Demagnetizing field

The demagnetizing field, also called the stray field (outside the magnet), is the magnetic field (H-field) generated by the magnetization in a magnet. The total magnetic field in a region containing magnets is the sum of the demagnetizing fields of the magnets and the magnetic field due to any free currents or displacement currents. The term demagnetizing field reflects its tendency to act on the magnetization so as to reduce the total magnetic moment. It gives rise to shape anisotropy in ferromagnets with a single magnetic domain and to magnetic domains in larger ferromagnets. ∇ × H = 0 , {displaystyle abla imes mathbf {H} =0,}     (1) ∇ ⋅ B = 0. {displaystyle abla cdot mathbf {B} =0.}     (2) B = μ 0 ( M + H ) , {displaystyle mathbf {B} =mu _{0}left(mathbf {M} +mathbf {H} ight),}     (3) H = − ∇ U . {displaystyle mathbf {H} =- abla U.}     (4) ∇ 2 U in = ∇ ⋅ M . {displaystyle abla ^{2}U_{ ext{in}}= abla cdot mathbf {M} .}     (5) ∇ 2 U out = 0. {displaystyle abla ^{2}U_{ ext{out}}=0.}     (6) U in = U out ∂ U in ∂ n = ∂ U out ∂ n + M ⋅ n . {displaystyle {egin{aligned}U_{ ext{in}}&=U_{ ext{out}}\{frac {partial U_{ ext{in}}}{partial n}}&={frac {partial U_{ ext{out}}}{partial n}}+mathbf {M} cdot mathbf {n} .end{aligned}}}     (7) E = − μ 0 2 ∫ magnet M ⋅ H d d V {displaystyle E=-{frac {mu _{0}}{2}}int _{ ext{magnet}}mathbf {M} cdot mathbf {H} _{ ext{d}}dV}     (7) E = μ 0 ∫ magnet 1 M 1 ⋅ H d ( 2 ) d V . {displaystyle E=mu _{0}int _{ ext{magnet 1}}mathbf {M} _{1}cdot mathbf {H} _{ ext{d}}^{(2)}dV.}     (8) ∫ magnet 1 M 1 ⋅ H d ( 2 ) d V = ∫ magnet 2 M 2 ⋅ H d ( 1 ) d V . {displaystyle int _{ ext{magnet 1}}mathbf {M} _{1}cdot mathbf {H} _{ ext{d}}^{(2)}dV=int _{ ext{magnet 2}}mathbf {M} _{2}cdot mathbf {H} _{ ext{d}}^{(1)}dV.}     (9) U ( r ) = − 1 4 π ∫ volume ∇ ′ ⋅ M ( r ′ ) | r − r ′ | d V ′ + 1 4 π ∫ surface n ⋅ M ( r ′ ) | r − r ′ | d S ′ , {displaystyle U(mathbf {r} )=-{frac {1}{4pi }}int _{ ext{volume}}{frac { abla 'cdot mathbf {Mleft(r' ight)} }{|mathbf {r} -mathbf {r} '|}}dV'+{frac {1}{4pi }}int _{ ext{surface}}{frac {mathbf {n} cdot mathbf {Mleft(r' ight)} }{|mathbf {r} -mathbf {r} '|}}dS',}     (10) H = H 0 − γ 4 π M 0 , {displaystyle mathbf {H} =mathbf {H} _{0}-{frac {gamma }{4pi }}mathbf {M} _{0},}     (11) H k = ( H 0 ) k − γ k 4 π ( M 0 ) k , k = x , y , z . {displaystyle H_{k}=(H_{0})_{k}-{frac {gamma _{k}}{4pi }}(M_{0})_{k},qquad k=x,y,z.}     (12) The demagnetizing field, also called the stray field (outside the magnet), is the magnetic field (H-field) generated by the magnetization in a magnet. The total magnetic field in a region containing magnets is the sum of the demagnetizing fields of the magnets and the magnetic field due to any free currents or displacement currents. The term demagnetizing field reflects its tendency to act on the magnetization so as to reduce the total magnetic moment. It gives rise to shape anisotropy in ferromagnets with a single magnetic domain and to magnetic domains in larger ferromagnets. The demagnetizing field of an arbitrarily shaped object requires a numerical solution of Poisson's equation even for the simple case of uniform magnetization. For the special case of ellipsoids (including infinite cylinders) the demagnetization field is linearly related to the magnetization by a geometry dependent constant called the demagnetizing factor. Since the magnetization of a sample at a given location depends on the total magnetic field at that point, the demagnetization factor must be used in order to accurately determine how a magnetic material responds to a magnetic field. (See magnetic hysteresis.) In general the demagnetizing field is a function of position H(r). It is derived from the magnetostatic equations for a body with no electric currents. These are Ampère's law and Gauss's law The magnetic field and flux density are related by where μ 0 {displaystyle mu _{0}} is the permeability of vacuum and M is the magnetisation. The general solution of the first equation can be expressed as the gradient of a scalar potential U(r): Inside the magnetic body, the potential Uin is determined by substituting (3) and (4) in (2):