equations:maxwell_equations
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equations:maxwell_equations [2018/04/02 12:10] jakobadmin [Interpretation] |
equations:maxwell_equations [2023/04/02 03:14] (current) edi [Concrete] |
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| ====== Maxwell Equations ====== | ====== Maxwell Equations ====== | ||
| - | //see also [[models:classical_electrodynamics|Classical Electrodynamics]] and [[models:quantum_electrodynamics|Quantum Electrodynamics]]// | + | //see also [[models:classical_electrodynamics|Classical Electrodynamics]] and [[models:standard_model:qed|Quantum Electrodynamics]]// |
| <tabbox Intuitive> | <tabbox Intuitive> | ||
| - | The Maxwell equations that electric charge never gets lost but is always conserved. In addition, they tell us how charged objects interact with each other. | + | The Maxwell equations tell us that electric charge never gets lost but is always conserved. In addition, they tell us how charged objects interact with each other. |
| There are in total 4 Maxwell equations and each tells us something important about electricity, magnetism and their interplay. | There are in total 4 Maxwell equations and each tells us something important about electricity, magnetism and their interplay. | ||
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| ---- | ---- | ||
| - | The static limit of Maxwell's equations is known as [[equations:coulombs_law|Coulomb's law]]. | + | The static limit of Maxwell's equations is known as [[formulas:coulombs_law|Coulomb's law]]. |
| - | + | ||
| + | ---- | ||
| + | |||
| + | **Relationship to Lorentz and U(1) Gauge Symmetry** | ||
| + | |||
| + | The diagram below maps the path from Lorentz symmetry and U(1) gauge symmetry to the Maxwell equations. For a detailed explanation see [[https://esackinger.wordpress.com//blog/lie-groups-and-their-representations/#sym_to_maxwell|Fun with Symmetry]]. | ||
| + | |||
| + | {{:equations:sym_to_maxwell.jpg?nolink}} | ||
| <tabbox Abstract> | <tabbox Abstract> | ||
| The Maxwell equation can be written more compactly with the help of the field strength tensor and its dual | The Maxwell equation can be written more compactly with the help of the field strength tensor and its dual | ||
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| - | The Maxwell equations can be written extremely compactly with the help of [[advanced_tools:differential_forms|differential forms]]: | + | <diagram> |
| + | |||||-|-|Duality|-|-@2| | ||
| + | ||!||||||| | ||
| + | ||!||| AA|| |AA=$A$ (potential)| | ||
| + | ||!||| |!@4| | | | | | | ||
| + | |Differentiation|| AA|-@2| BB |AA=$F$ (field; Faraday)|BB=$*F$ (dual Field; Maxwell) | ||
| + | ||!||| |!@4| | | | !@4| | | ||
| + | ||!||| AA|| BB |AA=$dF=0$ (identity based $\partial \partial =0$)|BB=$d *F=4\pi *J$ | ||
| + | ||!||| || | | | !@4| | | ||
| + | ||!@4||| |||| BB |BB=$d* J=0 $ (expressed as an identity based on $\partial \partial =0$) | ||
| + | </diagram> | ||
| + | |||
| + | The statement $d* J=0 $ or in a more familiar notation $\nabla \times J =0$ encodes the automatic conservation of source. | ||
| + | |||
| + | (Source: page 370 in Gravitation by Misner, Thorne, Wheeler) | ||
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| + | |||
| + | As already indicated in the diagram above, Maxwell's equations can be written extremely compactly with the help of [[advanced_tools:differential_forms|differential forms]]: | ||
| \begin{eqnarray} | \begin{eqnarray} | ||
| d\star \bf F\it_{(2)} &= \star \bf J\it_{(1)} \\ | d\star \bf F\it_{(2)} &= \star \bf J\it_{(1)} \\ | ||
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| <tabbox Derivation> | <tabbox Derivation> | ||
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| -->Derivation of the inhomogeneous Maxwell equations# | -->Derivation of the inhomogeneous Maxwell equations# | ||
| - | The Maxwell equations can be derived from the [[frameworks:lagrangian_formalism|Lagrangian]] | + | The Maxwell equations can be derived from the [[formalisms:lagrangian_formalism|Lagrangian]] |
| $$ \mathcal{L}_{EM} = -{1\over 4} F_{\mu \nu}F^{\mu \nu} - J^{\mu}A_{\mu} .$$ | $$ \mathcal{L}_{EM} = -{1\over 4} F_{\mu \nu}F^{\mu \nu} - J^{\mu}A_{\mu} .$$ | ||
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| <tabbox Interpretation> | <tabbox Interpretation> | ||
| + | **The Maxwell equation describe the conservation of magnetic flux and electric charge** | ||
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| or equivalently | or equivalently | ||
| $$ \nabla \vec B = 0, \quad \nabla \times \vec E + \partial_t \vec B = 0 , $$ | $$ \nabla \vec B = 0, \quad \nabla \times \vec E + \partial_t \vec B = 0 , $$ | ||
