Determine Constant K for Magnetic Flux Homework

[Denver Dang]

Homework Statement

Hello out there...

I've kinda figured this out, but I'm not quite sure how tbh.

I got this problem:
http://www.gratisupload.dk/download/41959/"

The length a is constant, but b varies in time like this:

$$\[b\left( t \right)=a\left( 1+{{\left( \frac{t}{\tau } \right)}^{2}}-2{{\left( \frac{t}{\tau } \right)}^{3}} \right),\]$$
where $\tau$ is a timeconstant. Besides that I know that for t < 0 then b = a, and for t > $\tau$ then b = 2a.

The magnetic fields produced by the current in the conductors (I₁ and I₂) gives a magnetic flux through the rectangular loop of:

$${{\Phi }_{B}}=K\cdot b\left( t \right)$$Determine the constant K.

Homework Equations

$$B=\frac{{{\mu }_{0}}I}{2\pi r}$$

$$d{{\Phi }_{B}}=BdA=\frac{{{\mu }_{0}}I}{2\pi }L\,dr,$$
where L is b(t)

The Attempt at a Solution

What I've done is as following:

$$\[{{\Phi }_{B}}=\int_{a}^{3a}{BdA}=\int_{a}^{3a}{\frac{{{\mu }_{0}}{{I}_{1}}}{2\pi }b\left( t \right)\,dr}+\int_{a}^{3a}{\frac{{{\mu }_{0}}{{I}_{2}}}{2\pi }b\left( t \right)\,dr}=-\frac{a{{\mu }_{0}}\left( {{I}_{1}}+{{I}_{2}} \right)\ln \left( 3 \right)\left( 2{{t}^{3}}-3\tau {{t}^{2}}-{{\tau }^{3}} \right)}{2\pi {{\tau }^{3}}}\]$$

Putting this equal the magnetic flux I know, and then solving for K, I get:

$$K=\frac{{{\mu }_{0}}\left( {{I}_{1}}+{{I}_{2}} \right)\ln \left( 3 \right)}{2\pi },$$
which supposedly is the correct answer according to my book.

But what I don't understand is, that when I tried the limits a to 2a, which seems more obviously to me, I get ln(2) instead of ln(3). So I don't understand why the limits should be a to 3a instead - if I've even done it correctly in the first place.

So I thought one of you might knew :)Regards.

 

[Denver Dang]

No one ? :(

 

[nickjer]

What happened to the 'r' in the denominator of your flux? You aren't dealing with a circle so you shouldn't use polar coordinates for your area. You will be dealing with cartesian coords since you have a rectangle.

Also, why did you choose the limits that you chose? In the figure that you posted it looks like r goes from a/2 to 3a/2.

And why are you adding the fluxes from both wires. I would double check this by drawing the magnetic field vectors from both wires to see what directions they are both pointing.