Re: Lenz's Law

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding Ludwik's comment concerning Lenz's Law;

> I will be introducing Lenz's "law" this week. Before
> formulating it in the most general (verbal) form I plan
> to focus on the negative sign in the Faraday's Law
> using a simple illustration.

The negative sign in Faraday's law, per se, has nothing, in itself,
to do with Lenz's law.  The negative sign in Faraday's law can be
considered as a convention relating the orientation of the B field
relative to the E field in the EM field strength tensor.  If the
sign of the derivative term in Faraday's law was reversed and the
sign of the curl term in Ampere's law was simultaneously reversed
then nothing at all pertaining to EM would have changed *physically*.
All that would have happened is that the conventional direction of
which way the B-field is said to be pointing in a given situation
would have been reversed.  No physical effects--including the
adherence of nature to Lenz's law--would have changed relative to how
nature is currently structured.

Lenz's law can be found in the composite *relative* sign difference
between the sign in Ampere's law, and the sign in Faraday's law.  It
doesn't matter which of these signs is positive and which is
negative.  All that matters for Lenz's law (and for nature) is that
these two signs are opposite to each other.  (BTW, whichever sign
appears in Faraday's law that *same* sign must also appear in the
Lorentz force law.)

> Two concentric circles
> will be drawn. The slightly larger circle will represent
> the primary loop; its current, I1, will be clockwise.
> The smaller circle will represent the secondary loop,
> in the same plane.
>
> Faraday discovered that the induced current, I2, is
> clockwise when I1 is decreasing and that it is
> counterclockwise when I1 is increasing. The current
> I2 is said to be a consequence of the induced emf
> (whose direction is the same as that of I2).

This is true regardless of which sign is chosen for Faraday's law (as
long a we choose a consistent definition for the sign of the current
and of the charge).

> Mathematically the induced emf is described as:
>
>        emf = - d(FLUX)/d(TIME)
>
> The negative sign is necessary because the current I2
> increases (from zero) when dI1 is negative and
> decreases when  dI1 is positive.

No, the negative sign is not necessary.  It can be removed with the
currents behaving the *same* way as before by just reversing the
definition of the way the B-field points.  Such a reversal will
show up as a simultaneuous sign change in Faraday's law, in Ampere's
law, *and* in the Lorentz force law.

> These are well
> established experimental facts.  Changes in I1 and the
> resulting changes in I2 are linked like action and
> reaction forces in mechanics.

I think the analogy may be closer to LeChatelier's Principle.

> When one is positive
> (for example, clockwise) the other is negative, and
> vice versa. That is why there is the negative sign in
> the above relation. I will also say that Faraday's Law
> is a description of a cause-and-effect chain:
>
> delta(I1) --> delta(FLUX) --> delta(emf) --> delta(I2)

Note, in your first arrow here, the relative direction of changes in
I1 and the Flux depends on the *definition* of which way the B-field
is said to point for a given current flow.  This rests on the sign
that appears in Ampere's law.

> The magnetically induced emf is not localized, as in a loop
> with a battery. The electric field, E, along our circular wire
> loop, is equal to emf / (2*Pi*r). The direction of the induced
> electric field is the same as the direction of conventional I2).
> Other illustrations will add depth to Faraday's Law and to
> Lenz's principle. The signs of delta(I1) and delta(I2) are
> always opposite; this has nothing to do with which direction
> (clockwise or couter-clockwise) is declared as positive in a
> particular illustration.
> Ludwik Kowalski

Besides having an overall ambiguity (to be settled by convention)
in the relative signs of the corresponding terms in Ampere's lwa and
Faraday's law, there is a ambiguity in the *magnitudes* of some of
the constants that appear in the equations of electromagnetism.
Different choices for these magnitudes results in a choice of a
different system of units to use in describing E&M.

