Problem 2:Conservation of Cars

Conservation of charge in one dimension is similar to conservation of cars on a road. Cars move along the road with car density λ and speed v which, in general, are functions of position.

(a) The continuity equation requires that λ divided by v at all points on the road is the same.

As car density increases, the velocity increases? hmm...

λv = α

where α is a constant. The more cars on the road, the slower we go.

This, however, is not vary true. with some constant α, and a velocity v = 70mph you could have a density of 100 cars per mile, or of 200 cars per mile. In addition, what happens when you have a mile of road which isn't driven on for a while? you could then assume α = 0; now there are three cars racing and they enter this section of road going 100 mph, because α is a constant it would still equal 0 but you cannot get 0 with a real λ and v. So my question is, what am I doing wrong?

(b) Nothing in the continuity equation needs to be modified if there is a factory which adds cars to the road and/or a parking garage which takes them off the road.

If we extend the definition of cars to include their raw materials, then maybe, but the creation of a car violates the conservation of cars. The parking garage could be more of a capacitor, but a junk yard might serve as a destroyer of cars which would also violate the conservation of car. One could make the argument that car created equals car destroyed, but historically, that would be wrong. The continuity equation is a statement of conservation, and it would have to change to say what goes in or gets created inside either stays, gets destroyed, or comes out, - not an easy tool to work with.

Even extending the definition to include their raw materials their would not be equality. The reason for this is their is no way to know how much raw material we have for cars. We have the earths resources' but we know not all of them will go into making cars. Additionally the raw materials will come from mining and other processes which would thus need to be taken into consideration. Going back to the historic argument, I would like to add an additional factor to the equation being population. We know the number of cars on the road is increasing over time, and thus their cannot be a conservation of cars, but we can get a conservation of cars per person. Thus we still have cars being made and destroyed but we now have something that is being conserved which is cars per person.

Student question...This may work better as a divergence theorem type of question...if the car factory or parking garage represents change in enclosed car, then we can do something like Dr. Kowalski did in Lecture 25...car flux across a loop around a segment of road = change of car enclosed in loop. Something like: $\int \Phi_C \bullet dA = \frac{\part C}{\part t}$ where Φ_C is the flux of car (car/s/unit length) and C is the quantity of cars.

And the differential form of the continuity equation is:

$\nabla \cdot \vec{I} = - \frac{\partial \rho_C}{\partial t}$

It may be useful too.

What are we actually trying to figure out? Is there a contridiction between the flux equation and the continuity equation? It isn't apparent to me.