6. Current
In this section the current term of Maxwell's equations is finally derived in a few simple steps. Thus we now have Maxwell's equations from our initial postulates. It is not exactly new information, but expected in view of experiment. However, it is important to appreciate the unique path and perspectives by which they were derived.
Eq.(5-9) is always true for currents which are caused by the motion of point charges, since dq/dt always exists at the microscopic level. If there were an ideally continuous loop of charge, dq/dt (in Eq.(5-8)) would be zero, but then the two remaining terms would cancel out and there would be no field1. This actually makes sense, since a loop of this sort would have nothing physically occurring that would indicate that a current existed. It is interesting that these last two terms can be equivalent. It can be shown by replacing the electric field in Eq.(5-9) by its source. In general, however, each element of charge can have a different velocity, thus
(1)
where r' is the position vector of the charge and r is the field vector. Since the magnetic field superposes this can be summed to find the total field. This leads to:
(2)
which is just the Biot-Savart law. In this theory, the Biot-Savart law is not a postulate, but is a derived law. The differential form then can be derived by the usual process2. This yields
(3)
Thus Eq.(4-7) becomes
(4)
We have shown that the usual magnetic curl equation can be considered merely a natural extension of Eq.(1-2). Which is of some importance in consideration of the particle-less space theorized here.
The current density J is in relation to absolute rest. If the protons are also moving, an opposite magnetic field is created which partially cancels the field created by the moving electrons. What survives is dependent on the relative current density related to the relative velocity between the electrons and protons in the conductor. So in a conductor, J is the relative current. Interestingly, if there is a free current, the affective magnetic field is still based on a relative J. (See Section 12).
1. If the loop of charge density is perfectly continuous, then despite a "current" the amount of charge in space never changes at any point on the loop. If, however, the charge is made up of point charges with intervening empty spaces between them, then at a constant position on the loop the amount of charge is increasing and decreasing in an ongoing process.
2. This derivation can be found on page 215 of Griffiths1 or page 173 of Jackson2.
Contents
Next