# Expression of electrostatic field

1. Sep 15, 2013

### bznm

1. The problem statement, all variables and given/known data

Why does this expression, given in cylindrical coordinates, rapresent an electrostatic field:
$${\bf E(r)}=\frac{\alpha}{z^2}{\bf u_r}-2 \frac{\alpha r}{z^3}{\bf u_z}$$

2. Relevant equations

3. The attempt at a solution
I can't understand why the expression rapresent an electrostatic field.
An electrostatic field is characterized by the fact that it depends only by r, isn't it?
I have tried to transform this expression in cartesian coordinates, but with no result (this is the firt time that I work with cylindrical coordinates). I'm thinking that I don't know how to say if a field is electrostatic or not.

Last edited: Sep 15, 2013
2. Sep 15, 2013

### TSny

Not necessarily.

Have you studied the concept of the "curl" of a vector?

3. Sep 15, 2013

### bznm

Curl? is it the rotor?

4. Sep 15, 2013

### TSny

Yes. Do you know how to express it in cylindrical coordinates?

5. Sep 15, 2013

### bznm

I have found it on internet

6. Sep 15, 2013

### TSny

If E is electrostatic, what can you say about the rotor of E?

7. Sep 15, 2013

### bznm

it is equal to zero... I got it! Thanks!!

8. Sep 15, 2013

### TSny

Good.

[I'm not sure if you are meant to worry about the following possibility. Sometimes you need to be careful working in cylindrical coordinates because the rotor is not defined on the z axis (r = 0). Also, your electric field is not defined on the z axis. Even if you change to cartesian coordinates, you will find that the rotor is not defined on the z axis for your field.

An electrostatic field should have the property that the line integral of the field around any closed path is zero. This will be the case if the rotor of E is zero at every point. Since the rotor is not defined along the z-axis in your example, it is necessary to show that the line integral of E around a closed path that encloses the z-axis is zero before you can really conclude that you have an electrostatic field. This came up in a recent question: https://www.physicsforums.com/showthread.php?t=709685

In your case, you can show that the line integral is zero for a path that encloses the z-axis.