Speaking of the Lorentz force, the minus sign comes in on the equivalent force on a monopole like so:

F = g(H - v x D) (using the traditional 'g' to be the magnetic charge), or, putting it in force density terms:

f = rho_g(H - J_g x D), where J_g is the monopole current density.

And this shows something interesting. Many physicists prefer to think of E and B as the "fundamental fields", because they give the force on a particle directly. Well, with monopoles, it would be just the opposite, with D and H giving the force. I prefer to think of E and H, and D and B as the "pairs that go together".

And this shows something interesting (at least to me -- duality relations always fascinated me to no end). The force on an electric charge goes as E and B. This means that the force on charges is reduced by a medium with a higher permittivity, but the magnetic force between electric currents is *increased* by higher permeability. Surround your wires with iron, and the magnetic forces are increased.

So in an electric charge-only universe, there is tendency to think of epsilon and mu as sort of working backwards to each other, and that's the (E, B) vs (D, H) pairing POV.

However consider monopoles, and you see that flips around. The magnetic force between monopoles would be decreased by higher mu, but the electric force between monopole currents would be increased by a dielectric material.

The Duality picture exposes a lot of cut symmetries. For example, the voltage of the one is the current of the other. A volt is an electric potential (or EMF), energy per electric-charge, but it is also magnetic current, magnetic charge per time. And likewise an amp, electric-charge per time is also energy per magnetic-charge.

So if want the equivalent magnetic unit, you just sort of flip volts and amps and coulombs and webers. What would the unit of "magnetic capacitance" be? Well, capacitance is charge per unit voltage. In electric terms, that's a coulomb per volt (which we call a Farad).

Let's flip it to get the magnetic equivalent. That's a weber per magnetic voltage, which is a weber per amp, or a Henry, the unit of electric current inductance!

And so you then immediately realize that a Farad is thus the unit of magnetic current inductance. (but note there is a subtle difference between these. Capacitors make divergent dipoles, while coils make solenoidal ones -- the units are the same, but they are different. A coil of magnetic monpole current would make a solenoidal E-field, and a monopole capacitor would make a divergent H field).

Resistance and conductance would be flipped exactly. Magnetic current resistance is voltage per current, which is an amp per volt for monopoles, or a mho. Thus magnetic conductance is in ohms.

I always thought of that as just slick and it fascinates me to no end.

The meaning of volts, amps, and all electrical units "rotates and flips" as you change your basis in the charge/field space of Duality.

-Richard

I need to correct you on something here. Sort of a flip isn't it.

Voltage is the amount of Electron Pressure in a power source, or a curcit.

Amperage is the Flow Rate (or current rate) of the electrons though a conductor or semi-conductor

Wattage is the factor of both counted together V*A = W


However you are right they do have a duality, Voltage (In an AC or RF circit can be stepped down, this increases the available current and vis versa.

If you double your voltage you halve your Amperage, halve you voltage you double your amperage.

__________________
"There is no problem that cannot be solved by a suitable application of high explosives" - US Army Demolitions School

http://worldsofothersuns.home.comcast.net/

We're talking about two different things, here. I'm speaking of magnetic monopoles, and potentials and currents of them, and how that fits it the elegant (and somewhat abstract for the uninitiated who can't spew Maxwell expressions by rote ) framework of Duality.

In electric terms, voltage is energy per unit charge, which is the dimension of both (scalar) potential and EMF. In SI units, which I'll use throughout, that's a joule per coloumb.

However, that is the same thing as a weber per second, and the curl E/Faraday's Law relation shows:

Integral(E dot dl) = -d(Phi_B)/dt.

That equates voltage which magnetic current (flux per time). Flux and charge have the same units -- indeed, they're the same thing really.

And likewise, the curl H equation shows that magnetic voltage, MMF, is given by:

Integral(H dot dl) = I + d(Phi_D)/dt.

The unit of MMF is an Amp. And you can verify that an amp can also be expressed as a joule per weber, and an A/m (unit of H) is a Newton per weber, or force per magnetic charge.

Let's look at that just for fun. And amp is a coulomb per second, electric charge per time. But an amp is a Watt per volt, or Joule per (volt-second). From the above relation, we know a volt-second is a weber, so there we have it, an Amp is a joule per weber, or magnetic voltage.

You can also see this from magnetic energy relations, such as

Energy = 1/2 LI^2 = 1/2(Phi_B)I --> I = 2*(Energy)/Phi_B, showing the units of I, electric current, are energy per magnetic flux/charge.

And finally, for those who want to get really deep, what is the product of magnetic and electric charge/flux? That is, what is a coulomb-weber? Get a piece of that action (very punny).


-Richard

Jackson (and other advanced, graduate level texts) go in the details of Duality. But if anyone is interested in discovering duality for themselves, here's something to get you started. It's more fun and meaningful when you figure it out for yourself, I think:

To start, write Maxwell as she would be in her full form with both monopoles and electric charges:

div D = rho_e; curl E = -J_m - dB/dt
div B = rho_m; curl H = J_e + dD/dt

D = epsilon*E, and B = mu*H

rho_e and J_e are the electric charge and current densities, and rho_m and J_m are the corresponding monopole densities. Note the minus sign on the curl E equation -- that's what Ken and I were talking about with Lenz's law.

Note I've written it in perfectly symmetric form without the constants as I prefer to keep the constants in the relations between the (E, H) and
(D, B) pairs.

Adding monopoles shows the perfect symmetry.

Now, consider a transform of the following form:

E' = aE + bH

H' = cE + dH

where a, b, c, d are just scalar constants, and
a^2 + b^2 = c^2 + d^2 = 1

Substitute that in for E and H and get a new set of Maxwell equations in E' and H'. Consider the relationship between (a, b) and (c, d). Think of E and H as orthogonal basis vectors in a plane. Choose (c, d) in terms of (a, b) so H' is orthogonal to E'. Thus a/b is the tangent of an angle of rotation in that plane.

What happens when you do that?

Now, pay careful attention to what happens to the *source terms* in that transform. Pay careful attention to how that a/b ratio affects the source terms.

That is Duality.

We could, if we wished, do a 90 degree rotation and say electric charge is now "magnetic monopoles", and E is H and B is D (but pay careful attention to the relative sign of D vs what we now call B under such a rotation -- you'll see we must preserve the minus sign on curl E). Everything would work just as it is now, except everything would "flip". We'd be measuring voltages in amps, and currents in volts. Capacitance would in henries and inductance in farads. Resistance(reactance, impedence) would be in mhos(siemen is the official name, now) and conductance(susceptance, admittance) in ohms.

A bar magnet would be a bar electret -- the earth would have an electric field, not a magnetic one. And there would be no electric monopoles.

At first blush, you may protest, No, no, that can't work, that's crazy! But trust me, it does. All physical interactions would be the same, the only difference is the *words* (and words for units) we use to describe it. That is Duality.

And, if we wanted to be even more complicated, we say do a 45 degree rotation. Electrons have equal amounts of electric and magnetic charge, producing a combination of divergent electric and magnetic fields. Currents produce a combination of solenoidal electric and magnetic fields.

The earth's magnetic field would now be both electric and magnetic. Bar magnets would have both components as well. That would make the bookeeping a bit more complex, but it would work exactly the same.

-Richard