Movatterモバイル変換

[0]ホーム

Jump to content

Magnetic monopole

Edit links

From Wikipedia, the free encyclopedia

(Redirected fromMagnetic monopoles)

Hypothetical particle with one magnetic pole

It is impossible to make magnetic monopoles from abar magnet. If a bar magnet is cut in half, it isnot the case that one half has the north pole and the other half has the south pole. Instead, each piece has its own north and south poles. A magnetic monopole cannot be created from normal matter such asatoms andelectrons, but would instead be a newelementary particle.

Inparticle physics, amagnetic monopole is a hypotheticalelementary particle that is an isolatedmagnet with only one magnetic pole (a north pole without a south pole or vice versa).^[1]^[2] A magnetic monopole would have a net north or south "magnetic charge". Modern interest in the concept stems fromparticle theories, notably thegrand unified andsuperstring theories, which predict their existence.^[3]^[4]The known elementary particles that haveelectric charge are electric monopoles.

Magnetism inbar magnets andelectromagnets is not caused by magnetic monopoles, and indeed, there is no known experimental or observational evidence that magnetic monopoles exist.

Somecondensed matter systems contain effective (non-isolated) magnetic monopolequasi-particles,^[5] or contain phenomena that are mathematically analogous to magnetic monopoles.^[6]

Name	Without magnetic monopoles	With magnetic monopoles
Name	Without magnetic monopoles	Weber convention	Ampere-meter convention
Gauss's law	$\nabla \cdot \mathbf {E} ={\frac {\rho _{\mathrm {e} }}{\varepsilon _{0}}}$
Ampère's law (with Maxwell's extension)	$\nabla \times \mathbf {B} -{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}=\mu _{0}\mathbf {j} _{\mathrm {e} }$
Gauss's law for magnetism	$\nabla \cdot \mathbf {B} =0$	$\nabla \cdot \mathbf {B} =\rho _{\mathrm {m} }$	$\nabla \cdot \mathbf {B} =\mu _{0}\rho _{\mathrm {m} }$
Faraday's law of induction	$-\nabla \times \mathbf {E} -{\frac {\partial \mathbf {B} }{\partial t}}=0$	$-\nabla \times \mathbf {E} -{\frac {\partial \mathbf {B} }{\partial t}}=\mathbf {j} _{\mathrm {m} }$	$-\nabla \times \mathbf {E} -{\frac {\partial \mathbf {B} }{\partial t}}=\mu _{0}\mathbf {j} _{\mathrm {m} }$
Lorentz force equation	$\mathbf {F} =q_{\mathrm {e} }\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)$	${\begin{aligned}\mathbf {F} ={}&q_{\mathrm {e} }\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)+\\&{\frac {q_{\mathrm {m} }}{\mu _{0}}}\left(\mathbf {B} -\mathbf {v} \times {\frac {\mathbf {E} }{c^{2}}}\right)\end{aligned}}$	${\begin{aligned}\mathbf {F} ={}&q_{\mathrm {e} }\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)+\\&q_{\mathrm {m} }\left(\mathbf {B} -\mathbf {v} \times {\frac {\mathbf {E} }{c^{2}}}\right)\end{aligned}}$

