Relationship between magnetic field lines and magnetic field

[a1234]

Homework Statement

Show that the density of magnetic field lines can be used as a measure of the size of the magnetic field.

Relevant Equations

psi = BAcos(theta)

As stated in the problem, I want to demonstrate mathematically that field line density is directly related to the magnitude of B. How would I be able to do this, other than simply using the flux equation and showing that for a higher flux in the same area, the magnetic field must be rise accordingly. Or would this be sufficient?

 

[Charles Link]

The density of flux lines in 3 dimensions is proportional to the magnetic field strength. Drawings that are two dimensional don't show the inverse square law properly that occurs from a magnetic pole, etc. I would need to think about exactly why a 3 dimensional model gives exactly what is needed for the conservation of flux, etc, but in any case it does.

Edit: I think it probably can be demonstrated by saying that the flux lines into any box are the same as the flux lines out of the box=if they go in, they must emerge, and by Gauss' law, they then obey ## \nabla \cdot B=0 ##, just like the magnetic field, but maybe someone else can comment on this.

 

[a1234]

Edit: I think it probably can be demonstrated by saying that the flux lines into any box are the same as the flux lines out of the box=if they go in, they must emerge, and by Gauss' law, they then obey ## \nabla \cdot B=0 ##, just like the magnetic field, but maybe someone else can comment on this.

How can we show that the individual field lines obey Gauss' law? How would this be different from showing that Gauss' law is satisfied for the field?

 

[vela]

As stated in the problem, I want to demonstrate mathematically that field line density is directly related to the magnitude of B. How would I be able to do this, other than simply using the flux equation and showing that for a higher flux in the same area, the magnetic field must be rise accordingly. Or would this be sufficient?

It may help to start with the definition of a vector field line. If parametric curve ##\vec x(s)## is a field line, it obeys
$$\vec x'(s) = \vec B(\vec x).$$ Perhaps someone else can explain how to define field line density mathematically.

 

[tech99]

I learnt physics using the centimetre-gram-second (cgs) system of units, where Gauss specified that 4 pi lines of force originate from a unit magnetic pole, and the magnetic flux density is expressed in lines per cm2. So I find it hard to put my heart into the question!