Bad Boys Rape Our Young Girls, But Violet Goes Willingly

I remember hearing a quip on the radio a few years ago, I
don't remember the context, but a young British recruit during
World War II was keen on becoming a radio technician. He asked his
commanding officer what he would have to know in order to join the
engineering team. The officer replied: "You only need to know Ohm's
law, but you need to know it very -- very -- well." Ohm's law, you
might recall, is V=IR where V is the Voltage, I is the Current, and
R is the resistance. Well, in this article, I'm going to try to
explain electricity, in particular the mysterious workings of a
flashlight, so that you will have a new appreciation for the
interesting stuff going on behind the scenes every time you flip a
light switch.
### How fast do the electrons travel in the wire?

If we think
about it for a while, and look a few things up, we can probably get
a pretty good estimate to the answer. Suppose we have a current of
one Amp. flowing down a copper wire whose cross sectional area is 1
square mm. (A pretty big current down a pretty small wire.) Now
then:

current = electrons per second =

electron density (electrons per cubic millimeter)** times**

area of wire (1 square millimeter)** times**

velocity of electrons (millimeters per second)

So we need to figure out the electron density of copper. Lookup up copper in the periodic table tells us that it's atomic weight is 63.55 grams per mole. The density of copper is 9 grams per cc or 9 x 10^-3 grams per cubic mm. The number of atoms in a mole is our old friend Avogado's (Avogadro's) number, or 6.02 x 10^23 atoms per mole, and finally there is one free electron per copper atom. Putting these together we get:

electrons density = electrons per cubic millimeter =

electrons per atom (1)** times**

atoms per mole (6.02 x 10^23)** times**

moles per gram of copper = 1 / grams per mole of copper = 1/63.55** times**

grams per cubic millimeter = 9 x 10^-3

Multiplying we get 8.4 x 10^19 electrons per cubic millimeter

As is customary in the world of physics, when faced with something as complicated as a flashlight, in order to understand it you break it down into simpler pieces. Now a flashlight basically consists of four things, a battery, a bulb, some wire, and a switch. Let's start with the piece of wire, and see if we can understand it first.

Even though we may have fond memories of our third grade science teacher, chances are she was lying to you when she told you about electricity, electrons, and wires. She may have said that the battery makes these tiny little things called electrons, which flow through the wire like water through a pipe, at the speed of light and dump their little buckets of energy into the light bulb, and then return weakened and exhausted back to the battery to get recharged. What is wrong with this picture? Well, it turns out just about everything. Later on, in high school physics this picture is refined, and you learn that a wire is full of atoms, and these atoms are surrounded by electrons, and when you attach a battery to the wire, the electrons jump from atom to atom, creating a current. Well even this picture isn't very good. The real scoop is that when a bunch of copper atoms are put together in a bulk material, like a piece of wire, the electron in the outermost "shell" is very loosely bound to its copper atom, and in fact is basically free to move about the metal wire. These electrons do this all the time, even before the battery is attached to the wire. It is just that their motion, which is due to the inherent thermal energy (ie temperature) of the wire, is random, and as a result there is no net flow in any one direction. It is as though the wire is filled with an electron gas, and in fact that is probably the best mental picture to have of a wire.

Okay, so the wire is a pipe filled with an electron gas, what is a battery? It is just the pump that makes the gas move. It neither produces or consumes electrons, nor does it shovel little buckets of energy onto their backs. A battery is an electron pump, nothing more and nothing less. The light bulb is like a small turbine, extracting energy from the circuit, and the switch is like a spigot, that turns the flow on and off. This is a pretty good picture of what is going on, but there are some problems if you look at it a little more closely. Here is one question that might be disturbing:

current = electrons per second =

electron density (electrons per cubic millimeter)

area of wire (1 square millimeter)

velocity of electrons (millimeters per second)

