B field at Center of Conducting Square

[ikentrovik]

Hi there I am just really confused as to how they arrived at the answer to this problem. I attached a picture of the entire question and the solution.

Homework Statement

Please see attached file.

Homework Equations

Biot Savart Law for Magnetism of a Wire where the length of wire is much larger than distance away from the wire where the B field is being measured is:

(μ/4π)(2I/r)

I: current
r: distance from wire

The Attempt at a Solution

How I approached the problem, I modeled each side of the square of length L as an individual wire such that each wire contributes a B field at the center of the square. Since each side is of length L. I calculated (using a right triangle) that 2L^2= D^2 where D^2 equals the length from corner to corner. So

Solving for D, D = (2L^2)^(1/2) = L(2)^(1/2). Since I want to locate the center I then divided this by 2.

So I figured that the center is located at [L sqrt(2)]/2

I simply then plugged this into the equation above for r. And then multiplied by 4 since there are four wires contributing a magnetic field B at that location.

My final answer: 4(μ/π)(I/(L sqrt(2)) but the answer is actually [(μ)(I)/(2πL)](sqrt(2))

In the explanation they give then end up doing some really complicated math and integrating the general Biot Savart Law to get their answer. I am confused as to why they would integrate. Are they integrating because each section of each of the wires (ΔL) is contributing a magnetic field at B? Why can't I simply use the equation that I used? I feel like a lot of the times I struggle with these problems because I do not know when to use the formula and when to integrate. This is an example of this. Please help me!

Thank you

 

[CAF123]

The point P is at a vertical distance L/2 from any of the wires. This is certainly comparable to the length of each of the wires, so the equation you posted cannot be used. You will have to use the general form of Biot-Savart, which indeed involves doing an integral.
http://en.m.wikipedia.org/wiki/Biot–Savart_law

 

[ikentrovik]

By this you mean that the point P is actually only a distance of L/2 from the centers of any of the wires right? Do we assume that we are measuring the contribution of each of the wires from the center of the wires?

 

[CAF123]

By this you mean that the point P is actually only a distance of L/2 from the centers of any of the wires right?

Yes

Do we assume that we are measuring the contribution of each of the wires from the center of the wires?

The question only wants you to consider the B field contribution from one of the wires. So, the first thing to do would be to define a coordinate system suitable for your chosen segment and express all the ingredients of the Biot-Savart equation relative to it. Note that Biot-Savart is a line integral, so you will be integrating over the segment that you choose.

Choosing the lower most horizontal segment to analyse, for example, a convenient coordinate system would be one with its origin in the middle of the segment, so that the ends of the segment are at a distance +L/2 and -L/2 from the origin.