Electric Field Strength- variations

[chanderjeet]

If the electric field lines are curved as opposed to straight, say from straight parallel lines parabolic curves develop, would there be any change in the field strength? ie would it be stronger when the lines are parallel than when they curve?

 

[TVP45]

Can you describe this in a bit more detail?

 

[chanderjeet]

i was trying to describe a diagram that I didnt quite understand and wasn't sure if the electric field strength would be smaller when curved.

It looks a bit like the path an electron would take in a uniform electric field. With the direction of the electron being initially perpendicular to the direction of the uniform field between the plates. The electron would then follow a parabolic curve. But on this diagram only the electric field lines are present, no plates etc. more or less showing the direction of electric field lines. Like the tangent at the curve, showing direction.

What i wanted to know, was if E is stronger when the lines are parallel as opposed to when then move off into the curve.

 

[Dale]

The strength of the E-field is indicated by how closely spaced the field lines are, not their curvature.

 

[Bob S]

The strength of the E-field is indicated by how closely spaced the field lines are, not their curvature.

E-field lines are always normal (perpendicular) to conductors at the surface, because a conductor cannot support tangential electric surface fields. So field lines at curved surfaces are always curved.
Bob S

[added] If the field lines originate from a concave surface of a conductor, they are leaning toward each other. In this case, is the electric field a maximum at the surface, or away from the surface, such as near the center of curvature for the surface? What about requirements due to the orthogonality of the electric field lines and the equipotential lines, analyticity, Cauchy theorem, Cauchy Riemann equations?
Bob S

 

[chanderjeet]

when they curve, wouldn't the spaces between the field lines increase as well?

 

[Dale]

No, the space between the field lines could just as easily decrease.

 

[chanderjeet]

ok thanks for your help...still not sure what this diagram means though.

where the field lines move off into the curve, an annotation says E (which i assume is the electric field strength) is small and where the field lines follow the initial straight path, where they lie parallel to each other, the annotation says E is large.

 

[Bob S]

No, the space between the field lines could just as easily decrease.

So E = -grad V = maximum doesn't have to be at a surface? It can be anyplace in space?
Bob S

 

[SystemTheory]

This reference says the width between the field lines is indicative of field strength, but note, this is relative to a diagram of a particular problem and the field strength or density is really determined by a calculation and can't be derived from a diagram.

http://www.iop.org/activity/education/Teaching_Resources/Teaching%20Advanced%20Physics/Fields/Electrical%20Fields/page_4802.html

 

[Bob S]

This reference says the width between the field lines is indicative of field strength, but note, this is relative to a diagram of a particular problem and the field strength or density is really determined by a calculation and can't be derived from a diagram.

I agree that the space between field lines is indicative of field strength. For a specific situation, can the electric field be stronger in a space (without free charges) than at a conducting surface where the field lines terminate on surface charges?
Bob S

 

[Dale]

ok thanks for your help...still not sure what this diagram means though.

where the field lines move off into the curve, an annotation says E (which i assume is the electric field strength) is small and where the field lines follow the initial straight path, where they lie parallel to each other, the annotation says E is large.

What is the spacing between the lines where the annotation says E is large? Are they close or far compared to where the annotation says it is small?

 

[chanderjeet]

slightly smaller where the annotation says E is large

 

[Longstreet]

In electrostatic situation, and curl of E = 0, then I think if you have curved E then it must diminish in some direction for the curl to vanish.

 

[Per Oni]

[added] If the field lines originate from a concave surface of a conductor, they are leaning toward each other. In this case, is the electric field a maximum at the surface, or away from the surface, such as near the center of curvature for the surface?
Bob S

In case of a concave surface the field lines are going to be wider spaced, a lot will end up on the sharper edges of the plate and towards the convex side. To see that this is so increase the curvature of the concave side until your surface forms a hollow cylinder or hollow sphere. There will be no field lines inside those at all.

I agree that the space between field lines is indicative of field strength. For a specific situation, can the electric field be stronger in a space (without free charges) than at a conducting surface where the field lines terminate on surface charges?
Bob S

I don’t think so. Field lines tend to spread out sideways so as to minimise the stored energy per unit volume.

 

[Bob S]

From Bob S
I agree that the space between field lines is indicative of field strength. For a specific situation, can the electric field be stronger in a space (without free charges) than at a conducting surface where the field lines terminate on surface charges?

I don’t think so. Field lines tend to spread out sideways so as to minimise the stored energy per unit volume.

I certainly agree with you. I have seen some proof, perhaps using the analaticity of electric field lines and equipotential lines. What is the proof?
Bob S

 

[SystemTheory]

I agree that the space between field lines is indicative of field strength. For a specific situation, can the electric field be stronger in a space (without free charges) than at a conducting surface where the field lines terminate on surface charges?
Bob S

Field and Wave Electromagenetics by David K. Cheng (1985).

Boundary Conditions at Conductor / Free Space Interface

$$E = \frac{\rho_{s}}{\epsilon_{0}}$$

"The normal component of the E field at a conductor-free space boundary is equal to the surface charge density (rho) on the conductor divided by the permitivity of free space."

Finding the E field at a point P located at radius R from a differential surface element ds is calculated by taking a surface integral (double integral notation not shown):

$$E = \frac{1}{4\pi\epsilon_{0}}\int \textbf{a}_{R}\frac{\rho_{s}}{R^{2}}ds$$

where this should converge to the boundary condition specified above at R = 0, but I'm not up to date on my double integral techniques and it appears to me that E might blow up to infinity as R approaches zero in the integral evaluation?

