That's A Weird Frame

Electricity and Magnetism Level 4

A conducting frame is composed by joining the curves \( f(x) = x^{3} \) and the line \( y = 0 \).

The conducting frame has 0 resistance. A uniform magnetic field of magnitude \( B \) Tesla is directed inwards as shown. A magical conducting rod parallel to \(y\)-axis starts moving from the origin along the positive direction of \(x\)-axis with intial velocity \( u\text{ m/s}^{-1}\).

This magical rod has special properties as follows:

1) It has a finite mass \( m \text{ kg}\) .

2) Its resistance per unit length \( \sigma \) varies as \( \sigma = \sigma_{0}(1+x^{4}) \dfrac{\Omega}{m} \) where \(x\) is the distance of the rod from the origin.

If the distance travelled by the rod till it comes to rest is \( L \)
Find \( \left \lfloor L \right \rfloor \)


Take \( m =1 \text{ kg}, B = 1\text{ T}, u = 2\text{ m/s}^{-1} ,\sigma_{0} = 1 \dfrac{\Omega}{m} \).


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