Maxwell's Equations

Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. From them one can develop most of the working relationships in the field. Because of their concise statement, they embody a high level of mathematical sophistication and are therefore not generally introduced in an introductory treatment of the subject, except perhaps as summary relationships.

These basic equations of electricity and magnetism can be used as a starting point for advanced courses, but are usually first encountered as unifying equations after the study of electrical and magnetic phenomena.

1. Gauss' Law for Electricity The electric flux out of any closed surface is proportional to the total charge enclosed within the surface. The integral form of Gauss' Law finds application in calculating electric fields around charged objects.

Integral form: (Both E and dA are vectors)
E·dA = q/ε₀ = 4πkq

Differential form
▽·E = ρ/ε₀ = 4πkρ

2. Gauss' Law for Magnetism

The net magnetic flux out of any closed surface is zero. This amounts to a statement about the sources of magnetic field. For a magnetic dipole, any closed surface the magnetic flux directed inward toward the south pole will equal the flux outward from the north pole. The net flux will always be zero for dipole sources. If there were a magnetic monopole source, this would give a non-zero area integral. The divergence of a vector field is proportional to the point source density, so the form of Gauss' law for magnetic fields is then a statement that there are no magnetic monopoles.

Integral form: (Both B and dA are vectors)
B·dA = 0

Differential form
▽·B = 0

3. Faraday's Law of Induction

The line integral of the electric field around a closed loop is equal to the negative of the rate of change of the magnetic flux through the area enclosed by the loop.

This line integral is equal to the generated voltage or emf in the loop, so Faraday's law is the basis for electric generators. It also forms the basis for inductors and transformers.

Integral form: (Both E and ds are vectors)
E·ds = – dΦʙ/dt

Differential form
▽×B = –∂B/∂t

4. Ampere's Law

In the case of static electric field, the line integral of the magnetic field around a closed loop is proportional to the electric current flowing through the loop. This is useful for the calculation of magnetic field for simple geometries.

Integral form: (E, ds, dA are vectors)
∮B·ds = (μ₀i) + (1/c²)(d/dt)(∫ E·dA)

Differential form
▽×B = (4πkJ/c²) + (1/c²)(∂E/∂t)
▽×B = (J/ε₀c²) + (1/c²)(∂E/∂t)

Symbols Used:
E = Electric field (electric field intensity) in volt per meter
  or newton pe coulomb
B = Magnetic field (Magnetic field density) in Tesla or Weber per square meter
D = Electric displacement field (electrid flux density) in coulombs per squre meter or
  Newton per volt-meter
H = Magnetic field strength in amps per meter
▽· = the divergence operator, units are per meter
▽× = the curl operator, units are per meter
∂/∂t = partial derivative with respect to time, units are per second
dA = differential vector element of surface area A, with infinitesimally
  small magnitude and direction normal to surface S. Units are square meter
ds = differential vector element of path length tangential to the path/curve
  unit is meters
ε₀ = permittivity of free space, in Farads per meter
μ₀ = permeability of free space, in henries per meter
ρ = total charge density in coulombs per cubic meter
Φʙ = Magnetic flux through surface B, in Webers or volt-seconds
i = electric current in amps
J = current density in amps per square meter
c = speed of light in m/s
k = 1/(4πε₀)
c = 1/√(μ₀ε₀)


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