In any case E should decrease with an increase of R away from the conductor surface.

Also see Gauss's law in reference to this thread, where the flux of the normal E field is constant for any surface enclosing a charge, thus as the radius to the surface increases, the normal flux density decreases. For a symmetrical problem the E field lines are easily visualized as being farther apart as the radius increases of the enclosing surface.

 

[Bob S]

Field and Wave Electromagenetics by David K. Cheng (1985).

Boundary Conditions at Conductor / Free Space Interface

$$E = \frac{\rho_{s}}{\epsilon_{0}}$$

"The normal component of the E field at a conductor-free space boundary is equal to the surface charge density (rho) on the conductor divided by the permitivity of free space."

Finding the E field at a point P located at radius R from a differential surface element ds is calculated by taking a surface integral (double integral notation not shown):

$$E = \frac{1}{4\pi\epsilon_{0}}\int \textbf{a}_{R}\frac{\rho_{s}}{R^{2}}ds$$

where this should converge to the boundary condition specified above at R = 0, but I'm not up to date on my double integral techniques and it appears to me that E might blow up to infinity as R approaches zero in the integral evaluation?

In any case E should decrease with an increase of R away from the conductor surface.

Also see Gauss's law in reference to this thread, where the flux of the normal E field is constant for any surface enclosing a charge, thus as the radius to the surface increases, the normal flux density decreases. For a symmetrical problem the E field lines are easily visualized as being farther apart as the radius increases of the enclosing surface.

Does this apply to the fields in a parallel-plate capacitor, where the field lines are parallel and equally dense everywhere, both at the plates, and everywhere in between?
Bob S

 

[SystemTheory]

If the permitivity (dielectric constant) is that of free space, I believe the boundary conditions and integral above would give the constant field of a parallel plate capacitor after applying superposition of the two E fields upon the space between the plates. Of course in a practical capacitor the uniform field assumes plate spacing is small and area large, so the field is constant near the center. Changing the dielctric permitivity requires similar relations defined for the displacement vector D, according to Cheng. I don't know if this helps with your question?

 

[SystemTheory]

Bob S. & others:

Excellent reference on basic electrostatic field problems and parallel plate capacitor:

http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/ElectricForce/FlatSheet.html

http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Capacitors/ParallCap.html

 

[Bob S]

Bob S. & others:

Excellent reference on basic electrostatic field problems and parallel plate capacitor:

http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/ElectricForce/FlatSheet.html

http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Capacitors/ParallCap.html

Neither discuss the basic problem we are discussing, which is capacitors with cupped plates, such that the field lines at the conductor surfaces face inward toward the center. In this case, is the electric field still maximum at the conductor surface?
Bob S

 

[Longstreet]

Just take the limit where a long pipe intersects perpendicular to the parallel plates. The field will be zero inside the pipe (because the potential is everywhere uniform there), except at the edge where you have fringing fields. Now every field line has to go between potentials (E=-dV/dx), so any field line originating inside the pipe cannot terminate on the plate it connects to (dV/dx=0). If the plate separation is much larger the the diameter of the pipe (ie, once again uniform E on the other plate), then there are field lines from within the pipe that must certainly go through a region of field strength much greater then where it originated.

 

[Per Oni]

Just take the limit where a long pipe intersects perpendicular to the parallel plates. The field will be zero inside the pipe (because the potential is everywhere uniform there), except at the edge where you have fringing fields. Now every field line has to go between potentials (E=-dV/dx), so any field line originating inside the pipe cannot terminate on the plate it connects to (dV/dx=0). If the plate separation is much larger the the diameter of the pipe (ie, once again uniform E on the other plate), then there are field lines from within the pipe that must certainly go through a region of field strength much greater then where it originated.

I got the impression that Bob S was talking of a concave plate in free space, ie with no other conductors nearby.
If we introduce say a needle conductor and there’s a pd between plate and needle then there’s an increase in field when going from plate to needle.
If the plate is placed in free space I don’t think there will ever be an increase in field strength past the surface no matter how you bend it.

 

[SystemTheory]

Bob S,

Typically I would sketch the problem in paint and post if it really needs attention.

I found a problem in my Field & Waves book which says a static E field can be shaped by a dielectric lens. I am not sure if a charged parabolic surface provides a focus point for a static E field that is stronger than the E field strength at the surface, but that's where I'd investigate the principle further.

 

[Longstreet]

Oh ic. yeah I agree.

 

[Bob S]

Bob S,

Typically I would sketch the problem in paint and post if it really needs attention.

I found a problem in my Field & Waves book which says a static E field can be shaped by a dielectric lens. I am not sure if a charged parabolic surface provides a focus point for a static E field that is stronger than the E field strength at the surface, but that's where I'd investigate the principle further.

I was talking about fields in free space without dielectrics. Can the field anywhere between two concave surfaces of a capacitor be greater than the fields at the surface of the plates?
Bob S

 

[Longstreet]

we want to find the maximum of $$E^{2}$$

speculate it is where the laplacian is non-zero.

something like:
$$\nabla^{2}(E\cdot E) = \nabla\cdot(\nabla(E\cdot E)) = \nabla\cdot(2( (E\cdot\nabla)E + E\times(\nabla \times E))) = 2(E\cdot\nabla)\nabla\cdot E = 2(E\cdot\nabla)\rho$$

$$\nabla \times E = 0, \nabla \cdot E = \rho$$

So it would have to occur at the charge surface? Hope that's correct but no source.