Rod on rails in a magnetic field

[johann1301h]

Homework Statement

A rod with mass of m runs on rails in a magnetic field. The rails are connected with a wire at x = 0
so that the rod and the rails together with this wire form a closed circuit. The distance between the rails is L, and the resistance in the rod is R.

We assume that the rails and wire are ideal conductors, so that the total resistance in the circuit is R. At time t = 0, the rod moves in positive x-direction at velocity v0.
[/B]
https://uio-ccse.github.io/fys1120/homework/fig-week09/stav_skinner.svg
a)
Calculate the emf in the circuit using Lorentz's force law, and indicate the contribution from the different parts of the circuit.

b)
Calculate the emf with Faraday's law. Does this match the emf you calculated in the previous assignment?

c)
Find the size and direction of the current in the circuit. (With the clock)

d)
Determine the movement of the rod for t > 0. Can you check if the answer is correct using energy conservation?

2. The attempt at a solution
I believe i have solved b) and c), and a) shouldn't be to difficult, my struggle is with task d). So any help here is much appritiated!

b)
emf = BLv0

c)
I = (1/R)BLv0

d)
I think the rod wil move and then stop following a differential equation? I think the force on the rod is given by F = ILB, but the problem is that I (current) depends on the velocity of the rod! Any ideas?

 

[kuruman]

The force is indeed ILB in magnitude. What is its direction? Can you write and then solve a differential equation for the motion of the rod? Its solution will tell you what the rod will do a long time after motion starts. Hint: The current is a function of time, I(t), not what you have in part (c) because the velocity v is a function of time.

 

[johann1301h]

I think I found the equation, see picture

 

[kuruman]

I think I found the equation, see picture

That's the correct equation. Can you derive it? Do you know what to do with it?

 

[johann1301h]

Yes, I found v(t). see picture

 

[kuruman]

Yes, I found v(t). see picture

Did you find $v(t)$ or a picture for $v(t)$? The question is asking you to "determine the movement for t > 0" That means also finding the position of the rod $x(t)$ for t > 0. Incidentally, if you have the solution in front of you, you will be well advised to know how to reproduce it if called upon to do so on a test. By "reproduce" I don't mean "memorize", I mean "figure out the strategy for solving this kind of problem and use algebra to apply the strategy so as to get a solution." That's what learning from studying is.