Name	Gaussian units	SI units (Wb)	SI units (A⋅m)
Maxwell's equations (assumingLorenz gauge)	${\begin{aligned}\Box \varphi _{\mathrm {e} }=&-4\pi \rho _{\mathrm {e} }\\\Box \mathbf {A} _{\mathrm {e} }=&-{\frac {4\pi }{c}}\mathbf {j} _{\mathrm {e} }\\\Box \varphi _{\mathrm {m} }=&-4\pi \rho _{\mathrm {m} }\\\Box \mathbf {A} _{\mathrm {m} }=&-{\frac {4\pi }{c}}\mathbf {j} _{\mathrm {m} }\\\end{aligned}}$	${\begin{aligned}\Box \varphi _{\mathrm {e} }=&-{\frac {\rho _{\mathrm {e} }}{\varepsilon _{0}}}\\\Box \mathbf {A} _{\mathrm {e} }=&-\mu _{0}\mathbf {j} _{\mathrm {e} }\\\Box \varphi _{\mathrm {m} }=&-{\frac {\rho _{\mathrm {m} }}{\mu _{0}}}\\\Box \mathbf {A} _{\mathrm {m} }=&-\varepsilon _{0}\mathbf {j} _{\mathrm {m} }\\\end{aligned}}$	${\begin{aligned}\Box \varphi _{\mathrm {e} }=&-{\frac {\rho _{\mathrm {e} }}{\varepsilon _{0}}}\\\Box \mathbf {A} _{\mathrm {e} }=&-\mu _{0}\mathbf {j} _{\mathrm {e} }\\\Box \varphi _{\mathrm {m} }=&-\rho _{\mathrm {m} }\\\Box \mathbf {A} _{\mathrm {m} }=&-{\frac {\mathbf {j} _{\mathrm {m} }}{c^{2}}}\\\end{aligned}}$
Lorenz gauge condition	${\begin{aligned}&{\frac {1}{c}}{\frac {\partial }{\partial t}}\varphi _{\mathrm {e} }+\nabla \cdot \mathbf {A} _{\mathrm {e} }=0\\&{\frac {1}{c}}{\frac {\partial }{\partial t}}\varphi _{\mathrm {m} }+\nabla \cdot \mathbf {A} _{\mathrm {m} }=0\\\end{aligned}}$	${\begin{aligned}&{\frac {1}{c^{2}}}{\frac {\partial }{\partial t}}\varphi _{\mathrm {e} }+\nabla \cdot \mathbf {A} _{\mathrm {e} }=0\\&{\frac {1}{c^{2}}}{\frac {\partial }{\partial t}}\varphi _{\mathrm {m} }+\nabla \cdot \mathbf {A} _{\mathrm {m} }=0\\\end{aligned}}$
Relation to fields	${\begin{aligned}\mathbf {E} =&-\nabla \varphi _{\mathrm {e} }-{\frac {1}{c}}{\frac {\partial \mathbf {A} _{\mathrm {e} }}{\partial t}}-\nabla \times \mathbf {A} _{\mathrm {m} }\\\mathbf {B} =&-\nabla \varphi _{\mathrm {m} }-{\frac {1}{c}}{\frac {\partial \mathbf {A} _{\mathrm {m} }}{\partial t}}+\nabla \times \mathbf {A} _{\mathrm {e} }\\\end{aligned}}$	${\begin{aligned}\mathbf {E} =&-\nabla \varphi _{\mathrm {e} }-{\frac {\partial \mathbf {A} _{\mathrm {e} }}{\partial t}}-{\frac {1}{\varepsilon _{0}}}\nabla \times \mathbf {A} _{\mathrm {m} }\\\mathbf {B} =&-\mu _{0}\nabla \varphi _{\mathrm {m} }-\mu _{0}{\frac {\partial \mathbf {A} _{\mathrm {m} }}{\partial t}}+\nabla \times \mathbf {A} _{\mathrm {e} }\\\end{aligned}}$

Name	Notation	Gaussian units	SI units (Wb or A⋅m)
Electromagnetic tensor	$F^{\alpha \beta }=(F^{01},F^{02},F^{03},\;F^{23},F^{31},F^{12})$	$(-\mathbf {E} ,\;-\mathbf {B} )$	$(-\mathbf {E} /c,\;-\mathbf {B} )$
Dual electromagnetic tensor	${\tilde {F}}^{\alpha \beta }=({\tilde {F}}^{01},{\tilde {F}}^{02},{\tilde {F}}^{03},\;{\tilde {F}}^{23},{\tilde {F}}^{31},{\tilde {F}}^{12})$	$(-\mathbf {B} ,\;\mathbf {E} )$	$(-\mathbf {B} ,\;\mathbf {E} /c)$
Four-current	$J_{\mathrm {e} }^{\alpha }=(J_{\mathrm {e} }^{0},J_{\mathrm {e} }^{1},J_{\mathrm {e} }^{2},J_{\mathrm {e} }^{3})$	$(c\rho _{\mathrm {e} },\;\mathbf {j} _{\mathrm {e} })$
Four-current	$J_{\mathrm {m} }^{\alpha }=(J_{\mathrm {m} }^{0},J_{\mathrm {m} }^{1},J_{\mathrm {m} }^{2},J_{\mathrm {m} }^{3})$	$(c\rho _{\mathrm {m} },\;\mathbf {j} _{\mathrm {m} })$
Four-potential	$A_{\mathrm {e} }^{\alpha }=(A_{\mathrm {e} }^{0},A_{\mathrm {e} }^{1},A_{\mathrm {e} }^{2},A_{\mathrm {e} }^{3})$	$(\varphi _{\mathrm {e} },\mathbf {A} _{\mathrm {e} })$	$(\varphi _{\mathrm {e} }/c,\;\mathbf {A} _{\mathrm {e} })$
Four-potential	$A_{\mathrm {m} }^{\alpha }=(A_{\mathrm {m} }^{0},A_{\mathrm {m} }^{1},A_{\mathrm {m} }^{2},A_{\mathrm {m} }^{3})$	$(\varphi _{\mathrm {m} },\mathbf {A} _{\mathrm {m} })$	$(\varphi _{\mathrm {m} }/c,\;\mathbf {A} _{\mathrm {m} })$
Four-force	$f_{\alpha }=(f_{0},f_{1},f_{2},f_{3})$	${\frac {1}{\sqrt {1-v^{2}/c^{2}}}}(\mathbf {F} \cdot \mathbf {v} ,\;-\mathbf {F} )$

