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# Charged Spheres

Consider two identical spheres A and B with charge Q each. Now a third identical neutral sphere C is touched with A then B then A then B ane so on .

Will the process terminate, I mean will there be a time when there is no charge redistribution.

Note by Kushal Patankar
2 years, 1 month ago

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Kushal Patankar

This sort of scenario arises in many places. The answer is that the final charges will be $$\frac {2Q} {3}$$. There are many ways to explain this, I think it can be explained with a fully qualitative reasoning(no math), but since you have asked for it:

Let us think about it this way, let at some point of time the spheres $$S1$$ and $$S2$$ have charges $$Q1$$ and $$Q2$$. The sphere we can move is $$S3$$ and it has a charge $$Q3$$. When we touch a sphere with the movable one, what happens is that the charges on the spheres become equal. If we touch $$S1$$, the charge on each sphere becomes $$(Q1+Q3)/2$$. Consider this series:

$$1,0.5,0.75,0.625,0.6875,...$$

In the above series, the following rule is applicable for the $$n^{th}$$ term:

$$a_n=(a_{n-1}+a_{n-2})/2$$

Observe that this is the series of the values of the charges(assume $$Q=1C$$) on each sphere after each step starting from before the movable sphere comes into contact with $$S2$$. Write the successive charges on each sphere after each contact, and you will see that we get the above series.

We define a generating function like below:

$$f(x)=a_0+a_1 x+a_2 x^2+ a_3 x^3+a_4 x^4+...$$

For the current discussion, we have:$$a_0=1,a_1=0.5$$.

Try to solve this generating function in order to get a closed form expression for $$a_n$$. I will leave this for you to do on your own. The answer we get is: $$a_n= \frac {2^{n+1}+(-1)^n} {3 \times 2^n}$$.

You can now figure out the necessary details when $$n \to \infty$$ · 2 years, 1 month ago

@Kushal Patankar · 2 years, 1 month ago

Thanks bro. I will do the rest now. Thanks again. · 2 years, 1 month ago

Initially when A is touched to C, the charge flows unless both have equal charge i.e. Q/2 . Subsequently when C is touched to B, charge flows until both posses equal charge i.e. 3Q/4. Moving further with this trend, there would come a point when the 3 balls posses equal charge I.e. 2Q/3 and there would be no further charge redistribution. I hope you get the point. Please point out my misconceptions if any. :) · 2 years, 1 month ago

Yes you are right . But can the neutral ball aquire a charge of 2Q/3

If yes then how many touchs are required.

Are they finite . · 2 years, 1 month ago

I think the touches tend to infinity. · 2 years, 1 month ago