Find k from magnetic field and magnetic flux.

[Ylle]

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/"

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:

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

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

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

Now determine k

Homework Equations

$$\[{{\Phi }_{B}}=BA\]$$

The Attempt at a Solution

I know the answer is supposed to be:

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

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.

 

[gabbagabbahey]

Homework Equations

$$\[{{\Phi }_{B}}=BA\]$$

This formula is only true if $\textbf{B}$ is uniform over the surface and normal to the surface. Are these two conditions met by the $\textbf{B}$ 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.

 

[Ylle]

Ahhh, I guess, since the field is in z-direction it's not uniform.
So what I need to do is:

$$\int$$$$\int B dx dy$$ with the limits 0 to L in both integrals, and the equal the flux I have, and solve for k ?

 

[gabbagabbahey]

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 $y$ and $y$ varies over the surface.

So what I need to do is:

$$\int$$$$\int B dx dy$$ with the limits 0 to L in both integrals, and the equal the flux I have, and solve for k ?

Yup.

 

[Ylle]

I see :)

Thank you very much.