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

What I actually need to find is the magnetic field at any point in space for an elliptical solenoid. I think the way to do this would be to find the magnetic field caused by an elliptical current loop, and then to use some kind of summation for multiple loops to get the field caused by a solenoid. If there's a better way and someone can point me in that direction, I would appreciate it. However, based on that I'm working on the magnetic field of the elliptical loop and could use some help verifying if my work is correct. I have access to MATLAB and will use that as necessary, but I need to set up the application of Biot-Savart to plug in to MATLAB.

2. Relevant equations

Biot-Savart Law: [tex]\bf{\vec{B}} = \int{\frac{{\mu _0 }}{{4\pi }}\frac{{Id\vec{\ell} \times {\bf{\vec{r}}}}}{{|\bf{\vec{r}}|^3 }}}[/tex]

3. The attempt at a solution

The first thing I worked on is getting the vector [tex]\vec{r}[/tex] from the wire element to the point being evaluated. The arbitrary point being evaluated is [tex](x_0,y_0,z_0)[/tex].

For a coil parallel to the x-y plane:

[tex]R(\theta)=\frac{a \cdot b}{\sqrt{b^2\cos^2(\theta)+a^2\sin^2(\theta)}}[/tex]

Where a is the major axis radius and b is the semi-major axis radius.

[tex]x=R(\theta) \cdot \sin(\theta)[/tex]

[tex]y=R(\theta) \cdot \cos(\theta)[/tex]

[tex]z=z[/tex]

The vector is [tex]\vec{r}=(x_0-x)\hat{i}+(y_0-y)\hat{j}+(z_0-z)\hat{k}=(x_0-R(\theta) \cdot \sin(\theta))\hat{i}+(y_0-R(\theta) \cdot \cos(\theta))\hat{j}+(z_0-z)\hat{k}[/tex]

I'm not sure about the [tex]d\vec{\ell}[/tex] term. Would it be [tex]d\vec{\ell}=\frac{-a\sin(\theta)\hat{i}+b\cos(\theta)\hat{j}}{\sqrt{a^2\sin^2(\theta)+b^2\cos^2(\theta)}}[/tex]?

These terms would then just be plugged into the Biot-Savart Law equation and the integral would be evaluated from the lower bound of 0 to to the upper bound which would be the circumference of the ellipse.

Does this look correct?

Thanks

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# Homework Help: Magnetic field of elliptical coil (at all points in space)

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