Expression of electrostatic field

[bznm]

Homework Statement

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

Homework Equations

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.

Many thanks for your help.

 

[TSny]

An electrostatic field is characterized by the fact that it depends only by r, isn't it?

Not necessarily.

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

 

[bznm]

Curl? is it the rotor?

 

[TSny]

Curl? is it the rotor?

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

 

[bznm]

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

I have found it on internet

 

[TSny]

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

 

[bznm]

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

 

[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.

Again, I'm not sure if you are meant to worry about this.]