Calculating constant field lines

[illuzioner]

Hi,

I a question related to an equation for a magnetic field vector,
B, from a dipole in cylindrical coordinates. This is the equation I found for the B field (thought I'm not 100% certain this is correct).

B = (u0*m/4*Pi*r^2)* {2 Cos[theta] , Sin[theta]}

where the first index is the r component and the second is the theta
component.

I am attempting to recreate the field lines as shown on page 3 in:
http://www.intalek.com/Index/Projects/Research/MagneticForcesandTorq.pdf

so I need to find r as a function of theta.

I can calculate the magnitude of the field vector at any point. However, if I take that magnitude, set it equal to a constant and solve for r as a function of theta, I'm not getting the shape of the curve I see in the above pdf file.

How can I do this? Is there any simple source code anyone knows of that can do this?

Thanks!

 

[nasu]

I am afraid you started the wrong way. The field lines are not lines of equal values of B.
You may be confusing it with equipotential lines.

 

[illuzioner]

I am afraid you started the wrong way. The field lines are not lines of equal values of B.
You may be confusing it with equipotential lines.

Okay, I accept that. It's been quite a while since I've worked with this and I'm rusty. Can you direct me to how I find the field lines as a function of theta for the purposes of plotting the lines?

 

[nasu]

The field lines are tangent to the direction of the field in each point.
The direction of the field at a given point can be calculated from
tan(alpha)=By/Bx
Where alpha is the angle between the vector B and the x axis.
You can start by finding alpha as a function of position, alpha(r,theta) or alpha(x,y).
This will gave you the orientation of the field an any point. And you will know the magnitude at any point.

Plotting a picture as the ones in your link is a little more difficult. You have somehow to decide how dense you want them or how many of them.

Note. Now I see that you call your post "calculating constant field lines". Is this what you want to calculate?
The field lines shown in your link are not lines of constant field lines but common filed lines.
It is possible to calculate lines along which the magnitude of B is constant, as you suggested in your first post.
Now I am confused. Which one you have in mind?

 

[illuzioner]

Actually I made the assessment that the lines plotted were those of constant B, but the diagram is what I am after. I'm not sure how else it would work. I"m not sure what you mean by common field vs. constant field.

If the field strength is not constant, then what inside the formula is? Something must be held constant to plot a single line. Theta isn't, r isn't, and alpha isn't. What relationship does one point in the line have to the next point?

With a constant field I can calculate r and alpha given theta and |B|. If none of these is constant, how do I know what the next point is? I could draw any arbitrary curve since there are no constraints, but there is that definite shape I am trying to mimic.

Do you follow my reasoning?

 

[nasu]

Actually I made the assessment that the lines plotted were those of constant B, but the diagram is what I am after. I'm not sure how else it would work. I"m not sure what you mean by common field vs. constant field.

If the field strength is not constant, then what inside the formula is? Something must be held constant to plot a single line. Theta isn't, r isn't, and alpha isn't. What relationship does one point in the line have to the next point?

With a constant field I can calculate r and alpha given theta and |B|. If none of these is constant, how do I know what the next point is? I could draw any arbitrary curve since there are no constraints, but there is that definite shape I am trying to mimic.

Do you follow my reasoning?

When I said "common" field lines I meant the common meaning: lines tangent to the direction of the field. This is what I refereed to in my first post and those are shown in your drawing. They are called field lines.
You could, in principle, calculate the lines (or surfaces) of equal magnitude of the field. These are not technically "field lines" and they will not produce the images that you would like to reproduce.

I agree, it's easier to calculate a locus (line, surface, etc) of equal value of the magnitude.

For actual field lines, I could suggest to approximate them numerically by a chain of short segments. You could start with a given point, calculate the direction of the B vector at that point and draw a small (let say of size "epsilon") line segment along that direction. Then move your current point to the other end of the segment and calculate the direction of B at the new point and so on. You will get a field line made from short segments. If you decrease the size of "epsilon" the line will be smoother.