Biot-Savart Law for Currents
Claimed by David Medrano
Biot-Savart Law
The Biot-Savart Law can be used for more than just single moving charges; it can also be used to calculate the magnetic field for a large number of charges. One notable reason to do so is to find the magnetic field of a portion of a wire where there can be many moving charges. When we use Biot-Savart Law to find the magnetic field of a short wire, we can apply it to a variety of shapes.
A Mathematical Model
First We start off with the original version of the Biot-Savart Law. [math]\displaystyle{ \vec B=\frac{\mu_0}{4 \pi } \frac{q\vec v\times\hat r}{r^2}. }[/math]
Because we are dealing with a portion of wire [math]\displaystyle{ \mathrm{d}\boldsymbol{\ell} }[/math] long with an Area A containing n moving particles with charge q, we find that the total number of moving charges is equal to |q|(nAv) which is also equal to I, the current in the wire. [math]\displaystyle{ B = \frac{\mu_0I}{4\pi}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2}, }[/math]
Because the shape of the current carrying wire can vary from a straight wire to a loop, we must integrate over the region of the wire.
[math]\displaystyle{ B = \frac{\mu_0I}{4\pi}\int_{\mathrm{wire}}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2}, }[/math]
Example
A Computational Model
The following link shows the magnetic field produced by small segments of wire in a loop individually. For a long straight wire, we see that there is a circular magnetic field surrounding the wire with current. The following link does a stepwise visual of the contributions of each part of the wire at an observation location a distance r from the wire.
We see that along the axis of the wire, each contribution not on the axis is negated due to symmetry and the resulting magnetic field is all along the wire.
Connectedness
==Applications==
History
See also
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Further reading
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References
https://www.grc.nasa.gov/www/k-12/airplane/thermo0.html http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thereq.html https://www.grc.nasa.gov/www/k-12/airplane/thermo2.html http://www.phys.nthu.edu.tw/~thschang/notes/GP21.pdf http://www.eoearth.org/view/article/153532/