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

In summary, the magnetic field at a point in the area enclosed by a current carrying loop cannot be calculated using a closed form expression due to the complexity and variability of the variables involved, such as the distance from the center and the radius of the wire. While Ampere's law may be useful for certain symmetrical problems, it does not apply to a circular current loop. Additionally, the magnetic field near the loop (wire) is very large, and as the wire radius decreases, the magnetic field approaches infinity. Therefore, it is not possible to find a uniform magnetic field at every point within the loop's circumference.
  • #1
Yashbhatt
348
13

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.
magfield.png
 
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  • #2
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
blue_leaf77 said:
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.
 
  • #4
Yashbhatt said:
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 .
 
  • #5
Qwertywerty said:
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.
 
  • #6
Yashbhatt said:
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 .
 
  • #7
Qwertywerty said:
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.
 
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  • #8
Yashbhatt said:
There must be some way
If your problem is for practical purpose, use your computer.
 
  • #9
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
rude man said:
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.
 
  • #11
Yashbhatt said:
No. I don't want inductance. Just the magnetic field.
I know. But, you can't have it! :frown:
 
  • #12
rude man said:
I know. But, you can't have it! :frown:

Is there any chance that my intuition about it being constant is correct?
 
  • #13
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:
  • #14
blue_leaf77 said:
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.
 
  • #15
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.
 
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  • #16
rude man said:
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?
 
  • #17
Look at the sentence below equation (9.8.8), and compare it to what rude man said about the elliptic integral.
Yashbhatt said:
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.
 

What is a magnetic field?

A magnetic field is a region in space where a magnetic force can be detected and measured. It is created by moving electric charges and is characterized by its direction and strength.

How is a magnetic field created by a loop?

A magnetic field is created by a loop when an electric current flows through the loop. The electric current produces a magnetic field that loops around the wire, creating a circular pattern.

What factors affect the strength of the magnetic field produced by a loop?

The strength of the magnetic field produced by a loop is affected by several factors, including the number of turns in the loop, the magnitude of the electric current flowing through the loop, and the distance from the loop.

How is the direction of the magnetic field determined for a loop?

The direction of the magnetic field produced by a loop is determined by the right-hand rule. If you point your thumb in the direction of the electric current, your fingers will curl in the direction of the magnetic field.

Can the magnetic field produced by a loop be manipulated?

Yes, the magnetic field produced by a loop can be manipulated by changing the number of turns in the loop, the magnitude and direction of the electric current, and the distance from the loop. This is why magnetic fields are used in various technologies, such as electromagnets and motors.

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