So we need to figure out the electron density of copper. Lookup up copper in the periodic table tells us that it's atomic weight is 63.55 grams per mole. The density of copper is 9 grams per cc or 9 x 10^-3 grams per cubic mm. The number of atoms in a mole is our old friend Avogado's (Avogadro's) number, or 6.02 x 10^23 atoms per mole, and finally there is one free electron per copper atom. Putting these together we get:

electrons density = electrons per cubic millimeter =

electrons per atom (1)

atoms per mole (6.02 x 10^23)

moles per gram of copper = 1 / grams per mole of copper = 1/63.55

grams per cubic millimeter = 9 x 10^-3

Multiplying we get 8.4 x 10^19 electrons per cubic millimeter

Now we are getting close. 1 Amp is 6.24 x 10^18 electrons per
second so we have:

6.24 x 10^18 = 8.4 x 10^19 * 1 square mm * velocity

solving for velocity we get that v = .073 millimeters /
second or .00016 miles per hour.

Surprised? This is
** much slower** than your average snail. Yet we've
always heard that electricity travels at the speed of light.
Something funny is going on here. One question that springs to mind
is how come the light turns on so quickly. If we have to wait for
the electrons to get there, we would wind up waiting a few minutes
at least, yet the light is on immediately after we flip the
switch.

Okay, so here is the dirty little secret. While the electrons do
indeed move very slowly through the wire, the electric field moves
at close to the speed of light. "What is an electric field?" you
ask. Its what surrounds every charged particle, like electrons and
protons, that causes other charged particles to experience a force.
We all believe in gravity right? Well the gravitational field is
what surrounds every particle of mass, and it causes other masses
to experience a force. The electric field is similar, though
perhaps a bit more complicated. While gravity always produces an
attractive force, electric fields produce forces which both attract
and repel. If an electron were to suddenly appear out of nowhere,
popping into existence, the electric field that it causes would
propagate outwards from it at the speed of light. This is just a
fact of nature, if you have some electric charges, they produce a
* force field* which act on other electric charges.
Furthermore this
* force field* propagates at the speed of light.

Now imagine that we closed the switch on the flashlight. The battery starts pushing electrons into the wire, and though these electrons are moving slower than a snail, the light comes on instantaneously, as far as our human perception is concerned. How can this be? After that first instant when the switch is closed, electrons are piling up somewhere. These excess electrons create an electric field, and if you look at fig 1 you can see that this field acts to push electrons away from the pileup. In other words, it tries to keep new electrons from flowing into the pileup, and it tries to push the already piled up electrons further on down the wire. Moreover, this is happening at the speed of light, everywhere in the wire where electrons are bunching up. The net effect of this is that immediately upon closing the switch, a uniform current is created in the wire, with the electric field responsible for smoothing out the flow. This happens so quickly that you can safely assume that the current is equal all the way around the circuit, not only in the case of a battery, but also in the case of that AC power coming out of a wall plug.

