Magnetic field of a bent copper wire

[Srun]

Homework Statement

A copper wire with cross-sectional area S=2.5 mm^2 is bent to make three sides of a square frame that can pivot about the axis 00' as shown below. The wire is located in a uniform vertical magnetic field B. If upon passing a current I=16 milliAmperes through the wire, the wire deflects by an angle Θ= 0 degrees, find value of B (in SI units).

Homework Equations

F = Force, measured in Newtons
I = current in wire, measured in amperes
B = magnetic field vector, measured in teslas
= vector cross product
L = a vector, whose magnitude is the length of wire (measured in metres), and whose direction is along the wire, aligned with the direction of conventional current flow.

The Attempt at a Solution

I can't seem to find what r and the unit vector for r is.

Any hints? Or am I looking at this in the wrong way?

 

[rhoparkour]

First off, B is already given, you are assuming the wire does not exert forces on itself (or at least, these contributions are irrelevant).

There is no one unit tangent vector L. You must think of the bent wire as being made of little pieces of straight wire and find the torque exerted by the magnetic field on the wire, ask for equilibrium. This will give you B in terms of this deflection angle.

Or simpler yet, the wire is made up of three straight segments. The only line segment which contributes a nonzero force to the "total wire" is the bottom straight wire.

 

[Hellabyte]

You know that when we run a current through the segments of wire it will cause a force on them. Using the right hand rule and the magnetic force law. draw these force vectors.

Now because the current is going up the wire in one segment and down it in another, try and think of something you can say about the total force from those two segments. I am talking about the two parts of the wire that go vertical. Better yet take both force vectors and add them and see if it agrees with your argument.(we don't care about torques from these two wire segments)

Now, we know that a force is going to be exerted on that bottom wire, and we can find the direction. What other force is going to be acting on that wire? find the point at which these two forces are in equilibrium, and solve for B.