-Combinations of Capacitors-Energy Stored in a Charged Capacitor

AP Physics C

Mrs. Coyle

Circuit Symbols used in Circuit Diagrams

Resistor

Capacitors in Parallel

•The potential difference across the capacitors is the same.

•The flow of charges ceases when the voltage across the capacitors equals that of the battery.•The total charge of the capacitors in series is the sum of the individual charges.

1 2V V V

total 1 2Q = Q + Q

Equivalent Capacitance for Capacitors in Parallel

Ceq is greater than any of the individual capacitors C.

It is as if the areas of the capacitors were combined.

total 1 2Q Q Q

eq 1 2C V C V C V

eq 1 2C = C + C + ....

Capacitors in Series

Electrons are transferred from the left plate of C1 to the right plate of C2 through the battery

An equal amount of negative charge is removed from the left plate of C2, leaving it with an excess positive charge

All of the right plates gain charges of –Q and all the left plates have charges of +Q

total 1 2Q Q Q

Equivalent Capacitance for Capacitors in Series

1 2

1 1 1

eqC C C

The equivalent capacitance of a series combination is always less than any individual capacitor in the combination.

total 1 2Q Q Q

1 2V V V

Ex: Find the equivalent capacitance, the charge and the voltage across in one of the 2μF capacitors.

2 F

2 F3 F 4 F

Ans: 8 μF, 6 μC

6 V

Ex 26.4: Find the Equivalent Capacitance

Answer: 6F

Work to Charge a Capacitor The work needed to transfer a charge, q,

from one plate to the other through the battery is:

The total work required to charge the capacitor from q=0 to Q is:

qdW Vdq dq

C

0

21

2

Q qW dq

C

QW

C

Energy Stored in a Capacitor The work done in charging the capacitor is

stored as electric potential energy U:

This applies to a capacitor of any geometry. The energy stored increases as the charge

increases and as the potential difference increases.

At a certain maximum voltage, corona discharge occurs between the plates.

221 1

( )2 2 2

QU Q V C V

C

Energy and Energy density for a Parallel Plate Capacitor

Energy density (energy per unit volume)

2 2o

2o

1 AU = (d E )

21

AdE2

d

U

o

V Ed

AC

d

21

( )2

U C V

E

2E o

Uu =

Ad1

u = 2

E