Interesting Electromagnetic Inductance Problem

[christof6869]

So I've been working on this for a while now, but I'm a little stuck. I think I see how to solve the problem below, just need a couple tips to get me over the hurdle.

Homework Statement

There are two infinite wires in the same plane as a rectangular loop. Geometries are displayed in the attached figure. The two wires have currents of known relationship to time I₁(t)= I_1i+ut and I₂(t)= I_2i+vt, where u, v, I_1i and I_2i are all time-independent values. The loop has a known resistance R.

What is the induced current in the wire loop?

Homework Equations

Magnetic field created by current in a wire
B=μI/2πr (r is the distance from the wire to the loop)

EMF= -A(dB/dt) (A is the area of the loop cd)

I_induced=EMF/R

The Attempt at a Solution

I know the current induced is the induced EMF divided by resistance of loop R. To find EMF, I need to use Faraday's law, where the EMF induced is the change in magnetic flux times change in time. Since area is constant, only magnetic field changing in time causes the change in flux.

Here's where I have my problem. I don't know how to quantify the magnetic field. Ampere's law let's me use the following relationship.

B=μI/2πr

I'm pretty sure that the field strength falls off as you move farther away from the wire, so it has distance dependence (thus the r variable) as well as time dependence. So which distance do I use? If I take wire 1, the distance from the loop is b, but I need to account for the distance b + c. Is there some sort of average magnetic field going through the loop?

 

[kuruman]

First you need to calculate a general expression giving you the magnetic flux through the loop. To do this you need to consider a strip of width dy and length d (the length of the loop) located at distance y from the wire. If dy is small enough (we are doing calculus here) you can pretend that the magnetic field is constant over the area of the strip. The flux through the strip is

$$d\Phi=BdA=\frac{\mu_0Id}{2 \pi y}\:dy$$

You can find the total flux by integrating with y running over the appropriate limits. Integration takes care of the fact that the B field is not uniform over the area of the loop.

 

[christof6869]

Ok, that totally makes sense now. By using infinitesimally small areas, the change in magnetic field as r increases becomes negligible. You are a life-saver :)