For reference, this is from Griffiths, introduction to ~~quantum mechanics~~ electrodynamics, p253-255

When deriving the ideal magnetic dipole field strength, if we put the moment m at origin and make it parallel to the z-axis,

the book went from the vector potential A

$$

A= \frac{\mu_0}{4\pi}\frac{\vec{m}\times \hat{r}}{r^2}

$$

to

$$

A = \frac{\mu_0}{4\pi}\frac{m\sin{\theta}}{r^2}\hat{\phi}

$$

Can someone explain how the single ##\phi## component come about? This to me seems to indicate ##r## has non-zero ##\theta## component and zero ##\phi## component. But I thought ##r## is any point?

Also, for a point in spherical coordinates, is ##\phi## value defined when ##\theta = 0##? Or when ##r=0##?

When deriving the ideal magnetic dipole field strength, if we put the moment m at origin and make it parallel to the z-axis,

the book went from the vector potential A

$$

A= \frac{\mu_0}{4\pi}\frac{\vec{m}\times \hat{r}}{r^2}

$$

to

$$

A = \frac{\mu_0}{4\pi}\frac{m\sin{\theta}}{r^2}\hat{\phi}

$$

Can someone explain how the single ##\phi## component come about? This to me seems to indicate ##r## has non-zero ##\theta## component and zero ##\phi## component. But I thought ##r## is any point?

Also, for a point in spherical coordinates, is ##\phi## value defined when ##\theta = 0##? Or when ##r=0##?

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