Magnetic field at the center of a cube of wires

[Schecter5150]

Homework Statement

Find the magnetic field at a point p in the center of the cube with side length 2b.

Homework Equations

Biot-Savart Law

The Attempt at a Solution

I attempted using the biot-savart law, but my answer contradicts what someone told us as a hint. Supposedly the magnitude of the field at the bottom-front edge of the cube is B = U_o*I*sqrt(2/3)/(4*pi*b).

I get U_o*I*-(Ux+Uz)/(4*pi*b) as the B-field experienced at p from the bottom-front edge. What might I be doing wrong?

I have dl = -dx*Uy
r(hat) = (-Ux+Uz)/sqrt(2)
integrated from -b to b for any given side

Any ideas as to where this difference is coming from? I would imagine by symmetry that sides with opposing currents will cancel out the B-fields at P (i.e. bottom front and back).

 

[zhermes]

This is a good problem.
I'd start out with the right hand rule, and draw in some vectors at the center to see where each segment directs the field

 

[Schecter5150]

Ok, that helps me visualize it a bit better. I'm still concerned about the magnitude of the side I first calculated though. I cannot see how they can get sqrt(2/3).

Using the RHR, my vectors match up for the first three sides on the base of the cube. However, when I calculate the vector for the side when the current travels upward, my vector doesn't make sense. I don't know whether to approach it as Uz*dz X (Ux-Uy) as my resultant vector is (Ux+Uy) which seems to make sense by the RHR but isn't obvious to me at this point. I also have the same concern with approaching from the top of the cube. For example, the top back side gives me (Ux+Uz) when I approach it via a right triangle of vectors, yet the RHR leads me to believe it is actually (Ux+Uz). Could you please clarify for me which of these is the correct approach?

In any side, I obtain the coefficient Uo*I/(4*pi*b) which makes sense due to symmetry. I am just mainly concerned about the advice I received prior to starting this problem, as I cannot see how they could obtain such a value.

EDIT: On second thought, would the vector for the bottom front side actually be in the (Ux-Uz) direction? The RHR leads me to believe so. I think I may have been trying to visualize vectors for the electric field instead.

Thanks

 

[zhermes]

What corner are you setting as the coordinate origin?

 

[zhermes]

Think about RHR and vector using pairs of parallel wires---that way your resultant vectors are in the cartesian directions (i.e. perpendicular to the faces of the cube).

And the root 2/3 surely comes from the geometry, namely the 1/root(3) from the pythagorean theorem.

 

[Schecter5150]

I've started using a coordinate system based with the origin at the left-front corner.

So with the biot-savart law with the bottom front I get:

dl = -Uy*dy
r(hat) = (-sqrt(2)*b*Ux-b*Uz)/(sqrt(3)*b)
r = (sqrt(3)*b)

so then for dl x r(hat) I end up with: ((-sqrt(2)*b*Uz + b*Ux)/(sqrt(3)*b)*dy

When I integrate from y=0 to y=2b I end up with:

B = (Uo*I/(6*pi))*(-sqrt(2)*Uz+Ux)/(sqrt(3)*b) .

This still doesn't match up with what I was told. Where am I going wrong? I can't think of a sensible way to show that the radius changes as it moves along the length of the wire.