Relationship between magnetic field lines and magnetic field

a1234
Messages
78
Reaction score
6
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?
 
Physics news on Phys.org
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.
 
Last edited:
Charles Link said:
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?
 
a1234 said:
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.
 
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!
 
Thread 'Need help understanding this figure on energy levels'
This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
Thread 'Understanding how to "tack on" the time wiggle factor'
The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
Back
Top