Flux through a circle not centered at origin.

[Kizaru]

I feel really dumb for this, but I keep getting strange answers so I may be forgetting something.

Homework Statement

Find the magnetic flux through a circle in the xz plane of radius "a" due to a wire. . The center of the circle is (b,0,0).

Homework Equations

Well I found the magnetic field to be proportional to
$$\frac{1}{\rho}$$

That field has only a component in the phi direction.

Where rho and phi are cylindrical coordinates.

The Attempt at a Solution

I can't figure out the limits of integration for anything unless it's rectangular coordinates, but those become very messy and I obtain an integral which is clearly wrong. I KNOW the answer should reduce to something "nice."

I know I'm missing something stupid here.

 

[fluidistic]

Where is the wire? I guess it's the z-axis?

 

[Kizaru]

Yes, it is. It's an infinite wire carrying current I in the z-hat direction. Basically using the result of a previous problem, I know what the B field is. I just chose not to include the rest of the stuff for that field in this question.

 

[fluidistic]

According to wikipedia, $$\Phi _B=\iint _S \vec B \cdot d \vec S$$.
I'm not able to solve the problem so I'd love someone to help us.
If you have the expression of B in all the circle, I think it's easy to solve the integral.

 

[Kizaru]

I obtained B and I know what the definition of the magnetic flux is. I also know the vector element of area for cylindrical coordinates results in an integral over just $$\frac{1}{\rho}$$

IE, I have

$$\Phi _B= k \iint _S \frac{1}{\rho} d\rho dz$$
where k is a constant (having trouble texing the constants).
The issue is I am unsure of what to make the limits of integration.

 

[fluidistic]

I obtained B and I know what the definition of the magnetic flux is. I also know the vector element of area for cylindrical coordinates results in an integral over just $$\frac{1}{\rho}$$

IE, I have

$$\Phi _B= k \iint _S \frac{1}{\rho} d\rho dz$$
where k is a constant (having trouble texing the constants).
The issue is I am unsure of what to make the limits of integration.

Are you sure you have a $$d\rho dz$$ as a differential area? You said you were working cylindrical coordinates, I'm confused.

 

[Kizaru]

Yes. The B field is only in the phi-hat direction. The differential element of area in cylindrical coordinates is on this page: http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

Since the B field is only in the phi-hat direction, the resultant dot product is exactly as above.

 

[fluidistic]

Ok... for the integration I think that rho goes from 0 to a but you've killed me with your dz.
Looking at http://en.wikipedia.org/wiki/File:Coord_system_CY_1.svg, I would have add to integrate a $$d\phi$$ from 0 to $$2\pi$$.

 

[Kizaru]

What? Phi is the angle that goes counterclockwise from the x axis. I would NOT be integrating over dphi because phi is constant = 0. The circle lies in the xz plane.

Draw the circle out on a piece of paper to see what I mean. Clearly rho does not go from 0 to a. Rho goes from (b-a) to (b+a) when z =0, and it goes from b to b when z = a. The limits of integration over the inner integral are clearly not constant.

 

[phsopher]

Well, you could divide the circle into infinitesimal horizontal slices and calculate what is the area dA of a slice as a function of x (through trigonometry). On the circle, B is proportional to 1/x so the flux through a slice would be just B(x)(dA/dx)dx which you could integrate along the x axis. However, this gives a complicated integral which I'm not sure how to solve.

 

[Kizaru]

I tried a similar approach and obtained an integral which Mathematica integrated to have imaginary parts != 0. I am curious if I could shift my origin (make a substitution) and then convert to spherical coordinates (probably the other order). The spherical coordinates integral (after the substitution) for r would be from 0 to a, and for theta it would be 0 to pi. After the origin shift, my B field will still be in phi-hat direction only due to the symmetry.

 

[fluidistic]

My error, I thought it was in the x-y plane, sorry about that.

 

[phsopher]

I tried a similar approach and obtained an integral which Mathematica integrated to have imaginary parts != 0.

It does that because it doesn't know the values of a and b, they might be such that the thing inside the square root is negative. If we assume that b>a>0, however, it's clear that the value of the integral is real. You have to use assumptions, though i don't know how to do that in Mathematica. Anyway, I got (assuming B(x) = B₀/x)

$$\Phi = \int_{b-a}^{b+a} \frac{2 B_0 \sqrt{a^2 -(b-x)^2}}{x}dx = 2 B_0 a \int_{\frac{b}{a}-1}^{\frac{b}{a}+1} \frac{ \sqrt{1 -(\frac{b}{a}-x)^2}}{x}dx$$

which according to Maple is

$$2 B_0 a \,{\frac {\pi \, \left( 1+\frac{b}{a}\,\sqrt {-1+{\frac{b^2}{a^2}}}-{\frac{b^2}{a^2}} \right) }{ \sqrt {-1+{\frac{b^2}{a^2}}}}}$$

which can be simplified to $$2 \pi B_0 (b - \sqrt{b^2-a^2})$$

Which isn't that bad in the end (assuming I didn't make a mistake). I'm not sure how to calculate the integral by hand though.

 

[Kizaru]

YES! Thank you. That's exactly what the result should simplify to (it's very similar to that of a toroid, after all a toroid is roughly just N of those). Thank you. You were right, I didn't make the assumptions in Mathematica, but I'm still quite a novice and don't know how to do that yet.

Thank you :)