Solve Homework Problem: Velocity & Force of Rod in Rails

[erisedk]

Homework Statement

Two long parallel horizontal rails, a distance d apart and each having a resistance λ per unit length, are joined at one end by a resistance R. A perfectly conducting rod MN of mass m is free to slide along the rails without friction (see the figure). There is a uniform magnetic field of induction B normal to the plane of the paper directed into the paper. A variable force F is applied to the rod MN such that, as the rod moves, a constant current flows through R.
Find the velocity of the rod and the applied force F as a function of the distance x of the rod from R.
http://www.zigya.com/application/zrc/images/qvar/PHEN12050806.png

Homework Equations

The Attempt at a Solution

The emf induced will be Bvd, where v is the velocity of the rod at that instant.
The current is i = emf/resistance = Bvd/(R+2λx).
I know F = F_B = idB = B²d²v/(R+λx)
However, I don't know how to get rid of v in the F expression, and I don't know how to express v in terms of x. So, my aim is to figure out v, because then I'll get F as well.
I figured since the current is constant, di/dx = 0
$\frac{d}{dx} \frac{Bvd}{R+2λx}$ = 0

$( \frac{dv}{dx}) (R+2λx) = (v)(2λ)$

$\int \frac{1}{v} \, dv = \int \frac{2λ}{R+2λx} \, dx$

$v = c(R+2λx)$ where c is an arbitrary constant.

I don't know how to find c. Please help.

 

[cnh1995]

F = B2d2v/(R+λx)

You have mass 'm' of the rod. You can replace F by m(dv/dt) and further using chain rule, F=mv(dv/dx).

 

[erisedk]

You have mass 'm' of the rod. You can replace F by m(dv/dt) and further using chain rule, F=mv(dv/dx).

This might be a really dumb question but isn't F_net = ma? And here F_net = 0 as the variable force F is equal and opposite to the Lorentz force at any instant? I got the answer with this approach though. I just have this stupid question.

 

[cnh1995]

This might be a really dumb question but isn't F_net = ma? And here F_net = 0 as the variable force F is equal and opposite to the Lorentz force at any instant? I got the answer with this approach though. I just have this stupid question.

If the net force were 0, the rod would move with a constant velocity and the current would be decreasing since the resistance of the loop is increasing with distance.

 

[cnh1995]

Instead of what I said in #2, I believe your approach in the OP is correct.
Here's how I think the problem should be..The rod is initially moving with some velocity u at some distance x_o from the starting point, which drives a current Bud/(R+2λx_o). A variable force is applied to maintain this current. I believe u and x_o need to be specified. If x_o is assumed to be 0, still u is unknown. If x_o and u are known, constant of integration in your solution attempt can be eliminated.

 

[erisedk]

If the net force were 0, the rod would move with a constant velocity and the current would be decreasing since the resistance of the loop is increasing with distance.

F = F_B = idB = B²d²v/(R+λx)

Wouldn't this be incorrect then? Because net force would be F - F_B, I guess. I wrote F (variable external force) = F_B (Lorentz force) assuming that they would be equal and opposite. However, I do agree with your conclusion that the current would then be decreasing and we'd have a constant velocity. I'm just not sure how to resolve F = F_B dilemma.