curl in electromagnetism

[physica]

I've been reading up on the concept of an electrostatic field. They say that if the mathematics of the field results in the curl=0. From what I gather this means that the field cannot be rotated. Does the curl=0 also indicate that it's conservative? If so how?

[elfmotat]

A field is defined as conservative if it can be written as the gradient of a potential function:

 

[math]\mathbf{F}=\nabla f[/math]

 

The definition is also equivalent to saying that its curl is zero, because:

 

[math]\nabla \times \mathbf{F}=(\nabla \times \nabla) f =0[/math]

 

The curl of a gradient is zero because partial derivatives commute. Physically, curl represents the amount of "circulation" in a field. Conservative fields are useful because the amount of work done in moving a point particle from one point in the field to another is independent of the path it takes. To see this:

 

[math]W=\int_a^b \mathbf{F} \cdot d\mathbf{x}=\int_a^b \nabla f \cdot d\mathbf{x}= f(b)-f(a)[/math]

Edited September 22, 2014 by elfmotat

[studiot]

Another way to look at the curl comes from fluid dynamics.

 

If curl(F) = 0 then there are no sources or sinks within the domain of the field.

 

Many problems can be reduced to 2D and for these curl(F) is a vector that points out of the plane of the vector field.

Edited September 22, 2014 by studiot

[physica]

I think I get it. Combining both your posts a conservative field is when it doesn't matter what path the particle takes. If there is no sources or sinks in the field then it wont matter what path the particle takes..... is this train of thought in the right direction?

[elfmotat]

Another way to look at the curl comes from fluid dynamics.

 

If curl(F) = 0 then there are no sources or sinks within the domain of the field.

 

Many problems can be reduced to 2D and for these curl(F) is a vector that points out of the plane of the vector field.

 

I believe you're thinking of divergence. Fields with no sources aren't necessarily conservative. For example, the E or B field of an electromagnetic wave: the curl of E is nonzero even in the absence of sources.

[studiot]

elfmotat, you're quite right and what's worse I've made that mistake before.

 

Very red face - I need a long holiday.

 

Sorry about that, physica.

[imatfaal]

physica - have a search through Walter Lewins course 803 at MIT online. he devotes a chunk of one lecture to nice analogies to get this fixed in your head

[Enthalpy]

In the case of an electrostatic field, the divergence is the density of charge (with permittivity and signs), so it's zero without charges and nonzero with charges. In both cases, the electrostatic field has zero curl. So curl and divergence are independent.

 

Outside electrostatics, a varying induction creates a curl in the electric field. This creates our electricity in generators at the power plant, where the rotating magnetic field induces a voltage in a closed loop (the electric circuit), hence with curl.