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Find k from magnetic field and magnetic flux.

  • Thread starter Ylle
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  • #1
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Homework Statement


Hello...

I got a problem I really can't figure out.
I have the scenario in this link: http://www.gratisupload.dk/download/41677/" [Broken]

Besides that I know that there is a magnetic field that works everywhere and is along the z-axis. This field, that depends on the y-coordinate and the time t is given as:

[tex]B=ky{{e}^{-{{t}^{2}}/{{\tau }^{2}}}}{{e}_{z}}$[/tex]
where tau is a positive time-constant, k is a constant with dimension T/m and ez is unit vector in the direction of the z-axis.

The magnetic field raises a magnetic flux through the circuit given by:

[tex]\[{{\Phi }_{B}}={{B}_{0}}{{L}^{2}}{{e}^{-{{t}^{2}}/{{\tau }^{2}}}}\][/tex]
where B0 is a positive constant with dimension T.

Now determine k


Homework Equations



[tex]\[{{\Phi }_{B}}=BA\][/tex]


The Attempt at a Solution



I know the answer is supposed to be:

[tex]\[k=2{{B}_{0}}/L\][/tex]

It seemed to good to be true if I just inserted the flux and the magnetic field into this equation, and then setting A = L2.

If I did that I got: k = B0 / y.

And I've been searching my book for examples and stuff I could use. But I can't come up with anything when I only have the magnetic field and magnetic flux. So I'm thinking there must be a trick that I'm not aware of :S

So can anyone point me in the right direction?


Regards.
 
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Answers and Replies

  • #2
gabbagabbahey
Homework Helper
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Homework Equations



[tex]\[{{\Phi }_{B}}=BA\][/tex]
This formula is only true if [itex]\textbf{B}[/itex] is uniform over the surface and normal to the surface. Are these two conditions met by the [itex]\textbf{B}[/itex] in your problem and the square surface bounded by the circuit you are given?

If not, you will need to use the more general definition of magnetic flux.
 
  • #3
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Ahhh, I guess, since the field is in z-direction it's not uniform.
So what I need to do is:

[tex]\int[/tex][tex]\int B dx dy[/tex] with the limits 0 to L in both integrals, and the equal the flux I have, and solve for k ?
 
  • #4
gabbagabbahey
Homework Helper
Gold Member
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6
Ahhh, I guess, since the field is in z-direction it's not uniform.
The reason the field isn't uniform over the surface, is because it depends on [itex]y[/itex] and [itex]y[/itex] varies over the surface.

So what I need to do is:

[tex]\int[/tex][tex]\int B dx dy[/tex] with the limits 0 to L in both integrals, and the equal the flux I have, and solve for k ?
Yup.
 
  • #5
79
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I see :)

Thank you very much.
 

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