Magnetic field due to a loop within the area of the loop

[Yashbhatt]

Homework Statement

This is a question me and my friend were wondering about. How can one calculate the magnetic field due to a current carrying loop at a point in the area enclosed by the loop.

For example, at point P as shown in the attached figure.

Homework Equations

$$B = \frac{\mu_{0}I[sin\theta_{1}-sin\theta_{2}]}{4\pi(r)}$$

The Attempt at a Solution

I feel there are just too many variables to arrive at a solution. Even if we consider it for a point at a fixed distance $x from the center, then too the distance to the loop changes as we integrate. However, I have a feeling that analogous to the shell theorem for gravitation/electrostatics, the field should be the same at every point within the area but I can't prove it.

 

[blue_leaf77]

Use Biot-Savart law. For a general point within the loop's circumference it's indeed difficult to find a closed form expression for the magnetic field.

 

[Yashbhatt]

Use Biot-Savart law. For a general point within the loop's circumference it's indeed difficult to find a closed form expression for the magnetic field.

Yeah. I know I have to use Biot-Savart's Law but I am not able to formulate the integral. Seems more like a question of Mathematics rather than Physics.

 

[Qwertywerty]

Yeah. I know I have to use Biot-Savart's Law but I am not able to formulate the integral. Seems more like a question of Mathematics rather than Physics.

I may be wrong , but I don't think you can find the magnetic field at any general point of a circular loop .

 

[Yashbhatt]

I may be wrong , but I don't think you can find the magnetic field at any general point of a circular loop .

What do you mean by we can't find? There must be some way. Also, I am not talking about just any point. I am talking about a point at a certain fixed distance from the center.

 

[Qwertywerty]

What do you mean by we can't find? There must be some way. Also, I am not talking about just any point. I am talking about a point at a certain fixed distance from the center.

I said there probably might not be a solution , and that I might be wrong . And yes , a general point is referring to the same point you speak of .

 

[rude man]

I said there probably might not be a solution , and that I might be wrong . And yes , a general point is referring to the same point you speak of .

There is no closed-form solution. B can, aside from the center point, be only approximated. The radius of the wire is also of import. As the wire radius decreases the B field approaches infinity by the wire.

 

[blue_leaf77]

There must be some way

If your problem is for practical purpose, use your computer.

 

[rude man]

IMO the only practical reason for computing B everywhere inside a loop is to find the inductance, since L = φ/i and φ = ∫B⋅dA. If you want inductance there are inductance calculators on the Web. Those formulas too are approximations and they too involve the wire radius in addition to the loop area.

 

[Yashbhatt]

IMO the only practical reason for computing B everywhere inside a loop is to find the inductance, since L = φ/i and φ = ∫B⋅dA. If you want inductance there are inductance calculators on the Web. Those formulas too are approximations and they too involve the wire radius in addition to the loop area.

No. I don't want inductance. Just the magnetic field.

 

[rude man]

No. I don't want inductance. Just the magnetic field.

I know. But, you can't have it!

 

[Yashbhatt]

I know. But, you can't have it!

Is there any chance that my intuition about it being constant is correct?

 

[blue_leaf77]

By constant, do you mean constant along the circle of constant radius from the center? If yes, then no, magnetic field doesn't behave exactly like electric field. If your problem has certain symmtery which allows the use of Ampere's law, you can indeed benefit from this symmtery type. Unfortunately, the circular current loop does not satisfy this symmetry required in the application of Ampere's law.

EDIT: in the first sentence it should have been "radius from the center from the wire", i.e. not the center of the loop. The magnetic field should of course be constant along circle of constant radius from the loop center, but it's just almost impossible to find a closed form expression.

 

[Yashbhatt]

By constant, do you mean constant along the circle of constant radius from the center? If yes, then no, magnetic field doesn't behave exactly like electric field. If your problem has certain symmtery which allows the use of Ampere's law, you can indeed benefit from this symmtery type. Unfortunately, the circular current loop does not satisfy this symmetry required in the application of Ampere's law.

EDIT: in the first sentence it should have been "radius from the center from the wire", i.e. not the center of the loop. The magnetic field should of course be constant along circle of constant radius from the loop center, but it's just almost impossible to find a closed form expression.

By constant I mean constant everywhere at every point within the circumference of the loop.

 

[rude man]

As I said, the mag field near the loop (wire) is very large, as you can see if you do Biot-Savart on a short arc of the loop very close to that arc so that most of the mag field is due to just the arc and not the rest of the loop. As the wire (not the loop) radius approaches zero the mag field approaches infinity! So no, there is no way you can rationalize that the B field is uniform within the loop area.
I know you're looking for B and not inductance L, but the two are closely related: L = Φ/i and Φ = ∫B(A) dA with i = current. I've shown that L is not exactly computable even with very advanced calculus (elliptical integrals) so it follows immediately that B(A) is in the same boat.

 

[Yashbhatt]

As I said, the mag field near the loop (wire) is very large, as you can see if you do Biot-Savart on a short arc of the loop very close to that arc so that most of the mag field is due to just the arc and not the rest of the loop. As the wire (not the loop) radius approaches zero the mag field approaches infinity! So no, there is no way you can rationalize that the B field is uniform within the loop area.
I know you're looking for B and not inductance L, but the two are closely related: L = Φ/i and Φ = ∫B(A) dA with i = current. I've shown that L is not exactly computable even with very advanced calculus (elliptical integrals) so it follows immediately that B(A) is in the same boat.

Have a look at this article : http://ocw.mit.edu/courses/physics/...ng-2007/class-activities/ch9sourc_b_field.pdfhttp://ocw.mit.edu/courses/physics/...ng-2007/class-activities/ch9sourc_b_field.pdf

Go to appendix 9.8(Magnetic Field off axis). Is this somehow helpful?

 

[blue_leaf77]

Look at the sentence below equation (9.8.8), and compare it to what rude man said about the elliptic integral.

Is this somehow helpful?

Yes, obviously this is helpful as you can directly use (9.8.7) and (9.8.8) along with a computer.