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How is magnetic field at a point due to current carrying wire calculated in 3D?

Let's say wire has current I placed along x-axis in direction x.

Would it just be $$\frac{\mu_0I}{2\pi y} \hat y + \frac{\mu_0I}{2\pi z} \hat z ?$$

I realized that this doesn't quite make sense as when there's a point in $z = 0$, this would give magnetic field of infinity. How could I get magnetic field in this case?

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For an infinitely long and infinitely thin wire placed along the $x$ axis, the equation for the magnetic field is $$ \mathbf{B} = \frac{\mu_0 I}{2\pi}\frac{1}{\sqrt{y^2+z^2}} \hat{\boldsymbol{\varphi}_{yz}}, $$ where $\hat{\boldsymbol{\varphi}_{yz}}$ is the azimuthal direction in the $yz$ plane, defined as $\hat{\boldsymbol{\varphi}_{yz}} = [-\sin(\arctan{\frac{z}{y}}) \hat{\mathbf{x}},\cos(\arctan{\frac{z}{y}}) \hat{\mathbf{y}},0].$

It looks like this:

enter image description here

and it is translationally invariant along the $x$ axis so any $x$ value would yield the same result. The magnitude of the vectors scale as the distance$^{-1}$ from the origin.

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  • $\begingroup$ Harder than it needs to be. $\endgroup$
    – ProfRob
    Commented Apr 29, 2020 at 8:24
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    $\begingroup$ Really? I thought this was the simplest way, as using Biot-Savart and involving spherical coordinates to derive one seemed nearly impossible. $\endgroup$ Commented Apr 29, 2020 at 10:51
  • $\begingroup$ Also @SuperCiocia, I would be really greatful if you could show how it was derived using Biot-Savart or other methods. $\endgroup$ Commented Apr 29, 2020 at 14:01
  • $\begingroup$ you use the standard field from a wire: $\mathbf{B}(r) = \frac{\mu_0 I}{2\pi r} \hat{\boldsymbol{\phi}}$ that you get from Ampere's law. In this typical case, you assume the wire is oriented along the $z$ axis, so $r$ is the distance from the $z$ axis in cylindrical coordinates, $r = \sqrt{x^2+y^2}$. Here I just cast the formula for a wire aligned along the $x$ axis, so my "$r$" is the distance from the $x$ axis which is $\sqrt{y^2+z^2}$. $\endgroup$
    – SuperCiocia
    Commented Apr 29, 2020 at 19:52

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