ampère s model b , gilbert s model h yield identical field outside of magnet. inside different.
two models used calculate magnetic fields of , forces between magnets. physically correct method called ampère model while easier model use gilbert model.
ampère model: in ampère model, magnetization due effect of microscopic, or atomic, circular bound currents, called ampèrian currents throughout material. net effect of these microscopic bound currents make magnet behave if there macroscopic electric current flowing in loops in magnet magnetic field normal loops. ampère model gives exact magnetic field both inside , outside magnet. difficult calculate ampèrian currents on surface of magnet, though easier find effective poles same magnet.
gilbert model: in gilbert model, pole surfaces of permanent magnet imagined covered so-called magnetic charge, north pole particles on north pole , south pole particles on south pole, source of magnetic field lines. if magnetic pole distribution known, outside magnet pole model gives magnetic field exactly. in interior of magnet model gives h-field. pole model called gilbert model of magnetic dipole. griffiths suggests (p. 258): advice use gilbert model, if like, intuitive feel problem, never rely on quantitative results.
magnetic dipole moment
far away magnet, magnetic field described (to approximation) dipole field characterized total magnetic dipole moment, m. true regardless of shape of magnet, long magnetic moment non-zero. 1 characteristic of dipole field strength of field falls off inversely cube of distance magnet s center.
the magnetic moment of magnet therefore measure of strength , orientation. loop of electric current, bar magnet, electron, molecule, , planet have magnetic moments. more precisely, term magnetic moment refers system s magnetic dipole moment, produces first term in multipole expansion of general magnetic field.
both torque , force exerted on magnet external magnetic field proportional magnet s magnetic moment. magnetic moment vector: has both magnitude , direction. direction of magnetic moment points south north pole of magnet (inside magnet). example, direction of magnetic moment of bar magnet, such 1 in compass direction north poles points toward.
in physically correct ampère model, magnetic dipole moments due infinitesimally small loops of current. sufficiently small loop of current, i, , area, a, magnetic dipole moment is:
m
=
i
a
{\displaystyle \mathbf {m} =i\mathbf {a} }
,
where direction of m normal area in direction determined using current , right-hand rule. such, si unit of magnetic dipole moment ampere meter. more precisely, account solenoids many turns unit of magnetic dipole moment ampere-turn meter.
in gilbert model, magnetic dipole moment due 2 equal , opposite magnetic charges separated distance, d. in model, m similar electric dipole moment p due electrical charges:
m
=
q
m
d
{\displaystyle m=q_{m}d\,}
,
where qm magnetic charge . direction of magnetic dipole moment points negative south pole positive north pole of tiny magnet.
magnetic force due non-uniform magnetic field
magnets drawn toward regions of higher magnetic field. simplest example of attraction of opposite poles of 2 magnets. every magnet produces magnetic field stronger near poles. if opposite poles of 2 separate magnets facing each other, each of magnets drawn stronger magnetic field near pole of other. if poles facing each other though, repulsed larger magnetic field.
the gilbert model predicts correct mathematical form force , easier understand qualitatively. if magnet placed in uniform magnetic field both poles feel same magnetic force in opposite directions, since have opposite magnetic charge. but, when magnet placed in non-uniform field, such due magnet, pole experiencing large magnetic field experience large force , there net force on magnet. if magnet aligned magnetic field, corresponding 2 magnets oriented in same direction near poles, drawn larger magnetic field. if oppositely aligned, such case of 2 magnets poles facing each other, magnet repelled region of higher magnetic field.
in physically correct ampère model, there force on magnetic dipole due non-uniform magnetic field, due lorentz forces on current loop makes magnetic dipole. force obtained in case of current loop model is
f
=
∇
(
m
⋅
b
)
{\displaystyle \mathbf {f} =\nabla \left(\mathbf {m} \cdot \mathbf {b} \right)}
,
where gradient ∇ change of quantity m · b per unit distance, , direction of maximum increase of m · b. understand equation, note dot product m · b = mbcos(θ), m , b represent magnitude of m , b vectors , θ angle between them. if m in same direction b dot product positive , gradient points uphill pulling magnet regions of higher b-field (more strictly larger m · b). b represents strength , direction of magnetic field. equation strictly valid magnets of 0 size, approximation not large magnets. magnetic force on larger magnets determined dividing them smaller regions having own m summing forces on each of these regions.
cite error: there <ref group=note> tags on page, references not show without {{reflist|group=note}} template (see page).
Comments
Post a Comment