A Free surface problem in electrohydrodynamics

hunt_mat
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If I have a constant bulk charge in a fluid, does that correspond to a specific surface charge at the free surface?
I'm thinking about the following problem. I have an electrified fluid with a constant charge density, $Q$, within the fluid. Will this necessarily yield a surface charge?

Would I have to compute it by looking at the displacement fields on either side of the interface? Would it change if the bulk charge within the fluid remains constant?
 
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Im reading this on a phone so maybe I am missing something. But the electric field is different inside and outside the medium, the discontinuity is proportional to the surface charge.
 
Yes. The picture I'm thinking about is where there is an electric field within the fluid as well as a constant charge density. There is also an electric field above the fluid. The surface charge is the difference between the electric displacements D. I want to know if given that there is a bulk electric charge, will that give rise to a surface charge that I have to take into account?
 
hunt_mat said:
Yes. The picture I'm thinking about is where there is an electric field within the fluid as well as a constant charge density. There is also an electric field above the fluid. The surface charge is the difference between the electric displacements D. I want to know if given that there is a bulk electric charge, will that give rise to a surface charge that I have to take into account?
Can you tell me what is the electric field inside the fluid and outside (at least an example)?
 
That is something to be found as part of the solution. I'm connecting the Euler equations with Maxwell's equations to find the resulting shape of the free surface.
 
The bulk charge does not intrinsically produce a surface charge. Consider the case of a sphere with uniform charge density. In that case the electric field is continuous at the surface so there is no surface charge.

That said, with a charged fluid there will be forces on the bulk fluid due to the charge. So that may lead to motion that does produce a surface charge.

The bulk charge and the surface charge are not directly linked, but they can be indirectly linked through the motion of the fluid.
 
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hunt_mat said:
That is something to be found as part of the solution. I'm connecting the Euler equations with Maxwell's equations to find the resulting shape of the free surface.
But do you at least have the electric field inside and outside the medium?
 
pines-demon said:
But do you at least have the electric field inside and outside the medium?
The OP posits a fluid medium with a constant bulk charge density and so it carries a nonzero total charge. Assuming that the fluid does not extend to infinity in any direction, it can therefore be completely surrounded by a closed surface. The integral of the normal electric field over this surface must then be nonzero by Gauss' Law. Hence, by necessity there exists an electric field outside the medium.
 
renormalize said:
The OP posits a fluid medium with a constant bulk charge density and so it carries a nonzero total charge. Assuming that the fluid does not extend to infinity in any direction, it can therefore be completely surrounded by a closed surface. The integral of the normal electric field over this surface must then be nonzero by Gauss' Law. Hence, by necessity there exists an electric field outside the medium.
Unfortunately, the fluid is infinite in extent.
 
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pines-demon said:
But do you at least have the electric field inside and outside the medium?
Yes. I have solved the result when there is no electric charge, I thought that the next logical step(for a mathematician that is) is to look at the case where there is a constant charge density within the fluid.
 
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Dale said:
The bulk charge does not intrinsically produce a surface charge. Consider the case of a sphere with uniform charge density. In that case the electric field is continuous at the surface so there is no surface charge.

That said, with a charged fluid there will be forces on the bulk fluid due to the charge. So that may lead to motion that does produce a surface charge.

The bulk charge and the surface charge are not directly linked, but they can be indirectly linked through the motion of the fluid.
Surely only the tangential part of the electric field is continuous at the boundary. You have to make a calculation for the normal component of the electric field.

If the forces aren't too large, then should there be no surface charge? Or, at the very least, the assumption of no surface charge is reasonable.

I'm also assuming that there is no charge below the fluid, so I can assume that the electric field is zero at the bottom?
 
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hunt_mat said:
Surely only the tangential part of the electric field is continuous at the boundary.
For a uniform ball of charge (eg a solid charged insulator) both the tangential and normal components are continuous at the boundary
 
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Dale said:
For a uniform ball of charge (eg a solid charged insulator) both the tangential and normal components are continuous at the boundary
Isn't the normal component of the \mathbf{D}=\epsilon\mathbf{E} continuous across the boundary if there is no surface charge?
 
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