Maxwell equations	Gaussian units	SI units (Wb)	SI units (A⋅m)
Ampère–Gauss law	$\partial _{\alpha }F^{\alpha \beta }={\frac {4\pi }{c}}J_{\mathrm {e} }^{\beta }$	$\partial _{\alpha }F^{\alpha \beta }=\mu _{0}J_{\mathrm {e} }^{\beta }$
Faraday–Gauss law	$\partial _{\alpha }{{\tilde {F}}^{\alpha \beta }}={\frac {4\pi }{c}}J_{\mathrm {m} }^{\beta }$	$\partial _{\alpha }{{\tilde {F}}^{\alpha \beta }}={\frac {1}{c}}J_{\mathrm {m} }^{\beta }$	$\partial _{\alpha }{{\tilde {F}}^{\alpha \beta }}={\frac {\mu _{0}}{c}}J_{\mathrm {m} }^{\beta }$
Lorentz force law	$f_{\alpha }=\left[q_{\mathrm {e} }F_{\alpha \beta }+q_{\mathrm {m} }{{\tilde {F}}_{\alpha \beta }}\right]{\frac {v^{\beta }}{c}}$	$f_{\alpha }=\left[q_{\mathrm {e} }F_{\alpha \beta }+{\frac {q_{\mathrm {m} }}{\mu _{0}c}}{{\tilde {F}}_{\alpha \beta }}\right]v^{\beta }$	$f_{\alpha }=\left[q_{\mathrm {e} }F_{\alpha \beta }+{\frac {q_{\mathrm {m} }}{c}}{{\tilde {F}}_{\alpha \beta }}\right]v^{\beta }$

