Recent content by sdefresco

I understand the idea of the method of images, and its clever use of uniqueness to determine V(x,y,z) for non-trivial systems. My question now is simply about guidance for obtaining the effective "image" of this system, as it is clearly more complicated than the 2-plane analogue (in which there...

The final result's consistency with the seemingly-contradictory assumptions we've made is a good point, thank you for pointing that out.

Ah. Thank you for the clarification, and again for the overall help on the problem. At least in classical mechanics, there are still limiting cases that I could check for consistency. In electromagnetism, much of the time I feel as if I cannot even make sense of the limiting cases themselves as...

Yes, it came out to be V=Es, as I mentioned in post #42, should I choose dl to run along s. This is clear even beforehand, as E is not dependent on s, right? Substituting in E=q/2πεoR2 gives the full answer, obviously (if that's the extra information you were asking for.

I'm not exactly sure how that might be done based on what you've said.

Right, that's about where I am. Would integrating ∫ρ dv across the gap (as a 3D disc using cylindrical coordinates) work? I would find volume charge of the space by using Gauss Law in differential form. If that is no good, I assume the line integral would be the next place to go (E dot dl). In...

Awesome. Thank a lot for the help, the both of you. Electromagnetism has always been more painful for me than any other field of physics so far in my education. Even quantum 1 (with Griffiths) isn't as perturbing. The final question is to find the potential difference between the hemispheres...

Conductors have always spun me in circles intuitively. So, |Egap|=|σ1|/εo=|σ2|/εo, if I'm correct. This could be concluded using the boundary condition that ΔE=σ/εo when crossing a surface, or by using Gauss' Law. Also, σ1=Qeff/A=(1/2)Q/(πR2).

It seems a bit counter intuitive that this could be true for electrically isolated objects. This would mean that σ1 and σ3 have the same sign, but that's impossible since E=0 would not be true in this case for the hemisphere. I feel like calling the objects "electrically isolated" (as specified...

Total charge of the system is +Q. I don’t know what the total charge on the caps is. +Q or some fraction of Q dictated by geometry, I suppose. They need to net to Q, so 1/2 Q each sounds right. I feel the caps’ charge need to be opposite in order for E=0 to be maintained within each of the...

I’m not understanding how that reveals information about the planar faces of the hemispheres. They don’t exist in a non-cut sphere analogue, so what happens to the charge on that?

By that logic, E=0 in the gap and sigma1,2 should be zero if they effectively don’t exist. That doesn’t seem right to me either.

The obvious answer of Q distribution amongst a conducting sphere is uniformly about the surface, which would be the equivalent of both of the caps’ surface charges. It’s when the sphere is cut that my logic starts to become second-guessed.

In total, yes, the E field from each of the caps should also net to zero. That seems to suggest sigma1,2=sigma3,4, but you’ve already stated that’s not a correct relationship. Likewise, |sigma3|=|sigma4|, but this doesn’t give a relationship between the four or with q. This set is due in a...