- #1
Daniel Gallimore
- 48
- 17
I'm familiar with the relationship [tex]\nabla\cdot\frac{\hat{r}}{r^2}=4\pi\delta(r)[/tex] in classical electromagnetism, where [itex]\hat{r}[/itex] is the separation unit vector, that is, the field vector minus the source vector. This is result can be motivated by applying the divergence theorem to a single point charge.
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.
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.