Maxwell's equations

Maxwell's equations are the set of four equations, attributed to James Clerk Maxwell (written by Oliver Heaviside), that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.

Formulation of Maxwell's equations in special relativity

In special relativity, in order to more clearly express the fact that Maxwell's equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwell's equations are written in terms of four-vectors and tensors in the "manifestly covariant" form:

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:J^eta = partial_lpha F^{lphaeta} ,!,

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and

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:0 = partial_gamma F_{lphaeta} + partial_eta F_{gammalpha} + partial_lpha F_{etagamma} ,!

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where J is the 4-current, F is the field strength tensor (Faraday tensor) (written as a 4 × 4 matrix), and partial_lpha = (partial/partial ct, abla) is the 4-gradient (so that partial_lpha partial^lpha is the d'Alembertian operator). (The α in the first equation is implicitly summed over, according to Einstein notation.) The first tensor equation expresses the two inhomogeneous Maxwell's equations: Gauss' law and Ampere's law with Maxwell's correction. The second equation expresses the other two, homogenous equations: Faraday's law of induction and the absence of magnetic monopoles.

Related Topics:
4-current - Tensor - Matrix - D'Alembertian - Einstein notation - Magnetic monopole

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More explicitly, J = (cρ, J) (as a contravariant vector), in terms of the charge density ρ and the current density J. In terms of the 4-potential (as a contravariant vector) ilde{A}^{lpha} = left(phi, mathbf{A} c ight), where φ is the electric potential and A is the magnetic vector potential in the Lorenz gauge left ( partial_lpha ilde{A}^lpha = 0 ight ), F can be expressed as:

Related Topics:
Contravariant vector - 4-potential

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:F^{lphaeta} = partial^lpha ilde{A}^eta - partial^eta ilde{A}^lpha ,!

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which leads to the 4 × 4 matrix (rank-2 tensor):

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:F^{lphaeta} = left(

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egin{matrix}

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0 & -E_x & -E_y & -E_z \

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E_x & 0 & -B_z & B_y \

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E_y & B_z & 0 & -B_x \

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E_z & -B_y & B_x & 0

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end{matrix}

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ight) .

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The fact that both electric and magnetic fields are combined into a single tensor expresses the fact that, according to relativity, both of these are different aspects of the same thing—by changing frames of reference, what seemed to be an electric field in one frame can appear as a magnetic field in another frame, and vice versa.

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(See Electromagnetic four-potential for the relationship between the d'Alembertian of the four-potential and the four-current, expressed in terms of the older vector operator notation).

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Different authors sometimes employ different sign conventions for the above tensors and 4-vectors (which does not affect the physical interpretation).

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Fαβ and Fαβ are not the same: they are the contravariant and covariant forms of the tensor, related by the metric tensor g. In special relativity the metric tensor introduces sign changes in some of Fs components; more complex metric dualities are encountered in general relativity.

Related Topics:
Contravariant - Covariant - Metric tensor - General relativity

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