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Two long wires are placed on a pair of fixed 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 a material of resistance $ρ$ per unit length. A magnetic flux density B is applied perpendicular to the rectangle so formed, and one wire is moved along the rails with a uniform speed v while the other is held stationary. It is seen that the force required to hold the stationary wire vanishes at a particular value of x. Find x

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$\frac{μ_{0} v}{2 π ρ}$

$\frac{μ_{0} v}{4 π ρ}$

$\frac{2 μ_{0} v}{2 π ρ}$

$\frac{μ_{0} v}{3 π ρ}$

answer is A.

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Detailed Solution

$i = \frac{Bdv}{ρ (2 d + 2 x)}, idB = \frac{μ_{0} i^{2} d}{2 πx}$

Force will be zero when magnetic field due to moving wire and the B (given) will cancel out.

$∴ B \Rightarrow \frac{μ_{0} i}{2 πx}, B = \frac{μ_{0}}{2 πx} (\frac{Bdv}{ρ (2 d + 2 x)})$

On solving, we get $[2 d + 2 x \approx 2 d], x \Rightarrow \frac{μ_{0} v}{4 π ρ}$

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