8.6 Magnetic Dipole Energy
All right. Now let’s consider the potential energy associated with the orientation of a magnetic dipole. Earlier we have seen that a current loop generates a magnetic field along its axis in upward direction if the current is flowing through this loop in counterclockwise direction. We have also defined a magnetic dipole moment vector such that if we have N number of turns of this loop and if the area surrounded by the current loop is A, the magnetic dipole moment vector was defined as N times i times A. Again, where N was the number of turns, A was the area vector associated with the surface surrounded by the current loop, and of course i was the current flowing through these turns.
We have seen that since the magnetic dipole moment vector is in the same direction with the area vector, and the area vector for a surface which is surrounded by a current loop, using right hand rule, holding the right hand fingers in the direction of flow of current, and keeping the right hand thumb in up direction gives us the direction associated with that area vector. And the μ is in the same direction with this area vector, therefore the magnetic dipole moment vector and the surface area vector were in the same direction.
We have seen that if we just take this system, current loop, and place it in an external magnetic field, then the external magnetic field generates a torque on that current loop and we obtain that that torque is equal to magnetic dipole moment vector crossed with the external magnetic field. So we can also call such a system as a magnetic dipole. Under the influence generated by this external magnetic field, the current loop rotates and aligns itself along the magnetic field lines.
If we compare this system with the electric dipole and the electrical interactions, as you recall, electric dipole was consisting of two point charges with equal and opposite signs separated by a very small distance of d. For such a system, we have defined dipole moment vector P and the magnitude of P was such that magnitude of the dipole charge times the separation distance of d.
Whenever we place such a dipole in an external electric field, we have seen that that external electric field generates a pair of forces on these two charges such that it causes a rotation of the dipole about an axis passing through its center. The associated torque was equal to P cross E. As you can see, they are very similar expressions. Here E was the external electric field that we place the electric dipole into that region. This system is what we called as electric dipole.
Therefore, if we’d like to rotate an electric dipole in an external electric field, or by the same token, magnetic dipole such as current loop inside of an external magnetic field, we have to do some work. If we make a note over here then we can say that if a magnetic field exerts a torque on a magnetic dipole, then work must be done to change the orientation of the dipole. Therefore the magnetic dipole must have a magnetic potential energy that depends on its orientation — of course let’s add it over here — orientation relative to the external magnetic field.
Earlier we have seen that for the electric dipole, this energy u, or as a function of the orientation angle