Two questions on electrostatics

1. Jun 19, 2006

r4nd0m

Hi,
I'm a bit stuck with some things in electrostatics.

My first problem:

in my textbook, when they try to derivate the formula for the potential of a point charge: $$V(b) = - \int E.d\mathbf{l} = -\frac{q}{4 \pi \varepsilon_0} \int_\infty^b \frac{1}{r^3} \mathbf{r}.d \mathbf{l}$$

they say that $$\mathbf{r}.d \mathbf{l} = r.dr$$
There's also a picture which looks like this:

http://img232.imageshack.us/img232/6044/charge3zj.jpg [Broken]

My question is: Why isn't $$d \mathbf{r} = d \mathbf{l}$$ ? Why does $$d \mathbf{r}$$ have the same direction as $$\mathbf{r}$$ ?

My second problem:

There is some law or theorem which says that there is no electric field inside a conductor. Can this be proved, or is it just an empirical law?

Last edited by a moderator: May 2, 2017
2. Jun 19, 2006

Galileo

The dl is a line element of the path you are taking to evaluate the integral (it doesn't matter what path you take), just some path from infinity to b.
$$\vec r$$ is a vector pointing in the direction of the electric field (if the charge is positive), thus in the radial direction.
The line element in spherical coordinates is something like $$d\vec l = dr\hat r+rd\theta \hat \theta+r\sin \theta d\phi \hat \phi[/itex], so [tex]\vec r \cdot d\vec l=rdr$$.

3. Jun 19, 2006

siddharth

Well, you need to be a bit more precise. The electric field is zero in a region completely surrounded by a conductor, provided there is no charge enclosed in that region.

It can be proved easily using the first uniqueness theorem.

4. Jun 19, 2006

r4nd0m

great, thank you very much for your help.