Magnetic field between two wires (vector sums)

[quantum13]

Homework Statement

Two parallel wires carry a current I and 2I in different directions. What is the magnetic field halfway between the two wires?

Homework Equations

Ampere's law
Int (B dot dA) = permissivity x enclosed current

The Attempt at a Solution

Draw a circle around wire 1
B x 2 pi r = mu x I
B = mu x I / 2 pi r
if d = distance between two wires, r = d/2


Where I'm confused is at the part where I add the two B vectors to find the total B. According to the rule of vector addition, I can add vectors to find the net vector. BUT I thought Ampere's law was supposed to describe all of the B at a point as proportional to ONLY the current inside and that currents outside the Amperian loop were not supposed to make any net contribution to B. So how can I add together two vectors when Ampere's law is supposed to describe a magnetic field with all the B vectors already added up? This is a more conceptual question. Thanks.

 

[tiny-tim]

Hi quantum13!

(have a mu: µ and a pi: π and btw, it's permeability for magnetism )

BUT I thought Ampere's law was supposed to describe all of the B at a point as proportional to ONLY the current inside and that currents outside the Amperian loop were not supposed to make any net contribution to B. So how can I add together two vectors when Ampere's law is supposed to describe a magnetic field with all the B vectors already added up? This is a more conceptual question. Thanks.

You have to look at the whole loop.

A loop C around a wire with currrent I₁ and field B₁ has an integral of ∫_C B₁.dl = 2πrB₁ = µ₀I₁.

The same loop has an integral of ∫_C B₂.dl for a different wire, but B₂ will not be constant, and if this wire lies outside C, the integral is zero.

 

[quantum13]

hooray another annoying and difficult realization in the land of physics :)


thanks