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

1. Aug 8, 2009

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.

Last edited: Aug 8, 2009
2. Aug 8, 2009

gabbagabbahey

Hi catheee, welcome to PF!

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

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

^^^Is this what you mean?

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?

3. Aug 8, 2009

catheee

yes! thats 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!

4. Aug 8, 2009

gabbagabbahey

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?!

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.

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?

5. Aug 8, 2009

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

6. Aug 13, 2009

gabbagabbahey

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

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?

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