Magnetic Moment Definition Verification/ Proof

[cubejunkies]

I saw the equation here http://en.wikipedia.org/wiki/Magnetic_moment#Current_loop_definition for the definition of the magnetic moment for a non-planar loop. Can someone tell me if there's a name for this equation $m= \frac { I }{ 2 } \int { \overrightarrow { r } } \times d\overrightarrow { r }$, if it's even right, and where I could find a derivation and/or proof of the equation? I've been fumbling over this for several hours now and I've gotten absolutely nowhere. I don't have any access to any texts on the matter, all I have is google and wikipedia at the moment.

Thanks

Anthony

 

[Jano L.]

It is really just a convenient definition. It assigns to some portion space (where the currents are) a quantity called magnetic moment. The motivation is, we want to find some approximate expression for the magnetic field due to these currents, and it turns out that in large distances, the magnetic moment is sufficient to find it. Try to get and look into D.J. Griffiths, Introduction to electrodynamics, sec. 5.4.3 or J.D. Jackson: Classical Electrodynamics, you should find the details there.

 

[Meir Achuz]

It follows from
$${\bf m}=\frac{1}{2}\int{\bf r\times j}d^3r$$
with the substitution $${\bf j}d^3r\rightarrow I{\bf dr}.$$
The derivation of the j equation takes about one page in an EM textbook.
It is on J. Franklin, "Classical Electromagnetism" on page 212.
Your equation is derived directly on page 210.
You should also know that $$\frac{1}{2}\int{\bf r\times dr}=$$
the area of the loop.

 

[andrien]

see the page 185 from here,I do not want to copy anything from that.so you can see here
http://books.google.co.in/books?id=8qHCZjJHRUgC&pg=PA794&dq=jackson+electrodynamics+page+187&hl=en#v=onepage&q=jackson%20electrodynamics%20page%20187&f=false

 

[sardar]

Hello Anthony,

Thank you for your question. The equation you mentioned is indeed the correct definition for the magnetic moment of a non-planar loop. It is called the "current loop definition" because it relates the magnetic moment to the current flowing through the loop.

To understand the derivation of this equation, we need to consider the behavior of a current-carrying loop in a magnetic field. When a current flows through a loop, it creates a magnetic dipole moment, which is a measure of the strength and orientation of the loop's magnetic field. This dipole moment is defined as the vector product of the loop's area vector and the current vector, as shown in the equation you mentioned.

To prove this equation, we can use the principles of electromagnetism, specifically the Biot-Savart law and the Lorentz force law. The Biot-Savart law states that the magnetic field at a point due to a small current element is directly proportional to the current, the length of the element, and the sine of the angle between the element and the position vector of the point. On the other hand, the Lorentz force law tells us that a current element in a magnetic field will experience a force perpendicular to both the current and magnetic field vectors.

Using these two laws, we can show that the torque (or turning effect) experienced by a current-carrying loop in a magnetic field is proportional to the product of the current and the area enclosed by the loop. This torque is directly related to the magnetic moment of the loop, which can be calculated by integrating over the entire loop.

I hope this helps to clarify the derivation of the magnetic moment equation for you. If you have any further questions, please do not hesitate to ask. Good luck with your research!

Best regards,