## A Problem of Wires on the Rails

It’s been a long time since I posted a cute physics problem so here’s one about magnetism for your edification and/or amusement.

*Two long wires are laid flat on a pair of parallel rails perpendicular to the wires. The spacing d between the rails is large compared with x, the distance between the wires. Both wires and rails are made of material which has a resistance ρ per unit length. A magnetic flux density B is applied perpendicular to the rectangle formed by the rails and the wires. One wire is moved along the rails with uniform speed v while the other is held stationary. Derive a formula to show how the force on the stationary wire varies with x and use it to show that the force vanishes for a value of x approximately equal to μ _{0}v/4πρ. *

*Give a physical interpretation of this result.
*

HINT: Think about the current induced in the wires…

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October 7, 2014 at 3:42 pm

Since no one else has said anything, I will.

There’s an induced emf due to Faraday’s law, which induces a current I = (Bvd) / (2 rho d) = Bv/(2 rho). (In the middle equality, the numerator is the emf and the denominator is the resistance.) The current in the moving wire produces a magnetic field. At the location of the stationary wire, that field has magnitude roughly mu0 I / (2 pi x). (“Roughly”, because that’s the field for an infinite wire.) A little fiddling with right-hand rules convinces one that the direction of this field is opposite the original field. When its magnitude equals B, the net magnetic field at the stationary wire is zero, so there’s no Lorentz force on this wire. Doing the algebra yields the answer given in the problem.

I don’t see any particularly nice physical interpretation. Sometimes, in these circumstances, it’s nice to go to a different reference frame, such as one in which the moving wire is stationary, or perhaps one in which the two are moving at equal speeds, but I don’t see that leading to anything enlightening in this case.