Magnetic field of a circular loop of wire

In summary, the problem involves finding the magnetic field at a distance a from the center of a circular loop of wire carrying current, where a is less than the radius of the loop, R. The formula for this is given by the Bio Savart law, which involves integrating the small current element, ds, which is always tangent to the loop. The angle between ds and the unit vector, r, does not change and can be expressed as ds x r = ds. The distance, r, from the loop to a can be determined using known variables, and the magnitude of the magnetic field varies when rotating over the angle, ϕ. Hence, the calculation of the magnetic field at a distance a from the center of the loop involves considering
  • #1
tjkubo
42
0

Homework Statement


I know how to find the magnetic field at the center of a circular loop of wire carrying current. If the radius of the loop is R, how do you find the magnetic field at a distance a from the center of the loop where a<R?

Homework Equations


[tex]
d{\mathbf{B}} = \frac{{\mu _0 }}{{4\pi }}\frac{{Id{\mathbf{s}} \times {\mathbf{\hat r}}}}{{r^2 }}
[/tex]

The Attempt at a Solution


The small current element ds is always tangent to the loop. r varies from R-a to R+a. The angle θ between ds and [tex]\mathbf{\hat r}[/tex] seems to vary from 90° to some maximum angle that depends on a.
Also, if you define ϕ to be the angle around P from the place you first start to integrate to ds, then [tex]ds\neq rd\phi[/tex].

This is as far as I can analyze. I have no idea what to do with the angles. I am guessing there is some kind of relationship between r and θ or between r and ϕ or between θ and ϕ that I can't see.
 
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  • #2
First thing you want to do is draw a picture and realize this is a highly symmetric problem. From symmetry you can determine the direction of [itex]\boldsymbol{B}[/itex], draw this vector. Then draw the x and y components. Due to symmetry can you tell what the magnitude of [itex]B_y[/itex] will be? Can you express [itex]B_x[/itex] in terms of B?

Why do you think the angle between [itex]d\boldsymbol{s}[/itex] and [itex]\mathbf{\hat r}[/itex] changes? It does not. So [itex]d\boldsymbol{s} \times \mathbf{\hat r}=ds[/itex].

How does the distance r from the loop to a depend on known variables and does the magnitude of B vary when you rotate over the angle [itex]\phi[/itex]?

Try to enter all information into the Bio Savart law now.
 
  • #3


Dear student,

Thank you for your question. I would like to provide you with a comprehensive response to help you understand the magnetic field of a circular loop of wire.

First, let's start with the basic equation for the magnetic field at the center of a circular loop of wire carrying current:

B = μ0I/2R

Where B is the magnetic field, μ0 is the permeability of free space, I is the current, and R is the radius of the loop.

Now, to find the magnetic field at a distance a from the center of the loop, we need to consider the magnetic field contribution from each small current element ds along the loop. This can be calculated using the Biot-Savart law:

d{\mathbf{B}} = \frac{{\mu _0 }}{{4\pi }}\frac{{Id{\mathbf{s}} \times {\mathbf{\hat r}}}}{{r^2 }}

Where μ0 is the permeability of free space, I is the current, ds is the small current element, r is the distance between the current element and the point where we want to find the magnetic field, and \mathbf{\hat r} is the unit vector pointing from the current element to the point where we want to find the magnetic field.

Now, let's consider the angle θ between ds and \mathbf{\hat r}. This angle varies from 90° to some maximum angle that depends on a, as you correctly stated. This maximum angle can be calculated using the Pythagorean theorem:

sinθ = a/r

Where a is the distance from the center of the loop and r is the distance between the current element and the point where we want to find the magnetic field.

Next, let's consider the angle ϕ. This angle is defined as the angle around the point P from the place where we first start to integrate to ds. This angle is related to the distance r as follows:

r = Rcosϕ

Where R is the radius of the loop and ϕ is the angle around the point P.

Now, we can combine these equations to find the magnetic field at a distance a from the center of the loop:

B = \frac{{\mu _0 I}}{{4\pi }}\int_{0}^{2\pi}\frac{Rcos\phi}{(R^2+a^2-2aRcos\phi)^
 

What is a magnetic field?

A magnetic field is a region in space where a magnetic force can be detected. It is created by the movement of electrically charged particles, such as electrons, and is represented by lines of magnetic flux.

How is a magnetic field produced in a circular loop of wire?

A magnetic field is produced in a circular loop of wire when an electric current flows through the wire. The direction of the magnetic field is perpendicular to the plane of the loop and follows the right-hand rule, with the direction of the field determined by the direction of the current.

What factors affect the strength of the magnetic field in a circular loop of wire?

The strength of the magnetic field in a circular loop of wire is affected by the amount of current flowing through the wire, the number of turns in the loop, and the radius of the loop. Increasing any of these factors will result in a stronger magnetic field.

How does the magnetic field change when the current in the loop is reversed?

When the current in the loop is reversed, the direction of the magnetic field also reverses. This means that the north and south poles of the magnetic field will switch places. However, the strength of the magnetic field will remain the same.

What is the application of a circular loop of wire in magnetic fields?

A circular loop of wire is commonly used in devices such as electromagnets, electric motors, and generators. It is also used in magnetic resonance imaging (MRI) machines, where the magnetic field is used to create detailed images of the body.

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