A pair of long, rigid metal rods, each of length L , lie parallel to each other

Question

A pair of long, rigid metal rods, each of length L, lie parallel to each other on a perfectly smooth table. Their ends are connected by identical, very light conducting springs of force constant k each(the figure (Figure 1) ) and negligible unstretched length. If a current I runs through this circuit, the springs will stretch.

At what separation will the rods remain at rest? Assume that k is large enough so that the separation of the rods will be much less than L.

Express your answer in terms of the variables I, L, k, and appropriate constants (0, and ).

Explanation / Answer

Spring force on rod = 2 k d

The magnetic force on the top rod is =F_B = i L B

where B = µ0 i / 2d

F_B = i L B = µ0 i^2 L / 2d

2 k d = µ0 i^2 L / 2d

d^2 = µ0 i^2 L / 4k

d = sqrt( µ0 L / 4 k ) I

Seperation = d = sqrt( µ0 L / 4 k ) I

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