Understanding Gauss' Shell Theorms?

[mgp]

So there are two theorems:

1. A shell of uniform charge attracts or repels a charged particle that is outside the shell as if all the shell's charge were concentrated at its center.

2. If a charged particle is located inside a shell of uniform charge, there is no net electrostatic force on the particle from the wall.

A. Is there any way to easily verify why this is true? It would seem like if you imagined a charged particle being very close to the wall, the electric field would behave different than if it were concentrated at the center as a point charge. Also, I tried something very simple with drawing a circle, placing a charge at point P 1 unit from the edge of the circle and distributing positive charge along the circle. I then added up the forces acting on P from radially opposite charges on the circle and compared this sum to the force i would get from just using a point charge in place of the circle. Of course, my answers did not match, even when I decided to mess around with using the inverse square of the distance. I'm sure this little diagram was flawed by some means, but I just do not quite understand how it is possible for even an extremely large sphere to be able to be equivalent to a mere point charge.

B. Why is there no net electrostatic charge on the particle due to the shell? One thing about this one I can understand is that a positive charge will direct its electric field vectors all radially outward. Which would positive charge on the outside of a shell should direct it inward as well as outward, right? So if electric field vectors are directed inward, they will all end up canceling because of opposite vector components and because its a uniform shell. Thus there would be no net force. However, if you imagined a positively charged particle inside a uniform positively charged shell, wouldn't the electric field of the particle get disrupted from the field from the shell? Wouldn't those electric field lines meet head on and thus affect the electric field of the point charge, thus affecting the electrostatic force from the point charge due to this charged shell?

C. Lastly, why is the electric field inside a conductor always zero, even if excess charge is added? I understand conduction electrons move about and an induced charge can be present, but in the end the net charge within a conductor is always zero right? If excess charge is applied, how is it neutralized by an equal and opposite charge?

 

[Valerie Park]

The first two theorems are true due to the properties of electrostatics, specifically Gauss' Law and Coulomb's Law. According to Gauss' Law, the electric field at any point inside a spherically symmetric charge distribution is zero. This means that when a charged particle is inside a shell of uniform charge, there is no net electrostatic force on it, as the field lines cancel out. Coulomb's Law states that the force between two charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. This means that for a charged particle outside a shell of uniform charge, the electric force it experiences is equivalent to that of a point charge at the center of the shell, as the inverse square law applies regardless of how the charge is distributed. As for why the electric field inside a conductor is always zero, this is due to the fact that all free electrons in a conductor move in response to an external electric field, causing an equal and opposite electric field to be produced inside the conductor. This equal and opposite field cancels out the external field, resulting in zero electric field inside the conductor.