What is the appearance of field lines in the electrostatic setting?

If a short and a long electrode are placed in parallel. The short one, connected to a constant high voltage, is placed above with a distance of $H$ from the long one. The long electrode is grounded and has a length of $L$, which is larger than but comparable with $H$.

Now, what is the rough appearance of electrostatic field lines in the setting, especially near the edge of the short electrode? What if the potentials of the two electrodes are interchanged?

It will be great if anybody could help to visualize the electric field. Thank you very much.

Answer

In the diagrams below I've attempted to sketch some examples of electric field lines for various electrode configurations, including the one your describe (sorry if they're a bit crude).

In the top diagram we have two same length parallel plates, one positively charged and one negatively charged. With the exception of the ends, you can see that the electric field lines are uniformly spaced. In this area we say that the electric field is homogeneous. At the ends we have edge effects. The field lines are more concentrated at the edges and are not uniform. We say the field in these regions is inhomogenous.

The bottom configuration is what you have described as one long and one short electrode. Once again, in the vicinity nearer the center the field is homogenous and inhomogeneous at the ends. The edge effect is not as pronounced in the vicinity of the lower plate as it is in the upper plate, due to the extension of the lower plate beyond the ends of the upper plate.

The diagrams in between show the field between a point charge and a plane, and between two point charges. For the point and plane, the field lines are concentrated near the point charge. For the two point charges, the field lines are concentrated near each point charge.

The density of the field lines in a particular region is roughly proportional to the strength of the electric field. The strength of the field is greatest in the inhomogeneous regions. This can have a significant effect on the dielectric strength of the medium in those regions.

For the diagrams below, as the voltage between the electrodes of each configuration is increased, dielectric breakdown of the air in between will initiate in the regions of greatest field density. That would be at the point electrodes in the middle two configurations, at the edges of both the bottom and top plate of the top diagram, and at the edge of the top plate in the bottom diagram.

The polarities of the electrodes should not effect the density of the field lines, only the direction of the electric field lines. By convention, the direction of the electric field is the direction of the force that a positive test charge would experience if placed in the field. So if we reverse the polarities in the diagrams below, the density of the field lines will be unchanged, but the direction of the lines will be the reverse of those shown.

for the last setting (short+long), is there any method to calculate or estimate the influence length of the inhomogenous side

I'm afraid the calculations can get complicated. I don't where to find it for the short and long plate example, but the link below calculates the edge effects of a parallel plate capacitor, which would be my top diagram. It may help you get an idea of what's involved. Hopefully what I covered here at least helped get you started.

https://web.stanford.edu/~ajlucas/Edge%20Effects%20in%20a%20Capacitor.pdf Hope this helps.

Hope this helps.