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Magnetic Dipole Moment Revision Notes

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16.5Magnetic Dipole Moment


In these revision notes for Magnetic Dipole Moment, we cover the following key points:

  • What is magnetic moment? How it is produced?
  • What is magnetic dipole moment? What type of quantity is it (vector or scalar)?
  • How to calculate the magnetic dipole moment?
  • How to calculate the magnetic torque in a coil in terms of magnetic dipole moment?
  • What are the similarities between magnetic and electric torque produced in a current carrying loop?
  • How to calculate the energy of a magnetic dipole?
  • How to calculate the work done to rotate a magnetic dipole?

Magnetic Dipole Moment Revision Notes

Magnetic moment is a vector quantity that describes magnetic processes. More specifically, it indicates the strength of a magnetic dipole, which represents the relationship between the external magnetic field and the strength of that field itself.

Generally, there can be two origins for the creation of the magnetic moment:

  1. The first one involves an electric and / or circular current having some current density distribution
  2. The second origin of magnetic moment creation involves particles having their own angular momentum, the so-called spins.

In most cases we are more interested in magnetic dipole moment μ rather than in magnetic moment m itself. In simple words, a magnetic dipole moment represents a magnetic north and south pole separated by a small distance.

In scientific terms, magnetic dipole moment, μ, is the quantity used to describe the torque of a coil due to the turning effect of magnetic field on this coil.

The direction of magnetic dipole moment is determined using the right hand rule. According to this rule, if we grasp the magnetic dipole moment vector with the right hand so that the thumb shows the direction of magnetic dipole moment, the four fingers show the direction of current.

In addition, the direction of magnetic dipole moment vector is normal to the plane of the loop.

The equation used to calculate the magnitude of magnetic dipole moment μ produced by a current carrying coil is

μ = N ∙ I ∙ A

where N is the number of turns in the coil, I is the current in the coil and A is the area of the coil. Obviously, the unit of magnetic dipole moment is [A ∙ m2].

Combining the equations

μ = N ∙ I ∙ A

and

τ = I ∙ B ∙ A ∙ sin⁡θ

we obtain for the magnetic torque in terms of the magnetic dipole moment

τ = μ ∙ B ∙ sin⁡θ

The vector form of the above equation is

τ = μ × B

A magnetic dipole in an external field has an energy that depends on the orientation of dipole moment in respect to the magnetic field direction. The energy U of magnetic dipoles in terms of the angle θ is:

U(θ) = -μ ∙ B

or

U(θ) = -μ ∙ B ∙ cos⁡θ

The sign minus is because the maximum value of energy is obtained for θ = 180° for which cos θ = -1. In this way, we obtain a positive value for the maximum energy.

If an external force rotates the coil (magnetic dipole) from an angle θ1 to another angle θ2, a resulting torque is produced. This action results in a work W done on the dipole by the applied torque. If the dipole is stationary before and after the change in the angle mentioned above, then the work done on the magnetic dipole is

W = Ufinal - Uinitial = U(θ2 ) - U(θ1)

where U(θ1) and U(θ2) are the values of energy for the angles θ1 and θ2 formed by the corresponding vectors of magnetic dipole moments and the magnetic field lines.

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