Two questions on electrostatics

In summary, the conversation discusses two problems related to electrostatics. The first is about deriving the formula for the potential of a point charge and the use of line elements in the integral. The second problem is about the law or theorem stating that there is no electric field inside a conductor. It is questioned whether this can be proven or if it is an empirical law. The conversation concludes with the explanation that the law can be easily proven using the first uniqueness theorem.
  • #1
r4nd0m
96
1
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: [tex]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} [/tex]

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

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

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

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?
 
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  • #2
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.
[tex]\vec r[/tex] 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 [tex]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[/tex].
 
  • #3
r4nd0m said:
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?

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
great, thank you very much for your help.
 

1. What is electrostatics and how does it differ from electricity?

Electrostatics is the branch of physics that deals with stationary or slow-moving electric charges. It differs from electricity, which involves the flow of electric charges through a conductor. In electrostatics, the charges are at rest and do not move.

2. What are the fundamental laws of electrostatics?

The fundamental laws of electrostatics include Coulomb's Law, which describes the force between two electrically charged particles, and Gauss's Law, which relates the electric flux through a closed surface to the charge enclosed by that surface. Additionally, the principle of superposition states that the total electric field at a point is the vector sum of the individual electric fields produced by each charge present.

3. How does the distance between two charges affect the electric force between them?

According to Coulomb's Law, the electric force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This means that as the distance between two charges increases, the electric force between them decreases.

4. What is an electric field and how is it related to electrostatic force?

An electric field is a region in which an electric force can be exerted on charged particles. The strength of the electric field at a point is equal to the force per unit charge experienced by a test charge placed at that point. In electrostatics, the electric force between two charges is directly related to the electric field between them.

5. Can electrostatic forces be shielded or canceled out?

Yes, electrostatic forces can be shielded or canceled out by introducing an opposite charge or a grounded conductor into the electric field. This can be done through methods such as grounding, which involves connecting a conductor to the ground to neutralize any excess charges, or using insulating materials to prevent the electric field from reaching a certain area.

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