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- Calculation of magnetic dipole moment of the Earth - Physics …
- Intuition for magnetic dipole moment - Physics Stack Exchange
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- How to calculate the magnetic dipole moment of a bar magnet?
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- What is doing the work on a magnetic dipole in a magnetic field?
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A View to a Kill (1985)
Magnetic dipole GudangMovies21 Rebahinxxi LK21
In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant.
It is a magnetic analogue of the electric dipole, but the analogy is not perfect. In particular, a true magnetic monopole, the magnetic analogue of an electric charge, has never been observed in nature. However, magnetic monopole quasiparticles have been observed as emergent properties of certain condensed matter systems. Moreover, one form of magnetic dipole moment is associated with a fundamental quantum property—the spin of elementary particles.
Because magnetic monopoles do not exist, the magnetic field at a large distance from any static magnetic source looks like the field of a dipole with the same dipole moment. For higher-order sources (e.g. quadrupoles) with no dipole moment, their field decays towards zero with distance faster than a dipole field does.
External magnetic field produced by a magnetic dipole moment
In classical physics, the magnetic field of a dipole is calculated as the limit of either a current loop or a pair of charges as the source shrinks to a point while keeping the magnetic moment m constant. For the current loop, this limit is most easily derived from the vector potential:
A
(
r
)
=
μ
0
4
π
r
2
m
×
r
r
=
μ
0
4
π
m
×
r
r
3
,
{\displaystyle {\mathbf {A} }({\mathbf {r} })={\frac {\mu _{0}}{4\pi r^{2}}}{\frac {{\mathbf {m} }\times {\mathbf {r} }}{r}}={\frac {\mu _{0}}{4\pi }}{\frac {{\mathbf {m} }\times {\mathbf {r} }}{r^{3}}},}
where μ0 is the vacuum permeability constant and 4π r2 is the surface of a sphere of radius r.
The magnetic flux density (strength of the B-field) is then
B
(
r
)
=
∇
×
A
=
μ
0
4
π
[
3
r
(
m
⋅
r
)
r
5
−
m
r
3
]
.
{\displaystyle \mathbf {B} ({\mathbf {r} })=\nabla \times {\mathbf {A} }={\frac {\mu _{0}}{4\pi }}\left[{\frac {3\mathbf {r} (\mathbf {m} \cdot \mathbf {r} )}{r^{5}}}-{\frac {\mathbf {m} }{r^{3}}}\right].}
Alternatively one can obtain the scalar potential first from the magnetic pole limit,
ψ
(
r
)
=
m
⋅
r
4
π
r
3
,
{\displaystyle \psi ({\mathbf {r} })={\frac {{\mathbf {m} }\cdot {\mathbf {r} }}{4\pi r^{3}}},}
and hence the magnetic field strength (or strength of the H-field) is
H
(
r
)
=
−
∇
ψ
=
1
4
π
[
3
r
^
(
m
⋅
r
^
)
−
m
r
3
]
=
B
μ
0
.
{\displaystyle {\mathbf {H} }({\mathbf {r} })=-\nabla \psi ={\frac {1}{4\pi }}\left[{\frac {3\mathbf {\hat {r}} (\mathbf {m} \cdot \mathbf {\hat {r}} )-\mathbf {m} }{r^{3}}}\right]={\frac {\mathbf {B} }{\mu _{0}}}.}
The magnetic field strength is symmetric under rotations about the axis of the magnetic moment.
In spherical coordinates, with
z
^
=
r
^
cos
θ
−
θ
^
sin
θ
{\displaystyle \mathbf {\hat {z}} =\mathbf {\hat {r}} \cos \theta -{\boldsymbol {\hat {\theta }}}\sin \theta }
, and with the magnetic moment aligned with the z-axis, then the field strength can more simply be expressed as
H
(
r
)
=
|
m
|
4
π
r
3
(
2
cos
θ
r
^
+
sin
θ
θ
^
)
.
