# Magnetic field between two wires (vector sums)

## 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.

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tiny-tim
Homework Helper
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. hooray another annoying and difficult realization in the land of physics :)

thanks