Determining Field from Div & Curl: Examples & Techniques

[latentcorpse]

My notes say that if we know the divergence and curl of a field then that uniquely determines the field.

Can somebody give me an example of how, given only the div and curl of a field, we can deduce the field?

I considered the electric field where we have,
\nabla \cdot \vec{E}=\frac{\rho}{\epsilon_0}, \nabla \times \vec{E}=0
but we can't actually establish E using only vector calculus can we? we need other techniques do we not? perhaps I'm just being silly?

 

[weejee]

I think the boundary condition needs to be specified.

 

[Brian_C]

The electrostatic case is easy to understand. To find the electric field at a point, you need to integrate the electric field contributions from each infinitesimal charge. Since the local charge density is found from the divergence of the E-field, you are essentially integrating the divergence over the volume of interest.

In practice, the charge density is not always known a priori. Sometimes the electric field or potential is specified along some boundary, which is where Neumann and Dirichlet boundary conditions come into play.

 

[latentcorpse]

yeah. can you just confirm a couple of things:

(i) the RHS of the divergence equation is easy enough to integrate given a charge density. To integrate the LHS however, I would use divergence theorem and then take it from there using an appropriate Gaussian surface yes?

(ii)In the method you just gave for finding the E field in post 3, you have only used the divergence equation, how would the curl one come into play?

cheers

 

[kof9595995]

to make sure it's electrostatic

 

[latentcorpse]

i.e. a non zero curl results in a non continuous tangential component of electric field at a conductor's surface and so the tangential component of electric field outside the conductor isn't necessarily 0 (as is guaranteed if the curl were 0) and so charges are moving on the surface - hence it's no longer electrostatics!

is that what you mean?

 

[kof9595995]

em...i would rather say a nonzero curl is resulted from a changing magnetic field, by faraday's law.So it won't be not electrostatics.

 

[Klockan3]

Actually electric fields and magnetic fields are the same thing if you consider standing still the same as moving in the time dimension.