Maxwell’s Equations, Quick and Easy
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A theorem of the differential calculus tells us that if we know the divergence and the curl of a vector-field, then we can reconstruct the vector-field itself. By the beginning of the Nineteenth Century natural philosophers had gained the knowledge of the divergence of the electric field in the form of Coulomb’s law. If they had possessed the more explicit form in Gauss’s law and if they had possessed the theory of Special Relativity, they could have obtained the curl of the electric field through discovering the existence of, and the curl and divergence of, a new field, the magnetic field.
In the differential form of Gauss’s law we have
In that equation the epsilon represents the electric permittivity of vacuum and reflects the fact that we use two different methods to measure electric charge and electric force. We also have the law of conservation of electric charge, which law we express through a continuity equation,
In that equation j represents the density of electric current flowing through a minuscule volume in which static electric charge exists with a density represented by the letter rho.
Every electric charge surrounds itself with an electrostatic potential, a voltage field that we represent with the Greek letter phi. The electric field simply follows the negative gradient of that potential. If a second electric charge comes into that potential field, it gains energy in accordance with
That increase in energy confers an increased mass upon the particle carrying q2 in accordance with Einstein’s formula, e=mc2. If the particle carrying q2 moves with some velocity v2, then the increased mass will also give the particle an increase in its linear momentum,
In accordance with Newton’s third law of motion, the particle carrying q1 must gain an equal amount of linear momentum in the opposite direction,
The law pertaining to conservation of energy tells us that e1=e2, so we can rewrite that equation as
in which A2 represents the electrotonic field emanating from q2.
We must thus add a term to the description of the electric force and in so doing we lay out the essence of Richard Feynman’s derivation of the existence and form of the magnetic force from Coulomb’s law and Special Relativity. We have, then,
For the true derivative of the electrotonic field we have
In going from the first line in that equation to the second I dropped two terms out of the vector identity associated with the operator v1@L because v1 has no derivatives: it’s essentially a constant over all space. We thus have Equation 7 as
in which B=LxA, the magnetic induction field. The third term on the right side of that equation is generally absorbed into the first term, by way of Equation 6, as a relativistic correction to the electrostatic potential.
Now we have the curl of the electric field,
In calculating that curl, the first and third terms on the right side of Equation 9 drop out because the curl of a gradient always equals zero.
The divergence of Equation 10 gives us
We get that result because the divergence of a curl always equals zero. Because the operations of divergence and time-differentiation commute with each other (that is, the order in which we apply them to B makes no difference in the result), we know thatL@B equals some constant. Comparing that last statement with the content of Equation 1 tells us that the constant represents the distribution of a charge-like source of the magnetic induction field. That cannot change, according to Equation 11, but, because it exerts a magnetic force upon any electrically charged objects passing through it, it must change (in accordance with Newton’s third law of motion). We can only resolve that dilemma by asserting that the distribution must have a form that does not exert a magnetic force, so now we know that
That’s no surprise because we defined B as the curl of the electrotonic field.
We can also modify Equation 10 by absorbing the vee-cross-bee term into the electric field. Thus we get a description of an electric field whose curl depends only upon changes in the magnetic induction field due to changes in the sources of that field and not upon changes in the relationship between a moving body and the magnetic induction field. We have, then,
which is Faraday’s law of electromagnetic induction.
Finally we divide the electric charge out of Equation 6 and calculate the double curl of what’s left to get
The first term on the right side of that equation vanishes because the velocity is solely associated with the motion of the particle generating the electrostatic field and, thus, occurs in the calculation as a constant field. Applying the appropriate vector identities, we transform that equation into
In going from the second line to the third I merely replaced the gradient of phi with the corresponding electric field. Referring to Equation 1 for the divergence of the electric field and noting that the product of the electric charge density and the velocity at which the charged particles move equals the electric current density, we have that equation as
In the second term on the right I exploited the fact that
Equation 7 tells us that
so Equation 16 becomes
In that equation the coefficient on the current density,
represents the magnetic permeability of vacuum and reflects the fact that we use two different methods to measure electric current and magnetic force.
Calculate the divergence of Equation 19. The left side vanishes because the divergence of a curl comes identically equal to zero. The first two terms on the right side vanish because their divergence, with substitution from Equation 1, expresses the conservation law pertaining to electric charge. That leaves only the third term on the right, the total time derivative of the Lorentz forcefield, which gives us a bit of a puzzle. Equation 19 looks like Ampere’s law, but Ampere’s law doesn’t contain an explicit reference to the Lorentz force.
At the beginning of "On the Electrodynamics of Moving Bodies" Einstein noted that the electromotive force induced in a wire has the same form, regardless of whether we conceive the wire moving relative to the magnetic field or we conceive the sources of the magnetic field moving relative to the wire. Observers in the two inertial frames implicit in that description must agree, certainly, that an electric current flows in the wire; they only differ on how it gets there. That analysis applies to our Lorentz-force term. The total time derivative on that term indicates that the change in the Lorentz force comes from both changes in the sources and movement of the observers in a spatially-varying field. But in a frame in which the observers are not moving the change in the field comes entirely from changes in the sources, which changes we indicate with the partial time derivative. With that understanding, that the observers don’t move but the sources of the field do, we can replace the third term in Equation 19 by the partial time derivative and then absorb it into the second term. So we have
which is Ampere’s law as modified by Maxwell.
Equations 1, 11,13, and 21 comprise Maxwell’s Equations. With those equations we can construct a description of the general electromagnetic field. We can then incorporate boundary conditions into that general solution to transform it into a properly specific solution of a given problem.
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