{\displaystyle \mathbf {H} ({\mathbf {r} })={\frac {|\mathbf {m} |}{4\pi r^{3}}}\left(2\cos \theta \,\mathbf {\hat {r}} +\sin \theta \,{\boldsymbol {\hat {\theta }}}\right).}
Internal magnetic field of a dipole
The two models for a dipole (current loop and magnetic poles), give the same predictions for the magnetic field far from the source. However, inside the source region they give different predictions. The magnetic field between poles is in the opposite direction to the magnetic moment (which points from the negative charge to the positive charge), while inside a current loop it is in the same direction (see the figure to the right (above for mobile users)). Clearly, the limits of these fields must also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used to calculate fields inside a magnetic material.
If a magnetic dipole is formed by making a current loop smaller and smaller, but keeping the product of current and area constant, the limiting field is
B
(
r
)
=
μ
0
4
π
[
3
r
^
(
r
^
⋅
m
)
−
m
|
r
|
3
+
8
π
3
m
δ
(
r
)
]
,
{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\left[{\frac {3\mathbf {\hat {r}} (\mathbf {\hat {r}} \cdot \mathbf {m} )-\mathbf {m} }{|\mathbf {r} |^{3}}}+{\frac {8\pi }{3}}\mathbf {m} \delta (\mathbf {r} )\right],}
where δ(r) is the Dirac delta function in three dimensions. Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole.
If a magnetic dipole is formed by taking a "north pole" and a "south pole", bringing them closer and closer together but keeping the product of magnetic pole-charge and distance constant, the limiting field is
H
(
r
)
=
1
4
π
[
3
r
^
(
r
^
⋅
m
)
−
m
|
r
|
3
−
4
π
3
m
δ
(
r
)
]
.
{\displaystyle \mathbf {H} (\mathbf {r} )={\frac {1}{4\pi }}\left[{\frac {3\mathbf {\hat {r}} (\mathbf {\hat {r}} \cdot \mathbf {m} )-\mathbf {m} }{|\mathbf {r} |^{3}}}-{\frac {4\pi }{3}}\mathbf {m} \delta (\mathbf {r} )\right].}
These fields are related by B = μ0(H + M), where
M
(
r
)
=
m
δ
(
r
)
{\displaystyle \mathbf {M} (\mathbf {r} )=\mathbf {m} \delta (\mathbf {r} )}
is the magnetization.
Forces between two magnetic dipoles
The force F exerted by one dipole moment m1 on another m2 separated in space by a vector r can be calculated using:
F
=
∇
(
m
2
⋅
B
1
)
,
{\displaystyle \mathbf {F} =\nabla \left(\mathbf {m} _{2}\cdot \mathbf {B} _{1}\right),}
or
F
(
r
,
m
1
,
m
2
)
=
3
μ
0
4
π
r
5
[
(
m
1
⋅
r
)
m
2
+
(
m
2
⋅
r
)
m
1
+
(
m
1
⋅
m
2
)
r
−
5
(
m
1
⋅
r
)
(
m
2
⋅
r
)
r
2
r
]
,
{\displaystyle \mathbf {F} (\mathbf {r} ,\mathbf {m} _{1},\mathbf {m} _{2})={\dfrac {3\mu _{0}}{4\pi r^{5}}}\left[(\mathbf {m} _{1}\cdot \mathbf {r} )\mathbf {m} _{2}+(\mathbf {m} _{2}\cdot \mathbf {r} )\mathbf {m} _{1}+(\mathbf {m} _{1}\cdot \mathbf {m} _{2})\mathbf {r} -{\dfrac {5(\mathbf {m} _{1}\cdot \mathbf {r} )(\mathbf {m} _{2}\cdot \mathbf {r} )}{r^{2}}}\mathbf {r} \right],}
where r is the distance between dipoles. The force acting on m1 is in the opposite direction.
The torque can be obtained from the formula
τ
=
m
2
×
B
1
.
