We are going to investigate how you would calculate the induced emf in a rectangular wire loop that lies parallel to a current-carrying wire having a variable current.

 

 

Experience tells us that we should use Ampere's Law to calculate the strength of the magnetic field passing through the wire. However, this field is not uniform, but varies inversely as the perpendicular distance from the wire.

 

 

Next, we need to calculate the flux passing through the loop, by referencing the equation

 

 

Finally, to calculate the induced emf, we would reference Faraday's Law

 

 

Let's begin by developing an expression for the flux passing through the loop. since the magnetic field is not uniform throughout the area, we need to build areas through which it is uniform and add these areas up to determine the total flux.

 

In the diagram, we see a strip that has an area of A = x(dy). If dy is small, the magnetic field will be uniform in this smaller area. Its flux contribution,  would be

 

 

Now all we need to do is add up all of these contributions to calculate the total flux through the loop. To do this we will set up an integral with limits from a to a+y.

 

       

 

Our next step will be to develop an expression for the induced emf using Faraday's Law.