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.

Detail

Charge density and the electric field

: abla cdot mathbf{D} = ho,

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where { ho} is the free electric charge density (in units of C/m3), not including dipole charges bound in a material, and mathbf{D} is the electric displacement field (in units of C/m2). This equation corresponds to Coulomb's law for stationary charges in vacuum.

Related Topics:
Electric displacement field - Coulomb's law

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The equivalent integral form (by the divergence theorem), also known as Gauss' law, is:

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: oint_A mathbf{D} cdot dmathbf{A} = Q_mbox{enclosed}

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where dmathbf{A} is the area of a differential square on the closed surface A with an outward facing surface normal defining its direction, and Q_mbox{enclosed} is the free charge enclosed by the surface.

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In a linear material, mathbf{D} is directly related to the electric field mathbf{E} via a material-dependent constant called the permittivity, epsilon:

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:mathbf{D} = arepsilon mathbf{E}.

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Any material can be treated as linear, as long as the electric field is not extremely strong. The permittivity of free space is referred to as epsilon_0, and appears in:

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: abla cdot mathbf{E} = rac{ ho_t}{arepsilon_0}

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where, again, mathbf{E} is the electric field (in units of V/m), ho_t is the total charge density (including bound charges), and epsilon_0 (approximately 8.854 pF/m) is the permittivity of free space. epsilon can also be written as arepsilon_0 cdot arepsilon_r, where epsilon_r is the material's relative permittivity or its dielectric constant.

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Compare Poisson's equation.

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The structure of the magnetic field

: abla cdot mathbf{B} = 0

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mathbf{B} is the magnetic flux density (in units of teslas, T), also called the magnetic induction.

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Equivalent integral form:

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: oint_A mathbf{B} cdot dmathbf{A} = 0

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dmathbf{A} is the area of a differential square on the surface A with an outward facing surface normal defining its direction.

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Like the electric field's integral form, this equation only works if the integral is done over a closed surface.

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This equation is related to the magnetic field's structure because it states that given any volume element, the net magnitude of the vector components that point outward from the surface must be equal to the net magnitude of the vector components that point inward. Structurally, this means that the magnetic field lines must be closed loops. Another way of putting it is that the field lines cannot originate from somewhere; attempting to follow the lines backwards to their source or forward to their terminus ultimately leads back to the starting position. Hence, this is the mathematical formulation of the assumption that there are no magnetic monopoles.

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A changing magnetic flux and the electric field

: abla imes mathbf{E} = -rac {partial mathbf{B}}{partial t}

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Equivalent integral Form:

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: oint_{s} mathbf{E} cdot dmathbf{s} = - rac {dPhi_{mathbf{B}}} {dt} where Phi_{mathbf{B}} = int_{A} mathbf{B} cdot dmathbf{A}

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where

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ΦB is the magnetic flux through the area A described by the second equation

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E is the electric field generated by the magnetic flux

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s is a closed path in which current is induced, such as a wire.

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The electromotive force (sometimes denoted mathcal{E}, not to be confused with the permittivity above) is equal to the value of this integral.

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This law corresponds to the Faraday's law of electromagnetic induction.

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Some textbooks show the right hand sign of the Integral form with an N (representing the number of coils of wire that are around the edge of A) in front of the flux derivative. The N can be taken care of in calculating A (multiple wire coils means multiple surfaces for the flux to go through), and it is an engineering detail so it has been omitted here.

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The negative sign is necessary to maintain conservation of energy. It is so important that it even has its own name, Lenz's law.

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This equation relates the electric and magnetic fields, but it also has a lot of practical applications, too. This equation describes how electric motors and electric generators work. Specifically, it demonstrates that a voltage can be generated by varying the magnetic flux passing through a given area over time, such as by uniformly rotating a loop of wire through a fixed magnetic field. In a motor or generator, the fixed excitation is provided by the field circuit and the varying voltage is measured across the armature circuit. In some types of motors/generators, the field circuit is mounted on the rotor and the armature circuit is mounted on the stator, but other types of motors/generators employ the reverse configuration.

Related Topics:
Electric motor - Electric generator - Field - Armature

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Maxwell's equations apply to a right-handed coordinate system. To apply them unmodified to a left handed system would mean a reversal of polarity of magnetic fields (not inconsistent, but confusingly against convention).

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The source of the magnetic field

: abla imes mathbf{H} = mathbf{J} + rac {partial mathbf{D}} {partial t}

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where H is the magnetic field strength (in units of A/m), related to the magnetic flux B by a constant called the permeability, μ (B = μH), and J is the current density, defined by: J = ∫ρqvdV where v is a vector field called the drift velocity that describes the velocities of that charge carriers which have a density described by the scalar function ρq.

Related Topics:
Magnetic field strength - Permeability

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In free space, the permeability μ is the permeability of free space, μ0, which is defined to be exactly 4π×10-7 W/A·m. Also, the permittivity becomes the permittivity of free space ε0. Thus, in free space, the equation becomes:

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: abla imes mathbf{B} = mu_0 mathbf{J} + mu_0arepsilon_0 rac{partial mathbf{E}}{partial t}

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Equivalent integral form:

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:oint_s mathbf{B} cdot dmathbf{s} = mu_0 I_mbox{encircled} + mu_0arepsilon_0 int_A rac{partial mathbf{E}}{partial t} cdot d mathbf{A}

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s is the edge of the open surface A (any surface with the curve s as its edge will do), and Iencircled is the current encircled by the curve s (the current through any surface is defined by the equation: Ithrough A = ∫AJ·dA).

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If the electric flux density does not vary rapidly, the second term on the right hand side (the displacement flux) is negligible, and the equation reduces to Ampere's law.

Related Topics:
Electric flux density - Ampere's law

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