Expression of a magnetic field

[physica]

Attached is the question and the result I have got so far. I have used cylindrical coordinates to get so far (my equation is clearly marked in the attachment). However, I am having trouble converting it to Cartesian coordinates. r would give Pythagoras's theorem which is not like the function I'm supposed to prove. I would be so grateful for some pointers. For starters is my function even remotely close?

 

Many thanks

 

 

[elfmotat]

The notation in that equation is a bit funny. Did you mean: [math]B_{\phi} \left( r \right ) = \frac{\mu_{0}}{2} J_z \, r[/math] ? Because if so that looks correct.

 

Now the question is, how can you relate B in cylindrical coordinates to B in Cartesian coordinates? Plugging in [math]r=\sqrt{x^2+y^2}[/math] will give you [math]B_{\phi}[/math] in Cartesian coordinate, but that's not what you're looking for. You're looking for [math]B_x[/math] and [math]B_y[/math].

Edited November 8, 2014 by elfmotat

[physica]

Thank you for the tip. Yes you guessed my function correctly sorry about the notation. From what I gather from your point you define the x vector and the y vector separately and then add them? I've now done this but what I don't get is that there is a y component on the x vector and a x component on the y vector. I also don't get how the x vector is negative. It looks like a result from a cross product with the z vector. I understand that the function I'm trying to prove makes sense as it imitates the right hand rule which is expected. Still struggling on how to get there mathematically. Thank you for your patience.

[elfmotat]

Say you're given the position vector r=(r,φ,z) and you want to find the components of the vector in Cartesian coordinates, (x,y,z). Try working that out first. Now by analogy, how would you find the components of some general vector A in Cartesian coordinates, given the same vector in cylindrical coordinates?

[physica]

I have done what you have suggested. I this has resulted in the vector being: (J_z)(x)e_x(y)e_y. However, the equation given in the question is (J_z)(-y)e_x(x)e_y. This seems to be a 90 degree rotation to the left. Is this right or am I being stupid? If it is the 90 degree rotation to the left is this valid because of symmetry? Or is this valid because the electric field is rotating round the wire? Thank you for your patience and sorry that I'm not grasping it quickly.

[elfmotat]

I have done what you have suggested. I this has resulted in the vector being: (J_z)(x)e_x(y)e_y. However, the equation given in the question is (J_z)(-y)e_x(x)e_y. This seems to be a 90 degree rotation to the left. Is this right or am I being stupid? If it is the 90 degree rotation to the left is this valid because of symmetry? Or is this valid because the electric field is rotating round the wire? Thank you for your patience and sorry that I'm not grasping it quickly.

 

Don't forget that φ increases in the counter-clockwise direction. That means e_φ is oriented in the counter-clockwise direction. An easier way to see it might be to take points where we do know B. For example, on the positive x-axis (where y=0) we have x=r, we know the magnitude of B is B_φ (because it's the only nonzero component), and B is oriented in the +y direction. In other words, we have:

 

[math]\mathbf{B} (y=0) = \frac{\mu_0}{2} J_z x \, \mathbf{e}_y[/math]

 

You can do the same thing on the positive y-axis as well, which is where the minus sign should come from. The should be easy to see, because B will be oriented in the -x direction.

Edited November 10, 2014 by elfmotat

[physica]

Thank you so much. This now makes sense and I have now solved the problem I was looking at