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.

(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.

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 car