# Charge distribution: electrostatic equilibrium in conducting sphere

Consider an uncharged sphere, made of electrically conducting material. We then charge the sphere with excess charges.

I understand that these excess charges repel each other till the electric field lines parallel to the surface of the sphere cancel out. Each charge experiences the superposition of electric fields provided by all other charges, and no net force is exerted on any charge. This results in an evenly distributed charge distribution on the surface of the sphere.

The system is in stable electrostatic equilibrium.

Consider that we take the same sphere with all the excess charges on the surface, and insert an additional charge into the center of the sphere, such that the system is in unstable electrostatic equilibrium.

Is this particular distrubution of charges (including the center charge) the only configuration of unstable electrostatic equilibrium?

Put another way, other than the distributions above, is there no other configuration of charges such that the closed system is in unstable electrostatic equilibrium?

Why yes/no?

• A charge placed at the center will be innunstable equilibrium. I did not get your question. Commented Jan 24, 2017 at 4:06
• Perhaps it should be neutral equilibrium? Commented Jan 24, 2017 at 7:09
• Farcher, I assume you're talking about the sphere with surface charges and center charge? I would think it's unstable, because any deviation from center results in a nonzero superposition of electric field vectors (considering distances to each equally distributed surface charge), hence a nonzero force on the center charge, and movement away from equilibrium. But I'm wondering if we can achieve at least one other equilibrium (of any kind: stable, neutral, or unstable) by using any amount of charges, and placing them anywhere in the sphere. Commented Jan 24, 2017 at 7:20

In the hollow interior of a conductive sphere, we can create the same type of unstable equilibrium situations that we know from empty space. For instance two positive point charges $$q$$ and a third one $$-q/4$$ precisely in the middle between them will be in (unstable) equilibrium in free space.
If this configuration is placed centered inside the hollow conducting sphere we will get some influence from the two negative mirror charges that appear outside the sphere for our two positive charges (the negative charge is centered so it has no mirror term). But by changing the charge ratio $$-1/4$$ ratio to a different value, things can be put in equilibrium again.
If the positive charges $$q$$ are close to the center, then only a slight influence from the mirror terms exists and only a slightly stronger negative central charge is needed. Solving the general case is left as an exercise for the reader.