# Off-Axis magnetic field ($B_z$ @ $\rho=0.5 R$) of a finite solenoid

I am doing mathematical modelling in MS Excel Spreadsheet for axial magnetic field (Bz) at different distances away from the centre for a finite model. I see small hump at the fringe (i.e. just outside the length of the solenoid core). What is the real explanation of this effect?

In some paper I see a vague reference which states: Quote "There is a bit of ripple along the edges inside the solenoid. As one gets very near to the turns of wire, the fields set up by the individual turns can be distinguished. Remember that the solenoid is wound using very thick wire, whose centers are about one-sixteenth of an inch apart. Actually, the rippling seen is more subtle than just the individual wires. It reflects some aliasing between the individual wires and the step size of the numerical procedure." Unquote

Can some one explain in more detail whether this is mistake in modelling or computational error or something very well established issue? Any logical and simple explanation of this "hump" or "ripple effect" would be deeply appreciated?

I have added the graph that I have generated using equation 36, page 671 of the paper link attached. The hump or ripple which I am referring to you can be seen on the wings.

• Welcome aronza to Physics Stack Exchange. "A picture is worth a thousand words", so showing the "hump" might clarify what you are asking about. Commented Feb 26 at 1:32
• How can I add a picture? New to the forum so if you can tell me so I can quickly provide more relevant details Commented Feb 26 at 5:20
• Commented Feb 26 at 6:12

The "hump" is incorrect. A static $$B_z$$ in free space can only get weaker the farther away from its sources you get, unless there is some ferromagnetic material nearby to shape the field. Also, the distance scale over which any changes occur is set by the distance from the sources or ferromagnets, so a roughly 1 cm wiggle is impossible when you 2.5 cm from the solenoid wires. (As far as I can tell from the paper, R = 5 cm.)
Although it is not necessary to completely understand its derivation when using an equation from a theoretical paper, it is important to note any caveats in the paper about when the equation is applicable. The paper you are citing clearly states in its Conclusion that the equation you are plotting is an approximation that is only valid for $$\rho\ll\mathrm{R}$$ or $$\rho\gg\mathrm{R}$$, so it is not surprising that it gives odd results for $$\rho\sim\mathrm{R}$$.