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I am trying to determine the external magnetic field of a solenoid at a given distance from its $z$-axis. Currently, I have been able to find the Biot-Savart law for a current loop: $$B_{z}=\frac{\mu_{0}}{4\pi}\frac{2\pi R^{2}I}{(z^{2}+R^{2})^{3/2}}.$$ How would I determine the external magnetic field strength of a solenoid from this equation?

Since each turn on a solenoid can be considered as one current loop, I presume there is a way to "add up" magnetic field strengths of a solenoid with $n$ number of turns (current loops). We can also roughly equate the $(z^{2}+R^{2})^{3/2}$ as just $z^{3}$ , when $z$ is significantly bigger than $R$.

I think integrating the formula would give the answer, but since I am only familiar with very basic calculus (i.e. basic definite integrals), I am not sure how to go about this. If anyone uses calculus, please be so kind to add the steps for my own understanding (no matter how simple !)

(Note, I only started 10th grade this fall, so please take this into account into your answers !)

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    $\begingroup$ There is an exact formula for the exterior field of a solenoid, but it is very complicated—involving integrals of special functions (which are themselves defined by integrals). The expression you have in your question is only the field of a single circular current loop on the axis of the loop, at $(0,0,z)$; just generalizing this expression to an arbitrary off-axis location takes an entire lecture in my graduate-level electrodynamics class. $\endgroup$
    – Buzz ♦
    Commented Sep 14, 2021 at 23:38
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    $\begingroup$ However, there are situations when it is possible, as you suggest, to get some reasonably approximations. Could you give some indication of what kind of situation you are interested in, to help determined whether there is a useful approximate expression that might be useful? $\endgroup$
    – Buzz ♦
    Commented Sep 14, 2021 at 23:38
  • $\begingroup$ In my scenario, I would want to determine the magnetic field strength slightly off axis (maybe between 1 to 5 degrees off axis). The non symmetry axis answer which Michael's post links seems to solve this,, but maybe there are simpler approximations ? $\endgroup$
    – WhiteKnightMatrix
    Commented Sep 15, 2021 at 8:41

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This is, it turns out, a deceptively hard question.

The equation you've found is actually only applicable for points on the axis of symmetry of a circular loop. You could, if you wanted, use this formula to find the magnetic field along the axis of a solenoid. The basic technique is described in the answers to this question.

The integral you get for points along the axis isn't that hard to perform (try it!). But if you wanted to find the magnetic field at a point off-axis, then the equation you found from the Biot-Savart Law wouldn't work in the first place. In fact, the full solution for points not on the symmetry axis turns out to be pretty heinous, only expressible in terms of ugly beasts called "elliptic integrals". Some of the details can be found over on this Mathematica.SE post, though the focus there is on getting a particular piece of software (Mathematica) to do the calculations for you.

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