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.

 

[paranormal]

for reaching out for clarification on this problem. The approach you took is a valid one and it does yield the correct result. However, the explanation you attached is using a more general and rigorous method to solve the problem. Let's break it down step by step.

Firstly, the Biot-Savart Law is a general formula for calculating the magnetic field at a point due to a current-carrying wire. It is given by:

B = (μ0/4π) * (I * dl x r)/r^3

Where μ0 is the permeability of free space, I is the current, dl is the infinitesimal length element of the wire, r is the distance from the wire, and x represents the cross product.

In the case of a long straight wire, the magnetic field at a distance r from the wire is given by:

B = (μ0/2π) * (I/r)

This is the equation you used in your approach, and it is valid. However, in this problem, we are dealing with four wires, each carrying a current I, and they are arranged in a square. This means that the magnetic field at the center of the square is the sum of the magnetic fields due to each individual wire.

Now, let's look at the explanation provided. The first step they took was to divide the square into four smaller squares, each with side length L/2. This is done to simplify the integration process, as the magnetic field due to each smaller square can be easily calculated using the Biot-Savart Law.

Next, they integrated over each of the smaller squares, using the Biot-Savart Law, to find the magnetic field at the center of each smaller square. This is where the integration comes in. Each infinitesimal length element of the wire (dl) is contributing to the magnetic field at the center of the smaller square, and the integration takes into account all of these contributions.

Once they have the magnetic field at the center of each smaller square, they add them all together to get the total magnetic field at the center of the larger square. This is the same result you obtained in your approach.

In summary, you can use the simpler equation for a long straight wire to solve this problem, as you did. However, the explanation provided is using a more general and rigorous method to solve the problem, which involves integrating over each smaller square. Both approaches yield the correct result, so you can