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)
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)
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)
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)
Symbols Used:
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