| - | encode the __conservation of magnetic flux__. | + | encode the __conservation of magnetic flux__. This interpretation comes about since the equation $\partial_\mu \tilde{ F}^{ \mu \nu} =0$ describes a conserved current for each index $\nu$. When we have a conserved current, we automatically have a conserved charge. In this case the conserved charge is the magnetic flux. |
| The __inhomogeneous__ Maxwell equations | The __inhomogeneous__ Maxwell equations | ||
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| encode the __conservation of electric charge__. ([[https://arxiv.org/pdf/1611.05759.pdf|Source]]) | encode the __conservation of electric charge__. ([[https://arxiv.org/pdf/1611.05759.pdf|Source]]) | ||
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| - | For a particularly nice visual interpretation see [[https://arxiv.org/abs/1709.08492|A pictorial introduction to differential geometry, leading to Maxwell's equations as three pictures]] by Jonathan Gratus | ||
| ---- | ---- | ||
| + | **The Maxwell equations tell us which degrees of freedom are non-physical** | ||
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| When in the 19th century people tried to understand how electromagnetism works they also figured this out. However, they made also another intriguing discovery. When writing down the laws which govern electromagnetism, it turns out that electric and magnetic fields are intimately linked, and that they are just two sides of the same coin. That is the reason to call it electromagnetism. | When in the 19th century people tried to understand how electromagnetism works they also figured this out. However, they made also another intriguing discovery. When writing down the laws which govern electromagnetism, it turns out that electric and magnetic fields are intimately linked, and that they are just two sides of the same coin. That is the reason to call it electromagnetism. | ||
| - | //In the early 20th century it then became clear that both phenomena can be associated with a single particle, the photon. But then it was found that to characterize a photon only two numbers at each point in space and time are necessary. This implies that between the six numbers characterizing electric and magnetic fields relations exist. These are known as [[equations:maxwell_equations|Maxwell equations]]// in classical physics, or as quantum Maxwell dynamics in the quantum theory. If you would add, e. g., electrons to this theory, you would end up with [[models:quantum_electrodynamics|quantum electro dynamics - QED]]. | + | //In the early 20th century it then became clear that both phenomena can be associated with a single particle, the photon. But then it was found that to characterize a photon only two numbers at each point in space and time are necessary. This implies that between the six numbers characterizing electric and magnetic fields relations exist. These are known as [[equations:maxwell_equations|Maxwell equations]]// in classical physics, or as quantum Maxwell dynamics in the quantum theory. If you would add, e. g., electrons to this theory, you would end up with [[models:standard_model:qed|quantum electro dynamics - QED]]. |
| <cite>http://axelmaas.blogspot.de/2010/10/electromagnetism-photons-and-symmetry.html</cite></blockquote> | <cite>http://axelmaas.blogspot.de/2010/10/electromagnetism-photons-and-symmetry.html</cite></blockquote> | ||
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| <-- | <-- | ||
| + | ---- | ||
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| + | For a particularly nice visual interpretation see [[https://arxiv.org/abs/1709.08492|A pictorial introduction to differential geometry, leading to Maxwell's equations as three pictures]] by Jonathan Gratus | ||
| <tabbox History> | <tabbox History> | ||
| - | For a nice write-up of the experiments and lines of thoughts that led to the Maxwell equations see Theoretical concepts in physics by Malcolm Longair. | + | * [[https://spectrum.ieee.org/tech-history/dawn-of-electronics/the-long-road-to-maxwells-equations|The Long Road to Maxwell’s Equations How four enthusiasts helped bring the theory of electromagnetism to light]] by James C. Rautio |
| + | * For a nice write-up of the experiments and lines of thoughts that led to the Maxwell equations see the corresponding chapter in the book Theoretical concepts in physics by Malcolm Longair. | ||
| + | |||
| </tabbox> | </tabbox> | ||
equations/maxwell_equations.1522663858.txt.gz · Last modified: 2018/04/02 10:10 (external edit)
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