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If I charge two capacitors which are connected parallel $[$the minus (-) of the one opposite to the minus (-) of the other and the plus(+) of the one opposite to the plus (+) of the other.$]$, will I get current (charge flowing) after adding a dielectric material to one of them ?

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The short answer is yes. Here is an easy way to see it: for simplicity assume that both capacitors have capacitance equal to $C$ and each have voltage $V$. Then, the charge on each capacitor is equal to $CV$ and the total charge is $Q=2CV$. Now, a material with dielectric constant $\kappa$ is inserted into one of them so that its new capacitance is $C'=\kappa\,C$. Since the total charge is conserved, but the capacitance of one of them has changed, the voltage also has to change, but they are connected in parallel so their new voltages are equal to each other. I.e. $$Q = 2CV = (\kappa\,C + C)V'$$. The new voltage is given by $$ V' = \frac{2}{\kappa+1}V$$ This means that the capacitors now have the following charges: $Q_1 = \kappa\,CV'$ and $Q_2=CV'$. Since these are both different from their original charges $Q=CV$, we know that $\Delta Q \ne 0$ and thus there was a current.

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Thank you a lot for your answer. Also another question born in my mind. Which direction will this current has when we have + opposite to + and - opposite to - ? – F.N Mar 24 '13 at 21:55

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