Magnetic fields from 3 current carrying wires

[vagabond260]

Homework Statement

Consider three straight infinitely long (parallel to each other), equally spaced
wires (with distance d), each carrying a current I in the same direction.
(a) Calculate the locations where magnetic field is zero.
(b) Sketch the magnetic field line pattern.
(c) If the middle wire is rigidly displaced a very small distance x (x << d) upward while
the other two wires are held fixed, describe qualitatively the subsequent motion of the middle
wire.

Homework Equations

Biot-Savart Law and Ampere's Law

The Attempt at a Solution

I know the magnetic fields are zero near the wires on the sides but how exactly do i calculate that in terms of "d"?

For part (c) I was thinking since the middle wire will be pulled on both sides (to the points where the magnetic fields are zero) with equal forces it should remain stationary. But then it's not lying on the same plane as the other two wires, so does that mean it will be pulled down to the same plane?

 

[supratim1]

Hi vagabond! welcome to PF!

what do you mean by 'I know the magnetic fields are zero near the wires on the sides'?

 

[vagabond260]

Hi vagabond! welcome to PF!

what do you mean by 'I know the magnetic fields are zero near the wires on the sides'?

I mean that the magnetic field from the wire at one end has to cancel the magnetic fields of the other two wires (and as the magnetic field strength falls off as 1/distance from wire) this distance has to be close to the wire on the sides.

 

[vagabond260]

suppose i have the following:

----------------------------------------1
A
----------------------------------------2
B
----------------------------------------3

the current is going from left to right; then the places of zero magnetic field have to be in regions A and B - near wire 3 and wire 1; but how exactly do i measure that distance

I'm trying to get a fraction of "d" but i get a value in terms of "pi" which doesn't make sense :S

 

[vagabond260]

Ok, I get a value of about (1-(sqrt(3)/3) d from the wires on the both sides where the magnetic field is zero - does that seem reasonable to anyone?

 

[Demon117]

First assume the three wires are coplanar. That being the case the two points must lie in the same plane. The distance from the point to the middle wire is simply x, and the distance from the point to the other two wires are d+x and d-x. Once you apply Ampere's law the solution will pop out algebraically.