Re: What is a simple Definition of Lorentz Contraction, or Lorentz force?

Jack,

Quite a barrage of questions!  I'll see if I can answer them in a
reasonable way.

First, I think I know the photograph to which you are referring, and (if I
am recalling correctly) it appeared as though the blackboard was partially
erased, changing the "E" into an "L".

Second, if I (or some other physicist) were to encounter a capital "L" in
an equation, the meaning might be immediately clear, but it might still
depend upon the context -- context that may not be obvious from looking at
some doodlings on a blackboard or a piece of paper.  For example, depending
upon the setting, "L" might refer to:
* A length (the easy answer)
* Angular momentum (probably the second-easiest answer, you can read about
it in any introductory physics text),
* The Lagrangian function (more obscure, but if you are really interested
you can read about Lagrangian functions in an advanced mechanics text)
* The luminosity of a star (OK, that one's a real stretch, but you can read
about it in any astronomy text)
And, of course, these are just the "standard" uses for "L", there's no
telling what someone may decide to define as "L" just to simplify a set of
equations, or whatever.

As a P.S. to this point, I'll note that "L = mc2" would imply (assuming "m"
is mass and "c" is the speed of light) that "L" has units of energy, and
only the Lagrangian function has units of energy.  But "L = mc2" would not
be a really meaningful equation in a Lagrangian sense, so I'm sticking to
my "partly-erased blackboard" explanation.  Or it might be a "doctored
photo" explanation.  Or....

Third, you asked what the Lorentz force was.  If you look in an
introductory physics text, you will find that if a charged object is moving
around in the presence of a magnetic field, then there will be a force on
that object -- the Lorentz force.  The exact expression is:
(F) = q(v)x(B),
where "q" is the electric charge on the object (perhaps measured in units
of Coulombs),
"v" is the velocity of the object (perhaps measured in meters/second),
"B" is the magnetic field (perhaps measured in units of Tesla),
"F" is the resulting force on the object (perhaps in units of Newtons),
the parentheses indicate that "F", "v" and "B" are vectors (having both a
magnitude and a direction),
and "x" indicates the so-called "cross product" of the vectors (v) and (B).
Compact, but full of information.  The dependence of the Lorentz force on a
vector cross product means that magnetic forces can (at first glance) be
quite strange.  An object moving along a magnetic field will have no force
upon it, while an object moving across a magnetic field will tend to be
driven in a circular of spiraling path.  Again, I encourage you to read
about the Lorentz force.

Fourth, the "Lorentz contraction" is a reference to an odd effect of
special relativity, "odd" in the sense that it runs counter to our everyday
low-speed intuition.  Just for the sake of saying it, "special relativity"
is Einstein's theory of how observations will differ between two observers
that are travelling along with a constant speed with respect to each other.
 The usual example would be two observers, one on a train platform and the
other on a train, as that train zips through the train station.  The two
observers can communicate (signs, walkie-talkies, whatever) and watch each
other conduct simple physics experiments (measuring the length of objects,
or measuring time intervals, for example).  Without dwelling on the details
-- which may be in one of the end chapters of your introductory physics
textbook -- the two observers will agree that everything they see (both in
front of them and moving along in front of the other observer) makes sense,
but some of the details are a bit off.  Specifically, each observer will
insist that the other observer's clock is running slow (that is, "one
minute" on the other guy's watch takes a little more than one minute on my
watch; in fact, that "minute" takes exactly sixty seconds multiplied by
some factor gamma; this gamma is a number slightly larger than one), this
effect is called "time dilation".  Both observers will agree on length
measurements that are NOT along the direction of motion: the height of the
train, or the height of the train station.  However, each observer will
insist that the lengths along the direction of motion are shorter for the
other guy (that is, if the observer on the train says that the train is
exactly 100 meters long, the observer on the platform will measure the
train as being shorter: exactly 100 meters divided by gamma), this effect
is called "length contraction" or "Lorentz contraction".  The size of both
effects are determined by gamma, which is defined as:
gamma = 1/sqrt( 1 - v2/c2 )
where "sqrt" is of course the square root,
"v" is the relative speed of the two observers (the speed of the "train"),
and "c" is the speed of light (300,000 km per second, or 186,000 miles per
second, if memory serves).
Some messing around with this equation will show that you need to reach a
relative speed of about 10% of the speed of light just to see a 1% effect
from special relativity, which really shows why we don't see big
relativistic effects in our everyday lives.  To illustrate this a bit
better, consider that the so-called "escape velocity" for a rocket to
escape Earth's gravity is 11 km/second; IIRC, the very fastest human-made
objects aren't much more than a factor of two or three faster than that. 
So relativitic effects are very tiny, and can be observed for spacecraft
(like the Voyagers), but there the effects get clouded by effects
associated with general relativity.  If you want to read more about special
relativity, beyond that found in end chapters of an introductory textbook,
you'll need to find a "modern physics" text, such as "Introduction to the
Structure of Matter" by Brehm and Mullin.

One final point concerns the famous equation "E = mc2". As it turns out,
this is an equation fragment, only strictly true sometimes.  What Einstein
found was a relationship between the kinetic energy of an object ("E"), its
momentum ("p") and its mass ("m"), in the form of the equation:
E2 = p2c2 + m2c4
where "c" is the speed of light.
For an object with no mass (like a photon), the equation becomes E=pc,
meaning that the energy and momentum of such objects are intimately
related.  Conversely, for an object with no momentum (not moving), the
equation becomes "E = mc2", meaning that in some sense mass and energy are
directly related for any object having a mass (meaning, the whole universe
as we know it in an everyday sense, aside from light).  This relationship
between mass and energy leads to understanding nuclear reactions, where
significant amounts of mass are "released" in the form of the energy of the
reaction products.  Again, if you are interested in reading more, you can
start with the end chapters in an introductory text, and then move on to a
"modern physics" text or an introductory nuclear physics text.

I hope this helps!  Good luck!
Aaron