Finding the magnitude of a magnetic field from a square loop

[Patdon10]

Homework Statement

A square loop, with sides of length L, carries current i. Find the magnitude of the magnetic field from the loop at the center of the loop, as a function of i and L. (Use any variable or symbol stated above along with the following as necessary: μ0.)

Homework Equations

magnetic field outside of a conductor:
u_0*I*r/A

The Attempt at a Solution

I got u_0*I*L/L^2

Not really sure what I should be doing differently? If it was in the shape of a circle it'd be easy, but because it's in a square, it's harder.

 

[epsilonjon]

I think the easiest way would be to try to solve the problem of finding the magnetic field produced by a straight current-carrying piece of wire of length L. Try to find the field at a point a distance x from the wire on its perpendicular bisector. You can do this by splitting the wire into infinitessimal lengths dl and then use the Biot and Savart law to calculate the field produced by dl. Then integrate along the length of the wire to find the total.

I think you should get

$$B = \frac{\mu_{0}I}{4\pi}\frac{L}{x\sqrt{x^{2}+(L/2)^{2} }}.$$
Now you've done the hard part it's just a matter of adding the fields from each of the 4 sides of the loop.

 

[Patdon10]

It seems pretty confusing, but I'll try it out and see what happens.

 

[epsilonjon]

It seems pretty confusing, but I'll try it out and see what happens.

I think that is the easiest way to do it. If you are trying to find the field from shapes like this then presumably you've covered the field from a straight current-carrying wire?

If it was in the shape of a circle it'd be easy, but because it's in a square, it's harder.

How would you do it for a circle?

 

[rwisz]

I would like to provide a more detailed and comprehensive response to this content. Firstly, it is important to note that the magnetic field from a square loop can be calculated using the Biot-Savart law, which states that the magnetic field at a point is proportional to the current and the distance from the current-carrying element. In this case, the square loop can be divided into smaller current-carrying elements, and the magnetic field at the center can be calculated by integrating the contributions from each element.

Now, in order to find the magnitude of the magnetic field at the center of the square loop, we need to consider the direction of the magnetic field at each point. Since the square loop is symmetrical, we can assume that the magnetic field at the center will be in the same direction as the magnetic field at any point on the loop. Therefore, we can use the equation for the magnetic field outside of a conductor, which you have correctly stated as B = μ0*I*r/A, where μ0 is the permeability of free space, I is the current, r is the distance from the current-carrying element, and A is the area of the loop.

However, in this case, the distance r is not constant, as it varies depending on the position of the current-carrying element on the loop. Therefore, we need to express r in terms of the variables given in the problem. The distance from the center of a square to any of its sides is L/2, therefore, the distance from the center of the loop to any of its sides will be L/2 as well. Since the magnetic field is perpendicular to the current and the side of the loop, the distance r will be equal to L/2.

Substituting this value for r in the equation for the magnetic field, we get B = μ0*I*(L/2)/A. Now, we need to find the area A of the loop, which can be calculated by multiplying the length of one side (L) by the length of the other side (also L), giving us A = L^2. Substituting this value for A, we finally get the magnitude of the magnetic field at the center of the loop as B = μ0*I*L/L^2.

In summary, to find the magnitude of the magnetic field at the center of a square loop, we first need to express the distance r in terms of the