# How does the magnetic field generated from a rectangular cross-sectional current-carrying conductor differ from a circular cross-sectional conductor?

I can find much information of cylindrical conductors (ie. regular wires), where $B=\frac{\mu_0 i}{2 \pi r}$ and $r$ represents the radius (or distance) from the centre of the conductor, however I haven't been able to find anything about a conductor with a rectangular cross-section.

I would have expected a different result due to fringing effects due to the square edges of the conductor, or at least a difference in which direction you go. I'm not sure if the Biot-Savart Law could be applied to this problem, or how to calculate the geometry of it.

• Which page in your reference do you feel suggests this for a rectangular wire? This is definitely not true in general, and your doubts due to fringing effects (and such) are correct. It seems like you're still referring to a straight wire's radially symmetric magnetic field. – Arturo don Juan Apr 5 '15 at 22:07
• @Arturo, I was referring to page 9, but I now see that it's talking about an arbitrary closed path around a regular wire, not an arbitrary conductor shape. – Adam M-W Apr 5 '15 at 22:10
• yeah you got it. – Arturo don Juan Apr 5 '15 at 22:15
• Actually, I only really need the magnitude of the field at points a specific distance from the centre of each side of the conductor along the line running perpendicular from the surface, so if there's a simplification that requires less calculation I could use here that would be helpful. – Adam M-W Apr 12 '15 at 1:21

The magnetic field is different, and is not the same result. The result using concentric circles actually makes use of the rotational symmetry of the system, but the same symmetry does not hold for the rectangular system. Don't expect a nice formula - these things can be difficult to calculate explicitly! Ampere's law and the Biot-Savart law still hold, but they may not be easy to apply. However, at large $r$ (imagine zooming out so that the whole rectangle looks like a point), your formula will be a very good approximation. You might be able to easily get further exact terms in your approximation by using the multipole expansion method.
If your wire is an actual thick wire with a volume current, then it's harder to say. In the case of a rectangular cross section with constant charge density, you can draw the potential as a squarish lump and visualize the electric field that way. You can do the same thing with a $B$-field, but now you need a vector potential. From certain equations for the vector potential and a certain curl identity you can see that a prescription for finding the fringe field is this: Draw (in 3D, with two axes being the plane of the wire cross section and another being your potential) the squarish lump that is the potential for the corresponding constant charge density problem, slice off loops of constant potential (which will be approximately squarish), and then traverse them one way or the other (according to the right-hand rule). This will be the direction of the magnetic field.
• @AdamM-W Oh, you want to know how to numerically calculate it? That might belong to a separate question. High precision would probably be totally useless for you, for engineering/practical calculations, because only the highest order term $\mu_0 I/(2 \pi r)$ will be significant. At most, you'd go to a second order term (in a "multipole expansion"). Anyways, the formula in the paper you linked is just the integral of the vector potential of a wire (the infinitesimal wire can be taken as an infinitesimal "current element"). – user12029 Apr 12 '15 at 3:41