Magnetic field and formulating an equation for a particle

[shyguy79]

Homework Statement

Consider a general point on the line joining the two wires and a distance d
away from the mid-point O.
By summing the magnetic field due to currents of magnitude i flowing in both the wires in the senses you found in (c), show that, on the line joining the two wires, the magnetic field strength is given by:

B= μ0iD / π(D2 −d2)

Homework Equations

B= μ0 i / π r

The Attempt at a Solution

My diagram is attached.

Remembering that B(r) = μ0i / 2π r

From the diagram above then the position r of the arbitrary particle A from the right hand conductor would be the distance to the mid point D plus the distance to A and could be expressed as (D+d). whereas its position from the left hand conductor is the distance to the mid point minus the distance to A as (D - d). The vector cross product of the fields since they point in the same direction and at right angles to O would mean that the resultant is the product. Putting this together then r = (D + d)(D - d) which then equates to r = (D2 - d2).

The distance between the two conductors is then (D+D) = 2D. Putting this together then the distance r can be expressed as the ratio: r = (D2 - d2) / 2D.

I have a way to substitute r into the original equation... does this make sense?

 

[jbsteele]

Now substituting the expression for r into the original equation then:B= μ0 i / π [(D2 - d2) / 2D] Simplifying this equation then:B= μ0iD / π(D2 −d2)