In electromagnetism, the magnetic moment or magnetic dipole moment is a vector quantity which characterizes the strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude of torque the object experiences in a given magnetic field. When the same magnetic field is applied, objects with larger magnetic moments experience larger torques. The strength (and direction) of this torque depends not only on the magnitude of the magnetic moment but also on its orientation relative to the direction of the magnetic field. Its direction points from the south pole to the north pole of the magnet (i.e., inside the magnet). The magnetic moment also expresses the magnetic force effect of a magnet. The magnetic field of a magnetic dipole is proportional to its magnetic dipole moment. The dipole component of an object's magnetic field is symmetric about the direction of its magnetic dipole moment, and decreases as the inverse cube of the distance from the object. Example magnetic moments for subatomic particles include electron magnetic moment, nuclear magnetic moment, and nucleon magnetic moment.
Definition, units, and measurement
Definition The magnetic moment can be defined as a vector relating the aligning torque on the object from an externally applied magnetic field to the field vector itself. The relationship is given by:
τ = m × B {\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} }
where τ is the torque acting on the dipole, B is the external magnetic field, and m is the magnetic moment. This definition is based on how one could, in principle, measure the magnetic moment of an unknown sample. For a current loop, this definition leads to the magnitude of the magnetic dipole moment equaling the product of the current times the area of the loop. Further, this definition allows the calculation of the expected magnetic moment for any known macroscopic current distribution. An alternative definition is useful for thermodynamics calculations of the magnetic moment. In this definition, the magnetic dipole moment of a system is the negative gradient of its intrinsic energy, Uint, with respect to external magnetic field:
m = − x ^ ∂ U int ∂ B x − y ^ ∂ U int ∂ B y − z ^ ∂ U int ∂ B z . {\displaystyle \mathbf {m} =-{\hat {\mathbf {x} }}{\frac {\partial U_{\text{int}}}{\partial B_{x}}}-{\hat {\mathbf {y} }}{\frac {\partial U_{\text{int}}}{\partial B_{y}}}-{\hat {\mathbf {z} }}{\frac {\partial U_{\text{int}}}{\partial B_{z}}}.}
Generically, the intrinsic energy includes the self-field energy of the system plus the energy of the internal workings of the system. For example, for a hydrogen atom in a 2p state in an external field, the self-field energy is negligible, so the internal energy is essentially the eigenenergy of the 2p state, which includes Coulomb potential energy and the kinetic energy of the electron. The interaction-field energy between the internal dipoles and external fields is not part of this internal energy.
Units The unit for magnetic moment in International System of Units (SI) base units is A⋅m2, where A is ampere (SI base unit of current) and m is meter (SI base unit of distance). This unit has equivalents in other SI derived units including:
A ⋅ m 2 = N ⋅ m / T = J / T , {\displaystyle \mathrm {A{\cdot }m^{2}} =\mathrm {N{\cdot }m/T} =\mathrm {J/T} ,}
where N is newton (SI unit of force), T is tesla (SI unit of magnetic flux density), and J is joule (SI unit of energy). In the CGS system, there are several different sets of electromagnetism units, of which the main ones are ESU, Gaussian, and EMU. Among these, there are alternative (non-equivalent) units of magnetic dipole moment:
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