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I'm programing simulation software for solenoid and hemholtz coils, where I have to calculate B(x,y,z). I have found dozens of variations of Biot-savart law for determining B on axis. But how about outside of the solenoid?

This is said to be outside of solenoid: $$ B(z) = \frac{\mu_0}{4\pi}\cdot\frac{iR^2}{2(R^2+z^2)^{\frac{3}{2}}} $$
But why only z coordinate? What about length of solenoid? In order to build simulation, I have to count with finite-length solenoid.

There are hints, that I should apply line integral (curve integral?) to the biot-savart law. But what is that curve? The electric wire? So I have to use eliptic integral to have something close to the solenoid? I'm quite lost in all of this.

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  • $\begingroup$ In fact, it has a closed form after the integral. please check this website. $\endgroup$
    – ryantang
    Jan 18, 2016 at 12:42

1 Answer 1

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The magnetic field of a single loop of current is exactly solvable both on- and off-axis, with the horrible disadvantage that the off-axis solution uses elliptic integral functions. The formula you quote seems to be the on-axis field of a single loop.

The exact solution is given in Jackson's Classical Electrodynamics, section 5.5, or in this webpage. From the latter, $$ B_z = \frac{\mu_0 I}{2\pi}\frac{1}{\sqrt{(a+r)^2+z^2}}\left[E(k) \frac{a^2-r^2-z^2}{(a-r)^2+z^2}+K(k)\right] $$ is the on-axis component, while $$ B_\rho = \frac{\mu_0 I}{2\pi}\frac{z/\sqrt{r^2-z^2}}{\sqrt{(a+r)^2+z^2}}\left[E(k) \frac{a^2+r^2+z^2}{(a-r)^2+z^2}-K(k)\right] $$ is the cylindrical polar radial component, where $a$ is the loop radius, $k=\sqrt{\frac{4ar}{(a+r)^2+z^2}}$, and $E(k)$ and $K(k)$ are complete elliptic integrals of the first and second kind, respectively.

This is right horrible but it is suitable for computational purposes. To find the field of a solenoid you should integrate this over an on-axis displacement of the ring. You could attempt this analytically (though I'm not aware of any exact solution) or numerically. The latter makes a bit more sense to me as your solenoid is of course a finite superposition of current loops (though of course those have a finite width).


Added:

The field above is for a single loop in the $x,y$ plane, centred at the origin. To perform the integration, you need to transform this result into the field of a loop around an arbitrary point on the $z$ axis. To do this, you need to change $z$ to the difference $z'=z-b$ and $r$ to the distance $$r'=\sqrt{x^2+y^2+(z-b)^2}=\sqrt{r^2+b^2-2bz}.$$ You also need to transform the field: the component $B_z$ is a cartesian component and transforms well, as does the cylindrical component $B_\rho$, but the spherical radial component $B_r$ from the previous version points away from the centre of the loop and needs to be transformed.

Once this is done, you integrate over $b$ from $-L/2$ to $L/2$. Note that this involves integrating nested square roots inside the elliptical integral functions, so it is unlikely to have a closed-form solution. I would urge you to consider a numerical integration scheme that views your solenoid as a finite (though possibly large) collection of current loops at different displacements $b$ and implementing the sum in your code.


If this is unacceptably complicated, and you have some specific limit in mind, look in Jackson. He gives expressions in the limit $k^2\ll1$ which he claims specialize well to the near-axis ($\theta\ll\pi$), near-the-centre ($r\ll a$) and far-field ($r\gg a$) cases, and do not involve elliptic integrals.

I hope this helps.

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  • $\begingroup$ Thank you for valuable answer! Should i integrate Br over Bx? That is: $B_{solenoid} = \int Br\ dBx$ but what will be the range (bounds) of integration? $\endgroup$
    – Fidilip
    May 12, 2013 at 12:51
  • $\begingroup$ @Fidilip Please see the revised post for details on the integration step from a loop to a solenoid. If you need further assistance, you should provide clearer details of what your problem is and what you expect to get out of the solutions you are asking about. $\endgroup$ May 12, 2013 at 18:41
  • $\begingroup$ I think the expression for $B_\rho$ is incorrect. The $\sqrt{r^2 - z^2}$ should be replaced with $r$. In this case, the expression for $B_\rho$ is actually an expression for $B_r$, where $r$ is the radial distance in cylindrical coordinates, not spherical (otherwise the $B_z$ expression is also incorrect). This means that the transformation given in the second part is also not quite right, since you're using $r$ to mean distance from the origin instead of the $z$ axis. $\endgroup$
    – user34722
    Mar 26, 2022 at 5:11
  • $\begingroup$ Also a word of caution to anyone using these functions in a program like Mathematica or Matlab, many software packages use a different convention for elliptic integrals than is used here. To get sensible answers you have to replace $k$ with $k^2$ everywhere. $\endgroup$
    – user34722
    Mar 26, 2022 at 5:13

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