Finding The Induced Current in This Loop

[Shakenbake158]

Homework Statement

Hey guys, I have a physics II test tomorrow on Electricity and Magnetism, and I cannot seem to figure out this question.
The rectangular loop in the figure has 2.2×10−2Ω resistance.
What is the induced current in the loop at this instant?
Picture:

Homework Equations

E = d(flux m)/dt

B = (mu_0)(I)/2(pi)(r)

flux m = integral (B * DA)

The Attempt at a Solution

However, the area is not changing, so I can pull that out of the integral.
Then I have to integrate:

(mu_not)(I)/2(pi)(r)

Everything is constant except 1/r, so I can pull everything out and be left with:

integral (1/r)dr = ln(r)


So now we have:

E = d/dt(A)*(mu_not*I)*(ln(r))/(2*pi)


How do I take the derivative of this with respect to time?
Did I do the other steps correctly?

 

[gneill]

The whole area of the loop is being displaced with respect to time (it's moving away from the wire). So if you have a formula for the flux at a given distance then you can find how it changes with distance. And, since know how the distance changes with time, what rule from calculus springs to mind?

 

[Shakenbake158]

The whole area of the loop is being displaced with respect to time (it's moving away from the wire). So if you have a formula for the flux at a given distance then you can find how it changes with distance. And, since know how the distance changes with time, what rule from calculus springs to mind?

Wait, I thought that the area was constant, since the area of the rectangular loop is not changing.

However, the B field is changing because the rectangular loop is getting further away.

 

[rude man]

Wait, I thought that the area was constant, since the area of the rectangular loop is not changing.

However, the B field is changing because the rectangular loop is getting further away.

That's right. So what's Farady say about the emf induced around a loop with changing flux?

 

[Shakenbake158]

That's right. So what's Farady say about the emf induced around a loop with changing flux?

Faraday's Law says that the induced EMF is equal to the changing flux. So do I take the derivtive of B?

 

[rude man]

Faraday's Law says that the induced EMF is equal to the changing flux. So do I take the derivtive of B?

Sure! Flux - area x B field. If the B field is non-uniform you have to intgerate B over the area. And if B changes with time you have to integrate AND consider how that integral changes with time. But area is always the same constant.

In any case emf = - N d(flux)/dt. Your N is of course 1.

(Exception: under certain moving-media circumstances that will not get you the induced emf but let's leave that for later unless you're really interested).