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## 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 = B

^{2}d

^{2}v/(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.