Capacitance II

(a) Parallel combination of capacitors-Capacitors connected in parallel combination have same potential difference across their terminals shown below in fig

where
V=potential difference across terminals of capacitors c1, c2, c3
and Q1, Q2 and Q3 resp. are charges on capacitors c1, c2, c3.
-In parallel combination system of capacitors is equivalent to a single capacitor of capacitance
C=Q/V= C1+C2+C3
where, C1=Q1/V
C2=Q2/V
C3=Q3/V
-Thus for capacitors connected in parallel combination their resultant capacitance C is the sum of their individual capacitances.
-Also for parallel combination of capacitors their resultant capacitance C is greater then the capacitance of greatest individual one.





(b) Series combination of capacitors
-Figure above shows three capacitors connected in series combination
-In series combination of capacitors potential difference across each capacitor is different but charge on each capacitor is same.
-The resultant capacitance of capacitors connected in series combination is the ratio of charge stored to the applied potential difference V and is given by
C=Q/V
or 1/C=V/Q
now V=V1+V2+V3
=Q(1/C1+1/C2+1/C3)
Thus,
1/C=1/C1+1/C2+1/C3
-Resultant capacitance Cof the capacitors connected in series combination is equal to the sum of reciprocals of their individual capacitances. Here in case of series combination C is less then the capacitance of smallest individual capacitor.

(c) Energy stored in capacitor-Energy stored in capacitor is
E=QV/2
or E=CV2/2
or E=Q2/2C
factor 1/2 is due to average potential difference across the capacitor while it is charged.
-Battery supply QV amount of energy during charging a capacitor but energy stored in capacitor is QV/2 , the another half of energy is transferred into the circuit resistance in the form of heat.
thus,
heat in the wire=energy supplied by battery-energy stored in the capacitor
=QV-QV/2 = QV/2

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