Answer :

Given: Length of the rectangular loop=l


Breadth of the rectangular loop=b


Current in the wire=i


Speed of loop=v


We have to find the emf induced in the loop by Faradays’s Law.


Faraday’s law states that whenever the magnetic flux through


a closed surface changes, there will be an induced emf prodced in


the loop that encloses the surface. The mathematical relation is



where is the magnitude of the induced emf, ϕB is the the magnetic


flux through the surface. The negative sign arises because the


induced emf will be produced such that it will oppose the change of


magnetic flux. The magnetic flux is given by



where B is the magnetic field and dS is a small area element on the


surface.


The wire which is near the loop is responsible for the magnetic field .


The wire is carrying current in the upward direction so by Fleming’s


right hand thumb rule, the magnetic field will be perpendicular to the


plane of the paper in the inward direction.



We have to now calculate the magnetic field acting across the loop.


Let us consider a small rectangular element of length b and width dx


at a distance of x from wire. The magnteic field on this element due


to the current carrying wire is given by Ampere circuital law.




(cos θ=1 because the vector and vector are both acting in ward in the plane of paper so θ=0°, here dl is a small current carrying element of the circular amperian loop)





The magnetic flux through this element will be





(cosθ=1 because the magnetic field across the area and the normal


vector of this area element both point in the same direction so θ=0°


so cosθ=1)


The total magnetic flux through the loop will be the flux through


Infinite such elementss from x=a to x=a+l





(taking constants out of the integral)





The flux through the loop is


The emf induced will be



(taking constants out of the differential)




(because as the rate of change of Distance from wire is the loop speed)




The magnitude of the induced emf is .


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