Creating a magnetic field (vector field)

[piscosour00]

Hi all,

I have a question for all of you. I've been wanting to make a 3D vector field that would represent a magnetic field (for fun) around some segment of wire with a constant current flowing through it. I'm assuming I have a parametric equation for the wire segment. The one equation that comes to mind is Biot-Savart's law:

$$\vec { B } =\frac { { \mu }_{ 0 }I }{ 4\pi } \int { \frac { d\vec { s } \times \hat { r } }{ { r }^{ 2 } } }$$

In practice, I've only ever used Biot-Savart's law to calculate in 2-D, and either the wire segment has been of infinite length, or we were just calculating the electric field at one-point, and the math has been nice. I want to generate a vector field that gives the magnetic field at all points around the wire. Does anyone know how to go about doing this ? What sorts of equations/techniques lend themselves to this ? Any nice examples people can point to (URLs) ?? Thank you.

 

[piscosour00]

Ok I've made some progress:

If I say that any point in R3 can be given by the position vector:

$$\vec { p } =[{{ x }_{ 0 }{ ,y }_{ 0 }{ ,z }_{ 0 } } ]$$

Since the vectors r and r-hat are vector that points from a point on the wire to the point in R3 at which you want to know the magnetic field, we can write the vector r as:

$$\vec { r } =\vec { p } -\vec { s }$$

And therefore r-hat is:

$$\hat { r } =\frac { \vec { p } -\vec { s } }{ { |\vec { p } -\vec { s } | } }$$

Where the vector s is the parametrization of the wire, and so Biot-Savart's law can be written as:

$$\vec { B } =\frac { { \mu }_{ 0 }I }{ 4\pi } \int { \frac { d\vec { s } \times \hat { r } }{ { r }^{ 2 } } } =\frac { { \mu }_{ 0 }I }{ 4\pi } \int { \frac { d\vec { s } \times (\vec { p } -\vec { s } ) }{ { |\vec { p } -\vec { s } | }^{ 3 } } }$$

This I can now integrate after evaluating the cross product, but it's not an easy integral to evaluate because of the nasty term in the denominator. Does anyone have any ideas for an easier way of evaluating this integral ? I tested this out in Mathematica for the simple case of an infinite wire running along the x-axis, and the resulting 3D vector plot looked right, but even Mathematica took a long time to evaluate the integral. Any ideas ?