(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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/" [Broken]

The lengthais constant, butbvaries in time like this:

[tex]\[b\left( t \right)=a\left( 1+{{\left( \frac{t}{\tau } \right)}^{2}}-2{{\left( \frac{t}{\tau } \right)}^{3}} \right),\][/tex]

where [itex]\tau[/itex] is a timeconstant. Besides that I know that fort< 0 thenb=a, and fort> [itex]\tau[/itex] thenb= 2a.

The magnetic fields produced by the current in the conductors (I_{1}andI_{2}) gives a magnetic flux through the rectangular loop of:

[tex]{{\Phi }_{B}}=K\cdot b\left( t \right)[/tex]

Determine the constantK.

2. Relevant equations

[tex]B=\frac{{{\mu }_{0}}I}{2\pi r}[/tex]

[tex]d{{\Phi }_{B}}=BdA=\frac{{{\mu }_{0}}I}{2\pi }L\,dr,[/tex]

whereLisb(t)

3. The attempt at a solution

What I've done is as following:

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

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

[tex]K=\frac{{{\mu }_{0}}\left( {{I}_{1}}+{{I}_{2}} \right)\ln \left( 3 \right)}{2\pi },[/tex]

which supposedly is the correct answer according to my book.

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

So I thought one of you might knew :)

Regards.

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# Homework Help: Determine a constant K

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