# What is the smallest ball that can carry one esu?

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1. Sep 24, 2015

### bobie

Suppose I want to collect 1 esu (2.081 billion charges) on a plate or on a ball, what is the smallest radius that will carry such a charge?
From data I gathered around a ball of 1 mm of radius would be large enough, can you confirm that? What material is best, steel, silver or non-conductive material?

Also, can I measure the charge on the plate/ball with an accuracy of five figures with a cheap instrument (less than 50-100 \$) ?
Thanks

2. Sep 24, 2015

### BvU

Hello bobie,

Why would there be a lower limit to this radius ?

3. Sep 24, 2015

### bobie

Hey BvU,

Isn't there a limit on the quantity of charges on any body? I read that on a 1m sphere you can put 10^16 charges, so I deduced that on a 1mm ball you can have10^9.
Do you know how to measure that charge?

4. Sep 24, 2015

### BvU

There is no limit in the continuum formalism. A sphere is a capacitor and if you put more charge on it, the voltage increases. On that voltage there is a practical limit: with things in the neigbourhood that are at a lower voltage you get spectacular lightning effects.

Measuring is easy in principle: calculate the capacitance and measure the voltage. But that becomes unpractical and unhealthy when you're in the MegaVolt range.

5. Sep 24, 2015

### bobie

All right, then, I want to measure the electrostatic force at home. I want to put 1 esu on two balls without any dischage at 1 cm distance.

What is the most suitable size?
How do I get 2 billion electrons on each ball with 4 significant figures?

6. Sep 24, 2015

### BvU

I take it you want +1 esu on one ball and -1 esu on the other ? heart to heart 1 cm apart ?
That limits the ball radius to 0.5 cm...
So then you know the voltage difference.

With $C = 4\pi\epsilon_0 R$ and $Q = CV$ you have all you need to find V.

Example: 1 esu on 1 mm $\ \Rightarrow$ $$\ \ V = {3.3356 \times 10^{-10} \over 4\pi \, 8.854\times 10^{-12}}\; {1\over 1 \times 10^{-3}} = 2998 \ V$$

(The 2998 has to do with the speed of light...)

Another ball with -1 esu 1 cm further heart to heart gives 6 kV / 8 mm = 750 kV/m

Link says 3 MV/m for air so no sparks expected.

But, as you see, a lower bound to the ball radius comes into the picture. Not because the charge itself on a ball is limited, but the voltage difference shouldn't exceed the breakdown voltage. In fact there is also an upper limit: close to 5 mm there will also be a high $\Delta V \over \Delta x$

Actually measuring the charge will be pretty difficult. And: the 4 significant figures is a nice idea, but probably hard to realize.

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7. Sep 24, 2015

### bobie

So, 2 balls 1 cm in diameter 1 cm apart should be safe, right? what material is irrelevant? but,

If we put negative charge on both balls, can't we avoid sparks and have a smaller radius?

Is there any trick by which I gen get 1 statcoulomb on the balls? What instrument can tell me there are (roughly) 2.081 billion charges?
Can you do that in a college lab?

8. Sep 24, 2015

### BvU

Good questions, but I really have no idea.

Material should be irrelevant.
Usually electrostatics experiments are done by rubbing isolators on cat skins and such. A bit more serious would be a home-made vanderwaals generator (or a cheapo one ).

 better make that a van de graaff generator -- boy where was I with my thoughts !!

With negative charges on both balls ( or positive on both ) at least you don't have the high $\Delta V\over \Delta x$ so you can have any radius you like and use huge voltages.

There must be sites with a lot of teaching resources on this subject.

For measuring quantitatively I have no bright ideas. Conventional measurements won't work (your two billion e are gone in a split second :)

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Last edited: Sep 24, 2015
9. Sep 25, 2015

### bobie

Thanks, that was really useful