magnetism

infinitely long wire carries a current I100A. Below the wire a rod of length L 10cm is forced to move at a constant speed v=5m/s long horizontal conducting rails. The rod and rails form a conducting loop. The rod has resistance of R 0.4ohms. The rails have neglibible resistance. The rod and rails are a distance a=10mm from the wire and in its non-uniform magnetic field as shown. What is the magnitude of the emf induced in the loop?

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asked by james
  1. Let x be the distance from the right end of the rails to the rod.
    Magnetic field of the long straight wire is
    B=μ₀I/2πr
    If the infinitesimal horizontal strip of length x and width dr, parallel to the wire and
    a distance r from it => area A = x dr and the flux is
    dΦ=BdA= (μ₀I/2πr)xdr
    Φ=∫ dΦ= (μ₀Ix/2π)∫dr/r (limits: from ‘a’ to ‘a+L’) = (μ₀Ix/2π) ln{(a+L)/a}
    ℰ=dΦ/dt = (μ₀I/2π) (dx/dt) ln{(a+L)/a} =(μ₀Iv/2π) ln{(a+L)/a}=
    =(4π•10⁻⁷•100•5/2π)• ln{(1+10)/1}=10⁻⁴•ln11=2.4•10⁻⁴ V

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    posted by Elena

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