Now lets try to really understand Ohm's law, which says that the
current in a wire is proportional to the voltage. This law isn't
really a law at all, not like the force law of gravity or the law
of conservation of energy or momentum. In fact it is really rather
astonishing that it should be true at all. Why? The voltage
produces an electric field in the wire, which produces a force on
the electrons. Now this force produces an
** acceleration**(remember Newton: F=MA) not a
velocity, thus we would expect the current to grow larger and
larger, as the electrons continue to accelerate, moving ever
faster. Instead Ohm's law is saying that this constant electric
field is producing a constant current, hence constant velocity.
Something funny is going on here. To understand this we need to
take a closer look at the goings on in the wire. Remember I said
that the electrons are like a gas, filling the wire. Well, these
electrons are constantly bumping into things, and bouncing off of
them. Think of it as follows: Imagine you are in a Ferrari, and you
can go from 0 to 100 in 6 seconds. However, you happen to be on a
road where there is a stop sign every three seconds. How fast, on
the average do you travel? In three seconds, you've accelerated
from 0 to 50. Your average speed during those three seconds was
25MPH. Then you had to slam on the brakes, screech to a halt, and
start all over again. Thus even though your Ferrari can do 100 in 6
seconds, your average speed is only 25MPH. Hell, you're not even
breaking the speed limit (at least on average.) In the general
case, we need to relate the distance you cover, say L, to your
acceleration, A, and the amount of time, T you are moving. At the
end of time T, your velocity V = A * T. The distance you've covered
is your average velocity over that time, which is V/2, times the
time T, so L = V/2 * T = 1/2 * A * T * T. Thus if we solve for T we
get that T = Square Root (2 * L / A). Finally this means that the
current, with is the average velocity of the electrons is 1/2 * A *
T = Square Root (L * A / 2). Take a breath for a moment and let's
see what we've discovered. The current is proportional to some
constant, namely Square Root (L/2) times Square Root(A). And A, the
acceleration, is proportional to the electric field, which is
proportional to the voltage. Oops. We must have made a mistake
somewhere, because that is
** not** Ohm's law. Ohm's law says the current is
proportional to the voltage, not the square root of the voltage.
What's wrong? Well in a gas, each molecule is constantly moving,
because of the thermal energy of the gas. Furthermore, the velocity
due to this thermal energy is very large compared to the drift
velocity we calculated above. In fact it is about 10^9 mm/sec or
almost 10 orders of magnitude (10^10) times bigger. Thus their
motion is totally random, and very fast. Therefore the time between
collisions is L / thermal velocity, not L / drift velocity. The
fact that this is 10 orders of magnitude greater than the drift
velocity means the drift velocity contributes almost nothing to the
time. Thus the average velocity between collisions is 1/2 A * T
where T does not depend on A at all. Thus the current is
proportional to A which is proportional to the voltage, and voila,
we have Ohm's law. In fact we can squeeze even more information out
of this observation. The amount of current flowing down a section
of wire is:

current density (I) = The number of atoms per unit volume (N)
** times**

the number of free electrons per atom (f)
** times**

the charge per electron (q)
** times**

the average velocity of the electron (1/2 A * T)

thus: I = 1/2 * N * f * q * A * T

now acceleration A = Force / mass of electron (m) = E / m

and time T = distance between collision L / thermal Velocity
V

so I = (N * f * L * q^2 / 2 m V) * E

What have we here? Well, we can see that the stuff in the
parentheses, which are all things that depend on the material the
wire is made of, is a first order formula for the conductivity.
Conductivity is the inverse of resistance, so if we stick all that
stuff on the other side of the equation we get:

V (voltage) = (2 m V / N * f * L * q^2) I .... Ohm's law

Which correctly predicts that the resistance should increase
as the thermal velocity of the electrons gets larger, or in other
words, resistance goes up with temperature, a well know fact.

Now the resistance of a metal like copper is essentially zero, but the resistance of the bulb is substantial. Thus there are a lot more collisions taking place in the filament than in the wire, and guess what, the filament gets hot. The electrical force is being converted into heat, and eventually light by the resistance of the filament, in other words, the flashlight produces light.

Now you might think we are done, and we are, except for the
title of this piece. What does all of this have to do with bad boys
raping our girls? Well, in any piece of electronics you are likely
to come across, you will find lots and lots of resistors. How can
you tell one resistor from another without taking the trouble of
measuring their resistance? Well, the manufacturers decided to
paint a color code on them. They chose the following colors:

Black = 0

Brown = 1

Red = 2

Orange = 3

Yellow = 4

Green = 5

Blue = 6

Violet = 7

Grey = 8

White = 9

Now this can be a little hard to remember, but if you know
that Bad Boys Rape Our Young Girls But Violet Goes Willingly, it
makes it a little easier.

Quote of the day:

*I wonder if other dogs think poodles
are members of a weird religious cult.*

**Rita Rudner**

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