Why does Ampere's law only apply to enclosed currents?

[lhluo]

In my textbook it tells us
∫B ds = I μ
(line integral and I = current enclosed)
It also states that the current not enclosed does not affect the magnetic field along the line.
I don't see a reason for other currents outside the loop not to affect the magnetic field

thanks for any help

 

[Simon Bridge]

Welcome to PF;
You only need to do the math for enclosed currents because the effects of all outside cancel out.
It is similar to finding the strength of gravity below the surface of a uniform sphere of mass - it depends only on the mass enclosed.

The details are in the derivation:
https://www.physicsforums.com/showthread.php?t=609430

 

[tiny-tim]

welcome to pf!

hi lhluo! welcome to pf!

I don't see a reason for other currents outside the loop not to affect the magnetic field

they do affect the magnetic field round that loop,

but they increase it in some places, and decrease it in others, and the total round the whole loop is always the same