Electrodynamics MOC

Maxwell’s equations

  1. Gauß’s law
  2. Gauß’s law for magnetic flux
  3. Faraday’s law of induction
  4. Ampère’s circuital law

Maxwell’s equations form the basis of Electrodynamics, along with the Lorentz force law. The differential and integral forms are translated using generalisations of the Fundamental theorem of calculus, namely Острогра́дский’s divergence theorem and Stokes’s theorem. Note that Gauß’s law for magnetic flux and Faraday’s law of induction are automatically satisfied by defining the fields in terms of Electric and magnetic potentials.

See also Maxwell’s equations in materials

Differential form

These use the mathematical language of Divergence and Curl.

  1. The divergence of an electric field at a given point (how much field lines move away from that point) is proportional to the charge density at that point.
⃗∇⋅⃗𝐄=𝜌𝑞𝜀0

^D1 2. The divergence of a magnetic field is ⃗𝟎 at all points, i.e. they do not diverge, since there are no monopoles. Thus a magnetic field is solenoidal.

⃗∇⋅⃗𝐁=0

^D2 3. Curl (tendency for field lines to orbit) in an electric field results in a change in a magnetic field, and vice versa.

⃗∇×⃗𝐄=−𝜕⃗𝐁𝜕𝑡

^D3 4. Electric current passing through a closed circuit results in the magnetic curl, and vice versa.1

⃗∇×⃗𝐁=𝜇0(⃗𝐉+𝜀0𝜕⃗𝐄𝜕𝑡)

Integral form

These use the mathematical language of the Double integral, Triple integral; and more specifically closed Circulation and Flux

  1. The nett number of electric field lines escaping through the boundary of some solid Ω is proportional to the amount of charge contained within that solid (contained charge can be calculated using an integral over Charge density).
Φ𝐸=⊂⊃∬𝜕Ω⃗𝐄⋅𝑑⃗𝐚=1𝜀0∭Ω𝜌𝑞𝑑𝜏′

^I1 2. The nett number of magnetic field lines escaping through the boundary of some solid Ω is always 0, i.e. the number of magnetic field lines leaving a space is equal to the number entering that space, since there are no monopoles.

⊂⊃∬𝜕Ω⃗𝐁⋅𝑑⃗𝐚=0

^I2 3. Electric current moving anticlockwise in a closed loop creates a change in magnetic flux inside the loop in the direction of negative orientation.

E=∮𝜕Σ⃗𝐄⋅𝑑⃗ℓ=−𝜕Φ𝐵𝜕𝑡=−𝜕𝜕𝑡∬Σ⃗𝐁⋅𝑑⃗𝐚

^I3 4. Nett current moving through a cross section solid conductor Σ or a change in electric flux cutting through Σ (working together additively) results in a magnetic field circulating around the conductor, and vice versa.

∮𝜕Σ⃗𝐁⋅𝑑⃗ℓ=𝜇0(𝐼Σ+𝜀0𝜕Φ𝐸𝜕𝑡)=𝜇0(∬Σ⃗𝐉⋅𝑑⃗𝐚+𝜀0𝜕𝜕𝑡∬Σ⃗𝐄⋅𝑑⃗𝐚)

Quantities

tidy | SemBr | en

Footnotes

  1. Note that partial term here is an adjustment that was made by Maxwell. ↩

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