In the figure below a rectangular loop of wire with length a

In the figure below, a rectangular loop of wire with length a, width b, and resistance R is placed near an infinity long wire carrying current i. The loop is then moved away from the wire at constant speed v.(Use any variable or symbol stated above along with the following as necessary mu_s, where all Quanties are in SI units.) When the center of the loop is at distance r, find the magnitude of the magnetic flux through the loop Find the current in the loop at distance r.

Solution

For an infinitely long wire with current I, the magnetic field at a distance r is given as:

B =  µoI / 2r

Now, we consider a small area of width dr at a distance of r from the wire. Hence the flux through this small area would be: d = B x adr

or d = µoIa dr / 2r

We can integrate the above expression for r from r- b/2 tp r + b/2 to determine the net flux through the rectangular loop

= d = µoIa dr / 2r = (µoIa / 2) * ln((2r + b) / (2 r - b))

a.) The net magnetic flux when the centre is at distance r from the wire is given as

= (µoIa / 2) * ln((2r + b) / (2 r - b))

b.) The emf induced would be given as the rate of change of the magnetic flux.

Therfore V = d /dt = (µoIa / 2) * [(b/(r² - b²/4)) v]

Hence, the current induced would be V/R = (µoIa / 2R) * [(b/(r² - b²/4)) v]