F = ma & Ohm law CONFLICT


Thread Starter

Rushi Shroff


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.