How to prove the radius of curvature at any point on a line?

[catheee]

In a magnetic field, field lines are curves to which the magnetic induction B is everywhere tangetial. By evaluating dB/ds where s is the distance measured along a field line, prove that the radius of curvature at any point on a line is given by

density symbol-->p= B^3 / [ B x( B * del) B]

where do i start with this?? I have absolutely no idea. please help with start with a direction to go.

 

[gabbagabbahey]

Hi catheee, welcome to PF!

This forum supports $\LaTeX$, which allows you to write equations in a clear manner...

density symbol-->p= B^3 / [ B x( B * del) B]

$$\rho=\frac{B^3}{\textbf{B}\times(\textbf{B}\cdot\mathbf{\nabla})\textbf{B}}$$

^^^Is this what you mean?

where do i start with this??

A good place to start might be with the (mathematical) definition of radius of curvature for a general curve...what equation(s) do you have for that?

 

[catheee]

yes! that's the equation I have, but its the only equation I was given and I was told to solve it, I don't see anything about the radius of curvature equation in the textbook I have. So, what you're saying is that I should get the radius of curvation equation and then use it to solve the problem?

btw, thanks for answering my question!

 

[gabbagabbahey]

yes! that's the equation I have,

Really?!

$\textbf{B}\times(\textbf{B}\cdot\mathbf{\nabla})\textbf{B}$ is a vector whereas $\rho$ and $B^3$ are both scalars...how exactly does one divide a scalar by a vector to produce another scalar?!

but its the only equation I was given and I was told to solve it, I don't see anything about the radius of curvature equation in the textbook I have.

Out of curiosity, what textbook is this problem from? It might be easier to see what they expect you to do if I scan through the text quickly.

So, what you're saying is that I should get the radius of curvation equation and then use it to solve the problem?

btw, thanks for answering my question!

Of course! How on Earth would you find what the radius of curvature for a certain curve is without knowing the definition of 'radius of curvature'? Surely, basic vector calculus is a prerequisite for studying whatever course this text is for?

 

[catheee]

im using introduction to electrodynamics by griffiths 3rd edition.
oh and sorry about double posting i didnt know that that was against the rules.
:)
im looking up the def of radius of curvature but it doesn't look like any of the equations would apply to this problem. Ughh

 

[gabbagabbahey]

im using introduction to electrodynamics by griffiths 3rd edition.

I am familiar with that text, but it won't be of much help for this problem.

im looking up the def of radius of curvature but it doesn't look like any of the equations would apply to this problem. Ughh

Okay, if I gave you the equation of some parameterized curve $\textbf{r}(u)=x(u)\hat{x}+y(u)\hat{y}+z(u)\hat{z}$, could you calculate the radius of curvature? Could you calculate the (unnormalized) tangent vector?

If so, then let $\textbf{r}(u)$ describe one of your field lines...what does the fact that $\textbf{B}\left(\textbf{r}(u)\right)$ is tangent to $\textbf{r}(u)$ tell you?