Why Are Volume Current Densities Zero on the Surface of Current-Carrying Objects?

[WWCY]

Hi all,

I have been reading Griffiths' book on Electrodynamics and have come across a point (image attached below) where he states that volume current densities are 0 on the surface of the current-carrying objects. He then uses these properties in pretty-important integrals.

However, I couldn't seem to find any explanation (physical or mathematical) in the book as to why it was the case. What exactly am I missing?

Thanks in advance!

 

[PeroK]

If you assume all the current is contained in some finite volume, then - as he says - you can always consider a larger volume, on which the surface integral must vanish

 

[WWCY]

Hi, thank you for your response.

If you assume all the current is contained in some finite volume, then - as he says - you can always consider a larger volume, on which the surface integral must vanish

Ah okay, it makes sense. However, if I choose to select the volume of integration such that it is exactly that of the current carrying body, will I still get $J=0$ for the surface integral?

 

[BvU]

No. Only the integral (containing $\vec {\bf J}\cdot\vec {d\bf a}\$) vanishes