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

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SUMMARY

Ampere's Law states that the line integral of the magnetic field B around a closed loop is proportional to the current I enclosed by that loop, expressed as ∫B ds = I μ. The discussion clarifies that while external currents do influence the magnetic field, their effects cancel out around the entire loop, resulting in no net impact on the magnetic field strength. This principle is analogous to gravitational effects within a uniform sphere, where only the mass enclosed contributes to gravitational force.

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lhluo
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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
 
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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
 
welcome to pf!

hi lhluo! welcome to pf! :smile:
lhluo said:
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 :wink:
 
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