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Determine a constant K

  • #1
148
1

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/" [Broken]

The length a is constant, but b varies 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 for t < 0 then b = a, and for t > [itex]\tau[/itex] 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:

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


Determine the constant K.


Homework 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]
where L is b(t)

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 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.
 
Last edited by a moderator:

Answers and Replies

  • #3
674
2
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.
 

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