Name	Gaussian units	SI units (Wb)	SI units (A⋅m)
Maxwell's equations	$\partial ^{\alpha }\partial _{\alpha }A_{\mathrm {e} }^{\beta }-\partial ^{\beta }\partial _{\alpha }A_{\mathrm {e} }^{\alpha }={\frac {4\pi }{c}}J_{\mathrm {e} }^{\beta }$	$\partial ^{\alpha }\partial _{\alpha }A_{\mathrm {e} }^{\beta }-\partial ^{\beta }\partial _{\alpha }A_{\mathrm {e} }^{\alpha }=\mu _{0}J_{\mathrm {e} }^{\beta }$
Maxwell's equations	$\partial ^{\alpha }\partial _{\alpha }A_{\mathrm {m} }^{\beta }-\partial ^{\beta }\partial _{\alpha }A_{\mathrm {m} }^{\alpha }={\frac {4\pi }{c}}J_{\mathrm {m} }^{\beta }$	$\partial ^{\alpha }\partial _{\alpha }A_{\mathrm {m} }^{\beta }-\partial ^{\beta }\partial _{\alpha }A_{\mathrm {m} }^{\alpha }=\varepsilon _{0}J_{\mathrm {m} }^{\beta }$	$\cdots ={\frac {1}{c^{2}}}J_{\mathrm {m} }^{\beta }$
Lorenz gauge condition	$\partial _{\alpha }A_{\mathrm {e} }^{\alpha }=0,\quad \partial _{\alpha }A_{\mathrm {m} }^{\alpha }=0$
Relation to fields (Cabibbo–Ferrari-Shanmugadhasan relation)	$F^{\alpha \beta }=\partial ^{\alpha }A_{\mathrm {e} }^{\beta }-\partial ^{\beta }A_{\mathrm {e} }^{\alpha }-\varepsilon ^{\alpha \beta \mu \nu }\partial _{\mu }A_{{\mathrm {m} }\nu }$ ${\tilde {F}}^{\alpha \beta }=\partial ^{\alpha }A_{\mathrm {m} }^{\beta }-\partial ^{\beta }A_{\mathrm {m} }^{\alpha }+\varepsilon ^{\alpha \beta \mu \nu }\partial _{\mu }A_{{\mathrm {e} }\nu }$	$F^{\alpha \beta }=\partial ^{\alpha }A_{\mathrm {e} }^{\beta }-\partial ^{\beta }A_{\mathrm {e} }^{\alpha }-\mu _{0}c\varepsilon ^{\alpha \beta \mu \nu }\partial _{\mu }A_{{\mathrm {m} }\nu }$ ${\tilde {F}}^{\alpha \beta }=\mu _{0}c(\partial ^{\alpha }A_{\mathrm {m} }^{\beta }-\partial ^{\beta }A_{\mathrm {m} }^{\alpha })+\varepsilon ^{\alpha \beta \mu \nu }\partial _{\mu }A_{{\mathrm {e} }\nu }$

Charges and currents	Fields
${\begin{pmatrix}\rho _{\mathrm {e} }\\\rho _{\mathrm {m} }\end{pmatrix}}={\begin{pmatrix}\cos \xi &-\sin \xi \\\sin \xi &\cos \xi \\\end{pmatrix}}{\begin{pmatrix}\rho _{\mathrm {e} }'\\\rho _{\mathrm {m} }'\end{pmatrix}}$	${\begin{pmatrix}\mathbf {E} \\\mathbf {H} \end{pmatrix}}={\begin{pmatrix}\cos \xi &-\sin \xi \\\sin \xi &\cos \xi \\\end{pmatrix}}{\begin{pmatrix}\mathbf {E'} \\\mathbf {H'} \end{pmatrix}}$
${\begin{pmatrix}\mathbf {J} _{\mathrm {e} }\\\mathbf {J} _{\mathrm {m} }\end{pmatrix}}={\begin{pmatrix}\cos \xi &-\sin \xi \\\sin \xi &\cos \xi \\\end{pmatrix}}{\begin{pmatrix}\mathbf {J} _{\mathrm {e} }'\\\mathbf {J} _{\mathrm {m} }'\end{pmatrix}}$	${\begin{pmatrix}\mathbf {D} \\\mathbf {B} \end{pmatrix}}={\begin{pmatrix}\cos \xi &-\sin \xi \\\sin \xi &\cos \xi \\\end{pmatrix}}{\begin{pmatrix}\mathbf {D'} \\\mathbf {B'} \end{pmatrix}}$

Units	Condition
SI units (weber convention)^[29]	${\frac {q_{\mathrm {e} }q_{\mathrm {m} }}{2\pi \hbar }}\in \mathbb {Z}$
SI units (ampere-meter convention)	${\frac {q_{\mathrm {e} }q_{\mathrm {m} }}{2\pi \varepsilon _{0}\hbar c^{2}}}\in \mathbb {Z}$
Gaussian-cgs units	$2{\frac {q_{\mathrm {e} }q_{\mathrm {m} }}{\hbar c}}\in \mathbb {Z}$

Movatterモバイル変換

Historical background

Early science and classical physics

Quantum mechanics

Poles and magnetism in ordinary matter

Maxwell's equations

In SI units

Potential formulation

Tensor formulation

Duality transformation

Dirac's quantization

Topological interpretation

Dirac string

Grand unified theories

String theory

Mathematical formulation

Grand unified theories

Searches for magnetic monopoles

"Monopoles" in condensed-matter systems

See also

Notes

References

Bibliography

External links