The four Maxwell's equations and the Lorentz force law (which together constitution the fundations of all the classical electromagnetism) are listed below:
(i) `oint vecB.vec(ds)=q//(in_0)`
(ii) `oint vecB.vec(ds)=0`
(iii) `oint vecE.vec(dl)=-d/(dt) int_svecB.vec(ds)`
(iv) `oint vecB.vec(dl)=mu_0I+mu_0d/(dt)int_s vecb.vec(ds)`
Lorentz force law: `vecF=q(vecE+vecvxxvecB)`
Answer the following question regarding these equation:
(a) Give the name (s) associated with some of the four equation above.
(b) Which equations above contain source `vecE and vecB` and which do not? what do the equations reduce to in a source-free region?
(c) Write down Maxwell's equations for steady (i.e. time independent) electric and magnetic fields.
(d) If magnetic monopoles existed, which of the equations would be modified? Suggest how they might be modified?
(e) Which of the four equations shown that magnetic field lines cannot start from a point nor end at a point?
(f) Which of the four equations show that electrostatic field lines cannot form closed loops?
(g) The equations listed above refer to integrals of `vecE and vecB`over loops/surfaces Can we write down equations for `vecE and vecB` for each point in space?
(h) Are the equations listed above true for different types of media: dielectrics, conductors, plasmas etc.?
(i) Are the equation true fora arbitrarily high and low values of `vecE,vecB,q,I`?

Explanation

(a) (i) Guss's law in electrostatics. (ii) Guss's
law in magnetostatics. (iii) Faraday's law of
electromagnetic induction (iv) Ampere's
Circutal law with Maxwell's modification.
(b) Equations (i) and (iv) contain the source q,
I: equations (ii) and (iii) do not. To obtain
equations in source free region, simply put
`q=0` and `I=0`. Then
`oint vecE.vec(ds)=0` [from eqn(i)]
and `oint vecE.vec(dl)=mu_0in_0d/(dt)int_svecE.vec(ds)`
[From eq(iv)]
(c) Maxwell's equations will be time
independent if the derivative of the physical
quantity involved with time is zero. Putting this
concept on the right hand side of equation (iii)
and (iv), we have Maxwell's equations as:
`oint vecE.vec(ds)=Q/(in_0) oint vecE.vec(ds)=0`
`oint vecE.vec(dl)=0 oint vecE.vec(dl)=mu_0I`
(d) Equation (ii) and (iii) would be modified .
Equation (ii) is based on the fact that monopoles
do not exist. If the monopoles exist, the right
hand side would contain a term say `q_m`
representing magntic dipole strength,
analogous to Gauss's law, in electrostatics, we would have
`oint vecE.vec(ds)=q_mxxcostant.`
Further equation (iii) would also be modified.
An additional term `I_m` representing the current
due to flow of magnetic charge would have to
be included on the right hand side of equation
(iii) analogous to the electric charge current of
equation (iv), we would have
`oint vecE.vec(dl)=I_mxxconstant-d/(dt)int_svecB.vec(ds)`
All this is of course, based on the expection of
symmetry of from of the equaitons for E and B.
Nature may never show up monopoles or else
even if monopoles exist, the actual modification
of Maxwell's equations might be very different.
(e) Equation (ii) only
(f) Equation (i) shows that the elctrostatic field
lines cannot from closed loops, as electrostatic field
field lines cannot pass through conductors.
(g) Yes, we can. Maxwell's equations can be
cast as different equations valid at every point
in space and at every instant.
(h) Maxwell's equations are true in all media.
But in macroscopic media, it is usually
convenient to write down equations for average
of `vecE and vecB` over regions which are small
macroscopically but large enough to contain a
very large number of atoms etc. The resulting
macroscopic Maxwell's equtions are of great
practical use.
(i) Maxwell's equations are the basic laws of
classical electromagnetism. They are true in all
media and for any value of E,B,q,I ect. whithin
the domain of validity of classical
electromagnetism. The precise domain of
validity is hard to specify and need not concern
us here.