@@ -34,19 +34,15 @@ Lots of improvements to make after several very productive conversations and whi
# Supplementary Materials
## 2D Simulations
## Extrapolation to 3D
#### Infinite Wire Approximation
Things get more complex when trying to extrapolate these 2D simulation results to 3 dimensions. For example, Biot-Savart tells us the contribution of the infinitesimal wire length dl, but what if the wire extends 5cm into the screen? In that case the sum of contributions from each element dl all contribute to the field strength at point A, but with increasing effective radii, and decreasing angle theta, which results in the cross product dl x r_hat also decreasing in magnitude. It turns out that if we integrate the contribution of an infinite wire on a single point A, we scale Bio-Savart by a factor of 2r. For small distances r relative to the length of the wire, this is a valid assumption, where true 3D FEA would likely be necessary for better resolution.
Things get more complex when trying to extrapolate 2D simulation results to 3 dimensions. For example, Biot-Savart tells us the contribution of the infinitesimal wire length dl, but what if the wire extends 5cm into the screen, and an adjacent magnet is only 2cm thick? In that case the sum of contributions from each element dl all contribute to the field strength at point A, but with increasing effective radius, and decreasing angle theta, which results in the cross product dl x r_hat also decreasing in magnitude. It turns out that if we integrate the contribution of an infinite wire on a single point A, we scale Bio-Savart by a factor of 2r. From my understanding, if we assume all magnetic materials scale infinitely into the page, this would result in linear scaling and could still provide useful relative approximations.
#### Coil Approximation
In the case of a coil, the distance r remains constant for an entire circumference of wire (assuming it isn't spiraling), and therefore we can scale Biot-Savart by
<pid="gdcalert4"><spanstyle="color: red; font-weight: bold">>>>>> gd2md-html alert: equation: use MathJax/LaTeX if your publishing platform supports it. </span><br>(<ahref="#">Back to top</a>)(<ahref="#gdcalert5">Next alert</a>)<br><spanstyle="color: red; font-weight: bold">>>>>> </span></p>
. This is also true for points not at the center of the coil, as the sum of distance to the edge of a coil is constant*** not sure about this.
In the case of a coil, center axis of the coil is at a fixed distance from the circumference of wire (assuming it isn't spiraling), and therefore we can scale Biot-Savart by 2*pi*r, but as we move towards the perimeter of the coil radially, the 3 dimensionality of the coil becomes more complex and nonlinear. I believe that unlike the infinite wire approcimation, this would not result in linear scaling and perhaps would result in error unless accounted for?