Assuming we use a system of units that does not define c = 1 by
the definition of the unit of length or of time, then it can be shown
that there are still *two* other *arbitrary* constants that can be
chosen that appear in the equations of electromagnetism such that
any (nonzero) choice for those constants results in merely a
different system of units.  Also, the *signs* of these two constants
(besides their magnitudes) are arbitrary as well.  Making an
unconventional choice for the signs of either or both of these
constants will have no effect on the behavior of Nature (including
Lenz's law).  It will only affect our conventional description of it.

Here I write the equations of electromagnetism *generically* in terms
of these two arbitrary constants (taken here as a & b) along with the
constant c (which is the speed limit of causation and is *not*
arbitrary once separate units of time and space are decided upon).
The differential form of these Maxwell equations are:

div(E) = a*[rho]    (where [rho] is the charge density)

curl(E) = -(b/c)*dB/dt (time derivatives are all taken as partials)

div(B) = 0

curl(B) = (1/(b*c))*(a*j + dE/dt) (j is the current density vector)

We also have the Lorentz force law for the EM force F acting on a
point charge q:

F = sgn(a)*q*(E + (b/c)*(v x B))

where v is the charge's velocity and x signifies the cross product.

It is important to realize that we are free to take the values of
the constants a & b to be *any* fixed nonzero real values we want.
Different choices for them will give different systems of
electromagnetic units.  If either a or b or both of them are taken
as negative numbers then Maxwell's equations and the Lorentz force
law might *look* funny with unconventional signs in weird places,
but they will still describe the exact *same* laws of nature, where
only the conventionally defined directions for E or B or both of
them may have been reversed.

We can use the above equations and specialize and solve them for some
important special cases.  For instance, we can find the electrostatic
force between two fixed charges Q & Q' separated by distance r as:

|F| = |a|*Q*Q'/(4*[pi]*r^2) .

We thus see that choosing different magnitudes for 'a' determines the
size of our conventional unit of charge.  Making 'a' negative will
reverse the conventional sign of the E-field relative to the charge
because the electric force per unit charge on a test charge is found
to be E*sgn(a).

We can also find the magnetostatic force (per unit length) between
two long thin parallel wires separated by distance r carrying
currents I & I' as:

|dF/dl| = 2*|a|*I*I'/(4*[pi]*r*c^2)

If we make 'b' negative then this reverses the direction of the B-
field relative to the E-field and relative to the charges and
currents.  Doing this means (among other things) that our former
right-hand rules involving the B-field all become left-hand rules.
Changing the magnitude of 'b' changes the relative magnitude of
the B-field relative to the E-field and amounts to a change in the
units the magnetic field is measured in.

In all conventional electromagnetic systems of units both constants
'a' & 'b' are taken as positive.  This makes the appearance of the
equations of E&M have a common form with common conventions for
what is positive relative to what is negative, with only the
corresponding magnitudes being different in different unit systems.

In the Gaussian system of units a == 4*[pi] and b == 1.

In the Lorentz-Heaviside system of units a == 1 and b == 1.

In the SI system of units a == (4*[pi]*10^(-7) N/A^2)*c^2 and b == c.
Since this is a cumbersome choice to substitute into the equations of
electromagnetism, the quantity 4*[pi]*10^(-7) N/A^2 is given new
notation: [mu]_0 == 4*[pi]*10^(-7) N/A^2, and another symbol,
[epsilon]_0 is also introduced that substitutes for a, i.e.
[epsilon]_0 == 1/a.  By introducing these symbols and using the
above chosen values of a & b all reference to c in the equations of
electromagnetism can be hidden from view.  Doing so gives the usual
SI form of Maxwell's equations and the Lorentz force law.

I have invented my own set of EM units that I like in preference to
all the above unit systems.  I call it the SU system (SU stands for
sensible units).  In the SU system all the mechanical units are the
ordinary mks units, but the EM units are defined by choosing
a == c and b == 1.  This choice maximally balances the unit system
between magnetic properties and electrical ones, and it brings out
as much of the built in relativistic invariances in the theory as
possible short of doing everything in Minkowski 4-vector space-time
notation.

David Bowman