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If I connect a sphere of capacitance $C_1$ to a charged sphere of capacitance $C_2$, will the charge be distributed evenly on both spheres (is the charge density going to be equal on both spheres?). If not, why?

I came across the following statement:

Connect a charged sphere to an other neutral sphere. How does the charge density change ? It depends on the capacity of the spheres.

That suggests the answer to my question is 'no'. To my knowledge, all that capacitance tells us is how much charge we need to put on an object to increase its potential by $1V$ (relative to some object placed at infinite distance from the object). How can capacitance influence the charge density of the spheres?

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  • $\begingroup$ You need to specifiy the distance between the spheres, wire diameter, etc. $\endgroup$ – Digiproc Jul 15 '16 at 22:14
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    $\begingroup$ What is the source of your quote? Didn't the source explain? $\endgroup$ – sammy gerbil Jul 16 '16 at 3:11
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If charges have to flow between sphere A and sphere B, you need a potential difference between the two. Since your question says that one of them is charaged and one of them is uncharged, there shall exist a potential difference between the two and hence there will be flow of charges but they needn't necessarily equalize. The amount of charge transfered depends on the capacitance.

Lets study the system mathematically.

  • Let $C_1$ be the capacitance of the uncharged sphere
  • Let $C_2$ be the capacitance of the charged sphere
  • Let $Q_2$ be the initial charge on the charged sphere

Let us assume that the spheres are initially kept far away from each. The uncharged sphere will be at $0$ potential since it does not have any charge. The charged sphere will have a non-zero potential since charges reside on its surface.

The potential of the charged sphere can be found by using the following formula, $$Q = CV$$ $$V_2 = \frac{Q_2}{C_2}$$

As said earlier, once the two spheres are connected, the charges will keep flowing as long as there exists a potential difference between the two.

When will the charges stop flowing? When the potential difference between the two spheres reduces to zero, i.e: both the spheres reach the same potential.

Let $q$ be the total amount of charge transfered from the charged sphere to the uncharged sphere. In other words, $q$ is the charge on the uncharged sphere after the two spheres are connected (since the uncharged sphere initially had no charge, all the transfered charge is the charge it holds).

After the transfer process, the initially uncharged sphere will now have $q$ amount of charge and the initially charged sphere will have $Q_2 - q$ amount of charge (by the law of conservation of charge).

Since the potential difference ceases to exist once they are connected (and after sufficient time has passed), we can equate the potentials of the two spheres and solve for $q$.

$$\frac{(Q_2 - q)}{C_2} = \frac{q}{C_1}$$

Solving the equation you get,

$$q = \frac{Q_2(C_1 + C_2)}{C_1}$$

Therefore, the amount of charges on the two spheres after they are brought to contact depends on their charge capacity (capacitance).

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