- #1

Daniel Gallimore

- 48

- 17

I stumbled across a similar relationship when playing with Stokes' theorem and the magnetic field of an infinite straight wire with a steady current [itex]I[/itex]: it is [tex]\nabla\times\frac{\hat{\phi}}{r}=2\pi\delta^2(r) \, \hat{z}[/tex]

The magnetic field of an infinite straight wire is [tex]B=\frac{\mu_0I}{2\pi r} \, \hat{\phi}[/tex] Stokes' theorem states [tex]\iint_S(\nabla\times B)\cdot d a=\oint B\cdot d \ell[/tex] The right side of the equation is [itex]\mu_0I[/itex]. The left side of the equation, however, is zero since [tex]\nabla\times\frac{\hat{\phi}}{r}=0[/tex] using the standard definition of the gradient in cylindrical coordinates. Using the relationship I introduced in the second paragraph seems to fix this problem. This relationship is also able to satisfy Stokes' theorem for a finite straight wire.

So far, I have only been able to find one tangential reference to this relationship in Ben Niehoff's response to the question "Is B with curl 0 possible?" If this formula is familiar to anyone, or if anyone can reference literature that specifically addresses this formula, I would greatly appreciate it.