Field of a Trianglular Solenoid

[Taulant Sholla]

Homework Statement

A coil of wire is formed into the shape of an isosceles triangle, as depicted in the diagram. Which of the following best describes the magnetic field inside the triangle?

Relevant Equations

b=uni

No calculators or equations are needed for this question. The correct answer is supposedly "The field is strongest at point Y" and I have no idea why.

I even coughed-up the following, but still can't see how this is the right answer.

 

[Taulant Sholla]

I'm guessing this might be an incorrect answer by the lack of responses? I certainly can't see a way at this choice is correct. Ideas?

 

[kuruman]

You know that for an infinite solenoid of radius $R$ the field is zero at distance $r>R$ from the axis. No solenoid is infinite, so at $r>R$ the magnetic field is approximately zero. The approximation becomes better and the field is closer to zero the closer you come to the solenoid. This is shown graphically in the pictures you scared up. Point Y is farthest from all solenoids therefore you are supposed to conclude that the field is strongest there, when you add the fields as vectors, where the infinite solenoid approximation is worst.

Having said that, let's see what Maxwell's equations have to say about this. One of these equations says that magnetic field lines form closed loops. Another one, a.k.a. Ampere's law says that if you take a line integral around a closed loop, the result is proportional to the current enclosed by the loop. So let's draw an arrow representing the magnetic field at point Y. It's part of a closed loop. Let's draw it as a closed oriented magnetic field line loop entirely inside the triangle. The integral $\oint \vec B \cdot d\vec l$ cannot be zero because the magnetic field is always along the contour, yet there is no current cutting through the plane of the loop. This is a contradiction. My initial thought when I saw this was that the field inside the triangle is zero.

I'm guessing this might be an incorrect answer by the lack of responses? I certainly can't see a way at this choice is correct. Ideas?

Please be more patient. It's Sunday after all.

 

[haruspex]

My initial thought when I saw this was that the field inside the triangle is zero.

Certainly, if we take the special case of an equilateral triangle and set Y at the centre then the field there is zero by symmetry.
So if the field is minimum at Y then it is also true that the field is zero everywhere in the triangle.

 

[Taulant Sholla]

You know that for an infinite solenoid of radius $R$ the field is zero at distance $r>R$ from the axis. No solenoid is infinite, so at $r>R$ the magnetic field is approximately zero. The approximation becomes better and the field is closer to zero the closer you come to the solenoid. This is shown graphically in the pictures you scared up. Point Y is farthest from all solenoids therefore you are supposed to conclude that the field is strongest there, when you add the fields as vectors, where the infinite solenoid approximation is worst.

Having said that, let's see what Maxwell's equations have to say about this. One of these equations says that magnetic field lines form closed loops. Another one, a.k.a. Ampere's law says that if you take a line integral around a closed loop, the result is proportional to the current enclosed by the loop. So let's draw an arrow representing the magnetic field at point Y. It's part of a closed loop. Let's draw it as a closed oriented magnetic field line loop entirely inside the triangle. The integral $\oint \vec B \cdot d\vec l$ cannot be zero because the magnetic field is always along the contour, yet there is no current cutting through the plane of the loop. This is a contradiction. My initial thought when I saw this was that the field inside the triangle is zero.Please be more patient. It's Sunday after all.

Thank you - very thought-provoking!