Charged Conductor: Understanding Force in a Cavity

[Kazza_765]

I've been studying gauss' laws etc. at uni and I was thinking about charged conductors. When a conductor is charged the charge distributes itself in such a way that the electric field inside the conductor is zero, and hence if there is a cavity inside the conductor, the net force on a charged particle within the cavity will also be zero.

Now my question is this. If we imagine a spherical, hollow, conductor with a positively charged particle contained within and then positively charge the conductor, nothing happens to the particle. But what happens if the conductor is not perfectly spherical but has a hole in it?

Logically I think that there will be a net force on the particle in the direction of the hole, and as the size of the hole approaches zero so does the net force on the particle. But if you think about the field lines, then every field line originating at the interior surface of the conductor must leave the cavity through that hole, which would mean that as the size of the hole decreases, the force on a particle passing through the hole would increase. I'm pretty sure the first explanation is the correct one, as I don't think visualising field lines is a good way to approach this problem, but perhaps someone could explain what actually happens in this situation.

 

[whozum]

Gauss' Law specifically states that the conductor must be closed. If there is a hole then there would be a net force on both the particle and the shell if they are both not fixed.

 

[thepatientmental]

Your understanding of charged conductors and the distribution of charge is correct. In the case of a perfectly spherical, hollow conductor, the charge will distribute itself in such a way that the electric field inside the conductor is zero, and therefore the net force on a charged particle within the cavity will also be zero. This is because the charge on the inner surface of the conductor will cancel out the charge on the outer surface, leaving no net electric field inside the conductor.

However, in the case of a conductor with a hole in it, the situation is slightly different. If the conductor is not perfectly spherical, the charge distribution will not be symmetrical and there will be a net electric field inside the conductor. This means that there will be a net force on a charged particle within the cavity.

Your intuition that the net force on the particle will be in the direction of the hole is correct. As the size of the hole approaches zero, the net force on the particle will also approach zero. This is because as the hole gets smaller, the electric field inside the conductor becomes more symmetrical and cancels out, resulting in a net force of zero on the particle.

You are also correct in thinking that visualizing field lines is not the best way to approach this problem. While it can be a useful tool for understanding electric fields, it is not always an accurate representation of the actual physical situation. In this case, the charge distribution and resulting electric field are what determine the net force on the particle, not just the visualization of field lines.

In summary, your understanding of charged conductors and the distribution of charge is correct. In the case of a conductor with a hole, the net force on a charged particle within the cavity will be in the direction of the hole, and as the size of the hole approaches zero, the net force will also approach zero. It is important to focus on the charge distribution and electric field, rather than just the visualization of field lines, when solving problems involving charged conductors.