# How do I tell if the field is a possible electrostatic field?

1. Feb 24, 2010

### theneedtoknow

If I have a picture of the field lines of a field, how do I tell if the field is a possible electrostatic field? What are some things to look for that would imply it either is or is not a possible electric field?

2. Feb 24, 2010

### quasar987

There are various criteria that comes to mind that would serve to disqualify a vector field as being an electrostatic one.

The idea is that the differential relations satisfied by an electrostatic field are local and as a result difficult to work with. But integrating them leads to more managable statements about the field.

For instance, starting from the differential relation that the field can be written as the gradient of a potential function ($\mathbf{E}=-\nabla V$) and integrating along any closed loop, the fundamental theorem of line integral yields the integral relation

$$\int_{\gamma}\mathbf{E}\cdot d\mathbf{r}=0$$

So you can start looking for loops for which this integral relation obviously fails.

What are other differential relations satisfied by an electrostatic field that beg to be integrated? The Maxwell equations of course! Derive analogous integral statements for those to obtain 2 more disqualifying criteria.

3. Feb 25, 2010

### theneedtoknow

So if I have a field that looks something like
I can tell it could be an e-field because, if I start at any point on a loop and make a trip all the way back to the starting point, i will pass through as many arrows head on as i will tail on, so it nets to 0?

4. Feb 25, 2010

### netheril96

Are you sure this is a field?

5. Feb 25, 2010

### theneedtoknow

Ok but say you draw field lines of a dipole, won't the E-field due to the dipole also have curl? (I know an E-field needs to have zero curl because E=-delV, only possible with zero curl, but just looking at the field lines of a dipole it looks....curly lol)

6. Feb 25, 2010

### Studiot

An electric field has one more thing not yet mentioned and not shown in your sketch.

At least one source and /or sink.

7. Feb 26, 2010

### Bob S

How do you explain completely closed electric field loops in vacuum without a physical source, such as Faraday's Law:

∫E·dl = -(d/dt)∫B·n dA

Like around a 60-Hz transformer magnetic core with an enclosed dB/dt?

Bob S

8. Feb 26, 2010

### Studiot

Bob,
As I understand it the OP offered a loop where the integral

$$\oint {E.dl}$$ is zero.

This happens with electric fields generated by the presence of charges.

9. Feb 26, 2010

### Redbelly98

Staff Emeritus
The OP did say it is a static field. So zero curl is required, or equivalently the integral given by studiot is zero.
Also, the divergence of E = ___? (Answer depends on whether charge is present.)

Last edited: Feb 26, 2010
10. Feb 26, 2010

### Studiot

Firstly, sorry about my presentation, I'm still struggling with Tex / Mathtype.

I should have added that the environment in which fields can turn right angles needs further amplification.

Also that magnetic fields can give rise to electric ones, as Bob noted, if we allow the integral to be non zero.

11. Feb 26, 2010

### Bob S

Wouldn't Curl E = constant qualify as a static electric field? So as long as dB/dt is a constant, the induced electric field is constant.

Bob S