An infinitely long wire carries a current I=100amp . Below the wire a rod of length L=10cm is forced to move at a constant speed v=5m/s along horizontal conducting rails. The rod and rails form a conducting loop. The rod has resistance of R=0.4 ohms . 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 in volts?

Question

An infinitely long wire carries a current I=100amp  . Below the wire a rod of length L=10cm  is forced to move at a constant speed v=5m/s  along horizontal conducting rails. The rod and rails form a conducting loop. The rod has resistance of R=0.4 ohms  . 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 in volts?

Elena · Answer

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