ and B are perpendicular. Torque is now a maximum' and I do not have a clear idea yet. I think we are in a situation liken to the one depicted here:

But here as you can see μ (normal vector) and B are not perpendicular. Could you explain this with more detail?

Thank you

asked Aug 2 in Physics Problems by JD_PM (490 points)
edited Aug 8 by sammy gerbil
Please check question for "magnetic dipole momentum and momentum." I don't know what this means. "Magnetic dipole moment" would be ok. For the 2nd "momentum" do you mean "torque"?

See similar questions : [magnetic moment of the loop](http://physics.qandaexchange.com/?qa=1880/) and [calculating magnetic moment and current induced](http://physics.qandaexchange.com/?qa=2039/)

1 Answer

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Best answer

... counterclockwise 45 degrees with respect to z axis means that you look along the +z direction from the origin. The plane of the loop then lies along the same direction as $B$ so $\mu$ and $B$ are now perpendicular. Torque is now a maximum. (See diagram below.)

The magnitude of $\mu$ is still the same ($0.016 Am^2$) because the current and area of the loop are the same. $B$ is still the same - its magnitude is $|B|=0.05T$. So the torque is now $\tau=|\mu| |B| \hat{k}=0.016\times 0.05 \hat{k} Nm=0.0008 \hat{k} Nm$.

The given answer is far too big. It should be the same order of magnitude as (a) but a little bigger.

In the diagram, the magnetic field $B$ (red) lies at $45^{\circ}$ to the x and y axes, and the rectangular loop (blue) initially lies in the yz plane in position $L_1$. (The +z direction points into the screen.)

After the $45^{\circ}$ counter-clockwise rotation about the z axis the rectangular loop lies in position $L_2$. The magnetic moment $u_2$ (green) is normal to the plane of the rectangle and is now perpendicular to $B$. In this position the torque on the rectangular loop $u_2 \times B$ is a maximum.

answered Aug 2 by sammy gerbil (26,096 points)
selected Aug 9 by JD_PM
Did not you miss $\sqrt{2}$?
No. The magnitude of $B$ is $0.05$. The magnitude of each of its x, y components is $\frac{0.05}{\sqrt2}$.
Okey Sr, I have updated my question. I made emphasis on b)
May you spare some time explaining b) with more detail please?
Done. Answer updated.
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