What is the force on a magnetic dipole in a coil at r = 0?

[unscientific]

Homework Statement

Homework Equations

The Attempt at a Solution

I found dB/dr to be constant near r = 0, so it's a straight line, implying B increases linearly with r near r = 0.

For the second part, I assume the setup is as such, if not placing the dipole with its axis parallel to the coil would make B = 0, and hence no force at all..Not sure what energy has to do in this case.

 

[MisterX]

Not sure what energy has to do in this case.

There is a relationship between potential energy and force.

 

[TSny]

I found dB/dr to be constant near r = 0, so it's a straight line, implying B increases linearly with r near r = 0.

Sounds good.

For the second part, I assume the setup is as such, if not placing the dipole with its axis parallel to the coil would make B = 0, and hence no force at all..Not sure what energy has to do in this case.

[Edited to correct a misstatement] Although there is only an r-component of field on the axis of the Maxwell loops (r-axis), off the axis there will be a radially inward field (perpendicular to the r-axis and toward the r-axis). A dipole in the form of a small current loop would experience a force due to this radially inward field.

But, it's easier to find the force by using the idea that force is the negative gradient of potential energy: F = -∇E. See if you can use this to find the force. [I see MisterX posted this suggestion before me. Sorry.]

 

[unscientific]

Sounds good.



[Edited to correct a misstatement] Although there is only an r-component of field on the axis of the Maxwell loops (r-axis), off the axis there will be a radially inward field (perpendicular to the r-axis and toward the r-axis). A dipole in the form of a small current loop would experience a force due to this radially inward field.

But, it's easier to find the force by using the idea that force is the negative gradient of potential energy: F = -∇E. See if you can use this to find the force. [I see MisterX posted this suggestion before me. Sorry.]

Then, wouldn't the force simply be F = - ∂U/∂r = m ∂B_ext/∂r at r = 0?

This is simply the earlier part of the question of finding ∂B/∂r at r = 0.

 

[TSny]

Then, wouldn't the force simply be F = - ∂U/∂r = m ∂B_ext/∂r at r = 0?

This is simply the earlier part of the question of finding ∂B/∂r at r = 0.

Yes, that's right.