{\displaystyle {\boldsymbol {\tau }}=\mathbf {m} _{2}\times \mathbf {B} _{1}.}
Dipolar fields from finite sources
The magnetic scalar potential ψ produced by a finite source, but external to it, can be represented by a multipole expansion. Each term in the expansion is associated with a characteristic moment and a potential having a characteristic rate of decrease with distance r from the source. Monopole moments have a 1/r rate of decrease, dipole moments have a 1/r2 rate, quadrupole moments have a 1/r3 rate, and so on. The higher the order, the faster the potential drops off. Since the lowest-order term observed in magnetic sources is the dipole term, it dominates at large distances. Therefore, at large distances any magnetic source looks like a dipole of the same magnetic moment.
Notes
References
Kata Kunci Pencarian:
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Daftar Isi
Calculation of magnetic dipole moment of the Earth - Physics …
Mar 9, 2024 · These calculations are from these notes from MIT OCW's 8.02 "Electromagnetism", on page 27. We can represent the location of a point ##P## on the surface of the Earth using spherical coordinates Let the ##z##-axis be the Earth's rotation axis, and the ##x##-axis passes through the prime...
Intuition for magnetic dipole moment - Physics Stack Exchange
Sep 15, 2023 · One way of creating a magnetic dipole is with a current loop and if you define the magnetic dipole moment as $\mathbf m=\frac{1}{2}\int \mathbf r \times \mathbf J dV$ then the formula for the magnetic field from this dipole has the same form as the electric field from an electric dipole (see wikipedia article on dipoles).
electromagnetism - Formula for force on a magnetic dipole
Mar 30, 2022 · It is a magnetic analogue of the electric dipole, but the analogy is not perfect. In particular, a true magnetic monopole, the magnetic analogue of an electric charge, has never been observed in nature. However, magnetic monopole quasiparticles have been observed as emergent properties of certain condensed matter systems.
Magnetic Dipole: How to plug into Maxwell's equations?
$\begingroup$ If magnetic charges would exist in Maxwell equations: $\mathbf{\nabla}.\mathbf{B} = 4 \pi \rho_m$ where $\rho_m$ is the magnetic charge density, then the magnetic dipole solution can be obtained exactly in the same way an electric dipole is obtained in the standard Maxwell theory; i.e., by taking two opposite infinitesimally ...
How to calculate the magnetic dipole moment of a bar magnet?
It is important to remember that although this may be the dipole moment of a bar magnet, a bar magnet has non-zero higher order moments whose contributions to the field fall off faster than the dipole $1/r^3$. The magnetic field of a bar magnet only matches a dipole field for distances much greater the dimensions of the magnet.
electromagnetism - Principle of equivalence between a magnetic …
So by a dipole field we mean a field whose potential looks like $\mathbf p \cdot \mathbf r /r^3$, and its source is an electric dipole. Of course the magnetic field has a vector potential rather than a scalar potential, so by a magnetic dipole field we should mean one where the vector potential is like $\mathbf A \sim \mathbf m \times \mathbf r ...
What is doing the work on a magnetic dipole in a magnetic field?
May 27, 2021 · The negative sign indicates the tendency of the dipole to be aligned with the magnetic field; a dipole transverse to its local magnetic field has more energy than when it's parallel to the field. Looking at your computation, it looks like the confusion lies in how you've defined $\theta$ .
What do we mean with magnetic monopole and dipole?
Every electron has two types of magnetic dipole moment, namely spin magnetic dipole moment and orbital magnetic dipole moment. There are several electrons in an atom and most electrons cancel out each other's magnetic moment. When all the electrons cancel out neatly, the total magnetic moment of the atom is zero.
magnetic fields - Why magnet is always a dipole? - Physics Stack …
Dec 5, 2021 · The dipole of permanent magnets is not a mystery if you know that electrons are electric charges and magnetic dipoles in one. There are two ways to macroscopically enhance these properties: If you separate the electrons, you get a macroscopic electric field around the body of the charge.
Difference between magnetic dipole interactions and spin …
The dipole-dipole interactions can be seemed as the classical prat of magnet since it treated the magnet as dipole like in electrodynamics, i.e. treat spin as circulating current. Even though spin cannot understood as classical picture, such interaction actually still exists.