Cartesian equation for the Magnetic field resulting from a single current loop?

[kilianod5150]

Hello

I am carrying out some analysis on the magnetic field generated over a 3D region by a single current loop. The present form of the equations is in cylindrical coordinates and is as follows
$$\vec{B}$$={Brc,0,Bz}
There is no angular component in this present from.
Note: The following website contains the formulas in question:
http://www.netdenizen.com/emagnet/offaxis/iloopoffaxis.htm

My question is as follows. How could one convert a complex cylindrical equation such as this to Cartesian coordinates? The main aim of this is to plot the fields in Matlab, if the equations were in cartesian form it would simply greatly my analysis.

The main problem I seem to encounter is that since there is no angle component, using conversions such as x=r*cos(theta) and y=r*sin(theta) do n0t seem to make sense as it would imply that there is only an x component and no y component.

I tried using Mathematica to convert the equations using the ConvertToCartesian command to no avail.

Any help with this problem would be greatly appreciated.

Regards

Kilian

 

[Meir Achuz]

Just use $$r=\sqrt{x^2 + y^2}$$.

 

[kilianod5150]

Thanks for the reply, I had thought of using that but i still have the problem where there is no y component of the magnetic field.
In other words converting B(rc,phi,z)=(Brc(rc,theta,z),0,Bz(rc,theta,z)) would result in B(x,y,z)=(Bx(x,y,z),0,Bz(x,y,z)). The By(x,y,z) component is zero, is this correct? Or is there a way to expand the rc unit vector into x and y parts?

 

[dgOnPhys]

The conversion between cylindrical and Cartesian components goes like

[URL]http://www.engin.brown.edu/courses/en3/Notes/Vector_Web2/Vectors6a/Vectors6a_files/eq0080MP.gif[/URL]

This is from
http://www.engin.brown.edu/courses/en3/Notes/Vector_Web2/Vectors6a/Vectors6a.htm"

 

[sardar]

Dear Kilian,

Thank you for your question. Converting equations from cylindrical coordinates to Cartesian coordinates can be tricky, especially when there is no angular component involved. However, it is not impossible.

One approach you can take is to use the conversion equations for cylindrical to Cartesian coordinates, which are:

x = r cosθ
y = r sinθ
z = z

In your case, since there is no angular component, we can assume that θ = 0. This means that the equations would simplify to:

x = r
y = 0
z = z

Now, for the magnetic field equation, we can substitute the values for x and y into the cylindrical equation to get:

\vec{B}={Brc,0,Bz} = {Bx,By,Bz}

This means that the magnetic field in Cartesian coordinates would be:

\vec{B}={Bx,0,Bz}

As you can see, the y component is now equal to 0, which makes sense since there is no angular component in the original equation.

I hope this helps with your analysis and plotting in Matlab. If you encounter any further difficulties, I would suggest consulting with a mathematician or physicist who specializes in electromagnetism for further assistance.

Best of luck with your research.

Sincerely,