Dear all,
I am a bit confused from the electric and magnetic charges and fields as used in RS, so I would appreciate if someone could help me by understanding it. My confusion results from the following:
In the book Structure of the physical universe Larson introduces the concept of magnetic charge and magnetic potential M. The magnetic potential he uses is probably different from magnetic vector potential commonly denoted as A. He explicitly mentions that the gradient of the magnetic potential is the magnetic field intensity (chapter XXIX). Later he makes it clear that that by magnetic field intensity (which is unfortunately in english a bit confusing because of it's usage for two distinct fields - in terms of mathematical fields) he termed magnetic induction commonly denoted as B. However
∇×B = j+dD/dt
Which is often a non-zero expression. From this would follow that ∇×(∇M) is not equal 0 - that is mathematically impossible...
Further more (and more importantly) he explicitly shows the dimensions of the electric charge q as t.s^{-1} and the dimensions of the magnetic flux φ as t^{2}.s^{-2}. However the product of the two qφ should have the dimensions of action [coulomb× weber]=[joule×seconds] which in reciprocal system is t^{2}.s^{-1} whereas if we multiply their dimensions we get the dimensions of mass t^{-3}s^{-3}.
If you can clarify this please go into detail as much as you can.
Thank you very much.
Jan
Electric and magnetic fields in RS
Electromagnetism
Larson did not get the concept of charge quite right due to the way he assigned units to capacitance. See pages 169-170 (Electrical Storage) of Basic Properties of Matter, where he discusses the difference between charge as energy (Q=t/s) and charge as electric quantity (q=s). Also see Table 30 on page 218 and Chapter 20 for what he updated since writing Structure of the Physical Universe. (Click on BPOM title for a PDF of the book, available for free on http://reciprocalsystem.org )
I did not understand why ∇×(∇M) must be equal to zero. Can you clarify?
This is addressed in BPOM, because "charge" here is the "quantity of charge" which has units of space (s), not energy (t/s). So the equation becomes s x t^{2}/s^{2} = t^{2}/s, units of action.Further more (and more importantly) he explicitly shows the dimensions of the electric charge q as t.s^{-1} and the dimensions of the magnetic flux φ as t^{2}.s^{-2}. However the product of the two qφ should have the dimensions of action [coulomb× weber]=[joule×seconds] which in reciprocal system is t^{2}.s^{-1} whereas if we multiply their dimensions we get the dimensions of mass t^{-3}s^{-3}.
I did not understand why ∇×(∇M) must be equal to zero. Can you clarify?
Every dogma has its day...
Curl of gradient
Curl of gradient is for all well-behaved functions zero because of geometrical considerations. Loosely explained the gradient is how steep is your path to the "hill". Curl is something like how fast a wihrlpool rotates. Obviously a path to a hill can not rotate so a non-zero rotation out of gradient would mean that when you reach the top of the hill you are on the beginning of the hill which is in ordinary geometry absurd. It can be shown for arbitrary polynomial and that means also all smooth functions we normally imagine. (However there are functions that do not obey and perhaps can even exist in nature...)
Thank you, for your answer, it's clear now.
Jan
PS: I fixed some bugs in this post
Thank you, for your answer, it's clear now.
Jan
PS: I fixed some bugs in this post