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

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1. Aug 13, 2015

Yashbhatt

1. The problem statement, all variables and given/known data
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.
2. Relevant equations
$$B = \frac{\mu_{0}I[sin\theta_{1}-sin\theta_{2}]}{4\pi(r)}$$

3. 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.

2. Aug 13, 2015

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.

3. Aug 14, 2015

Yashbhatt

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.

4. Aug 15, 2015

Qwertywerty

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

5. Aug 15, 2015

Yashbhatt

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.

6. Aug 15, 2015

Qwertywerty

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 .

7. Aug 16, 2015

rude man

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.

8. Aug 16, 2015

blue_leaf77

9. Aug 16, 2015

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.

10. Aug 17, 2015

Yashbhatt

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

11. Aug 17, 2015

rude man

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

12. Aug 18, 2015

Yashbhatt

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

13. Aug 18, 2015

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.

Last edited: Aug 18, 2015
14. Aug 18, 2015

Yashbhatt

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

15. Aug 18, 2015

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.

16. Aug 24, 2015

Yashbhatt

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

17. Aug 24, 2015

blue_leaf77

Look at the sentence below equation (9.8.8), and compare it to what rude man said about the elliptic integral.
Yes, obviously this is helpful as you can directly use (9.8.7) and (9.8.8) along with a computer.