Question

A rectangular metallic loop of length l and width b is placed coplanarly with a long wire carrying a current i (figure). The loop is moved perpendicular to the wire with a speed v in the plane containing the wire and the loop. Calculate the emf induced in the loop when the rear end of the loop is at a distance a from the wire. solve by using Faraday's law for the flux through the loop and also by replacing different segments with equivalent batteries.
Figure

Answer 1

The magnetic field at a distance x from the the current-carrying wire is given by

B = μ0i2πx

Area of the loop = bdx

Magnetic flux through the loop element:

dϕ=μ0i2πxbdx

The magnetic flux through the loop is calculated by integrating the above expression.

Thus, we have

ϕ=∫aa+lμ0i2πxbdx   =μ0i2πb∫aa+l(dxx)   =μ0i2πxln(a+la)

The emf can be calculated as:

e=−dϕdt=ddt[μ0ib2πlog(a+la)]  =−μ0ib2πaa+l(va−(a+l) va2)  

Answer 2

Answer:

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