Work done in rotating a current carrying loop

[zorro]

Homework Statement

The arrangement is as shown in the figure.
Find the work done to increase the spacing between the wire and the loop from a to 2a.

The Attempt at a Solution

I calculated the potential energies in initial and final configurations using U=-M.B (all vectors)

I got

I have a problem finding out the work done from this.
Work done by a conservative field is defined as the negative of Uf - Ui .
But my book just uses W.D. = Uf - Ui without any negation.
Where is my mistake?

 

[hikaru1221]

I assume that you calculated Ui and Uf correctly

First, I don't think magnetic field is conservative field. But for the interaction between an external B-field and a magnetic dipole in particular, it happens to be "conservative" in a way that the force on the dipole is: $$\vec{F}_B=grad(\vec{m}.\vec{B})$$. It's kind of contradictory, and I have no explanation. Perhaps it is consistent with its twin - electric dipole - whereas the force on electric dipole inside an E-field is $$\vec{F}_E=grad(\vec{p}.\vec{E})$$, which shows the unification and relativity of B-field and E-field. But I'm no expert.

Back to your main problem. Let's take an analogous example from gravitational field. When you lift a book from height h1 to height h2, the work done by gravity is mg(h1-h2) or Ui - Uf, and the work done by you to lift it is mg(h2-h1) or Uf-Ui. So you see the difference? The expense of the field itself is always Ui-Uf, while what you give to / take from the field in compensation is Uf - Ui. The sum of those two is zero, and the law of energy conservation is safe.

 

[zorro]

grrrrr...I solve big problems and forget small things . Yes you are right, since we are doing work to increase the spacing, it should be Uf - Ui.
Thanks

 

[zorro]

Regarding conservative nature of magnetic fields, here in this case the current forms a closed loop. Hence the field is conservative ( even if its non-uniform )

 

[hikaru1221]

Not really. The magnetic field is always non-conservative. The potential energy U you calculate is, in fact, the energy of interaction between external B-field and the loop, while the total energy of B-field of the system = energy of the loop + energy of the external B-field + U. That a field is non-uniform has nothing to do with whether it is conservative or not.
Anyway, I'm no expert, so I don't have an explanation on this for you.