Calculating Energy to Rotate Magnetic Dipole in Solenoid

[dibilo]

i received this problem from my lecturer

A solenoid is wound with 2200 turns per meter, and carries a current 2.5A. Take the axis of the solenoid to the solenoid to be parallel to the z-axis. A magnetic dipole moment given (m)=(20A.m^2)i+ (30A.m^2)k is located inside the solenoid.

i)how much energy would be required to rotate this magnetic dipole until (m) is parallel to the x-axis?

what i did was to find the magnetic dipole using |m|=sprt(20^2+30^2)= 36A.m^2

then i find the B field to be 6.9x10^-3 k

Uinitial = -(m).B= -(mxBx + myBy + mzBz), where by Bx and By =0 due to the solenoid being // to z-axis = -mzBz = -0.21J

Here is where my problem lies...

i wrote
Ufinal = -(20i).(6.9x10^-3) = 0 , due to Bx = 0 and mz = 0

but my tutor wrote on my worksheet that i should be using 36A.m^2 instead of 20, while in the 1st part, when finding Uinitial, i can use mzBz (30 x 6.9x10^3).

can anyone help me out on this? please as far as possible explain as simple as you can why is this so? thank you very much

 

[OlderDan]

I am having some difficulty following this but I think I see it. The final energy is zero, so the "error" is of no consequence, but the final configuration has the dipole aligned with the x-axis, and the strength of the dipole is 36, not 20. You have rotated it to a new orientation. The x-component is now the only component.

 

[thepatientmental]

Hi there,

Thank you for sharing your problem and explaining your thought process. It seems like you have a good understanding of the concepts involved, but there are a few things that can be clarified to help you understand why your tutor suggested using 36A.m^2 instead of 20.

First, let's review the formula for the potential energy of a magnetic dipole in a magnetic field: U = -m·B, where m is the magnetic dipole moment and B is the magnetic field. In this case, the magnetic dipole moment is given by (20A.m^2)i + (30A.m^2)k, which represents the magnitude and direction of the dipole. This means that the magnetic dipole moment is not a constant value, but rather a vector that can change depending on its orientation.

Now, let's consider the initial and final orientations of the magnetic dipole. In the initial orientation, the magnetic dipole moment is parallel to the z-axis, which means that only the z-component of the dipole moment (30A.m^2) contributes to the potential energy. This is why you correctly used 30A.m^2 in your calculation for Uinitial.

However, in the final orientation, the magnetic dipole moment is parallel to the x-axis. This means that only the x-component of the dipole moment (20A.m^2) contributes to the potential energy. Therefore, in order to find the potential energy in this final orientation, we need to use the x-component of the dipole moment and not the total magnitude of the dipole moment.

In summary, the reason why your tutor suggested using 36A.m^2 instead of 20 in your calculation for Ufinal is because the final orientation of the magnetic dipole only involves the x-component of the dipole moment, which is 20A.m^2.

I hope this explanation helps to clarify your confusion. Keep up the good work and don't hesitate to ask for help when needed. Good luck with your studies!