CHAPTER 4 : CAPACITOR

34Topic --- Capacitor

4.1 Capacitance & Capacitors In Series
& Parallel

4.2 Charging & Discharging Of
Capacitors

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34Topic --- Capacitor

OVERVIEW

Capacitor Capacitance In series
In parallel
Capacitors

Charging & discharging
of capacitor

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34Topic --- Capacitor

4.1 CAPACITANCE &
CAPACITORS IN SERIES &

PARALLEL
(a) Define and use capacitance,

(a) Find the effective capacitance of capacitors
of various arrangements by using the following
formulae:
Series : 1/ Ceff = 1/C1 + 1/C2 + 1/C3 + …
Parallel : C eff + C1 + C2 + C3 + ….

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34Topic --- Capacitor

Capacitor
(Condenser)

A device that is Consist of two Uses: Photo-
capable of conducting flash, on-off
storing electric plates separated switches,
charges or by a small air gap smoothen
electric or a thin direct current
potential insulator (d.c) voltages
energy (dielectric) such
as mica, ceramics OR
or even oil
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34Topic --- Capacitor

Capacitance, C – Measurement of the
ability of capacitor to store charges

the ratio of the charge on either plate to the Scalar quantity,
potential difference between them Unit: CV– 1 or F

When the The charges stored, Q The greater 1 farad, F = the charge of
capacitance of a is directly capacitance of a 1 coulomb stored on
capacitor, C is each of the conducting
proportional to the capacitor, the plates as a result of a
constant, potential difference, more charge is
potential difference of 1
V across the required volt between the two
conducting plates plates

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43Topic --- CaTpheamcaigtnoitrude of the charge Q on each plate is the

Capacitors same
In Series

C1,V1 The potential difference across each capacitor C1, C2
C2,V2
C3,V3 +Q1 and C3 are V1,V2 and V3 respectively
−QI

+Q2
−Q2 The total potential difference V is given by
+Q3
−Q3

Ceff, +Q Where and
V
−Q capacitors 6
in series

3CaToppiac c4itor-s-- CaTphaecpitooterntial difference across each capacitor

In Parallel is the same as the supply voltage (V)

+Q1 +Q2 +Q3 Ceff, +Q
−QI −Q2 −Q3 V −Q
C1,V1 C2,V2 C3,V3

The charges stored by each capacitor C1,C2 and C3 are Q1,Q2 and Q3 respectively

The total charge Q on the effective capacitor is given by

where 7
and capacitors
in parallel

34Topic --- Capacitor

Example 4.1

Calculate the total capacitance in (a), (b) and (c).

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34Topic --- Capacitor

Example 4.2

Determine the potential difference across and
charges on capacitors X, Y and Z.

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34Topic --- Capacitor

• Capacitors Y and Z are in
parallel, so VY = VZ

Thus V = VX + VY,
VY = 12 – 4 = 8 V
VZ = 8V

For capacitor Y,
QY = CYVY = 8 x 10-6 C

For capacitor Z,
QZ = CZVZ = 1.6 x 10-5 C

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34Topic --- Capacitor Example 4.3

Determine the Figure 4.1
effective capacitance OR
of the configuration
shown in Figure 4.1. All 11
the capacitors are
identical and each has
a capacitance of 2 μF.

34Topic --- Capacitor

Solution:

⚪ Label all the capacitors in the circuit.

⚪ To calculate the
effective capacitance,

it is easier to solve it
from the end of the

circuit (left) to the
terminal (right).

⚪ Capacitors C1, C2 and C3 are connected in series, then

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34Topic --- Capacitor

⚪ Capacitors Cx and C4 are connected in parallel, then

⚪ Capacitors Cy, C5 and C6 are connected in series, then

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34Topic --- Capacitor

⚪Capacitors Cz and C7 are connected in parallel, then
the effective capacitance Ceff is given by

OR

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34Topic --- Capacitor

In Figure 4.2, C1= 100 μF, C2 = A Example 4.4
200 μF and C3 = 300 μF. The B
applied potential difference D
between points A and B is VAB
= 8.0 V. Calculate Figure 4.2

(a) the charge on each
capacitor

(b) the potential difference
across each capacitor

(c) the potential difference
between points A and D.

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34Topic --- Capacitor

Solution:

AA D

D

BB
a. Capacitors C1 and C2 are connected in parallel then C12 is

Thus the effective capacitance Ceff in the circuit is given by

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34Topic --- Capacitor

a. The total charge Q stored in the effective capacitance Ceff is
Since the capacitors C12 and C3 are connected in series then the
charge stored in each capacitor is the same as the total charge.

The potential difference across the capacitor C3 is

thus the potential difference across the capacitor C12 is given by

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34Topic --- Capacitor

a. Since the capacitors C1 and C2 are connected in parallel then
the potential difference across each capacitor is the same as
V12.
Therefore

and

b. The potential difference across each capacitor is given by
c. The potential difference between points A and D is given by

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34Topic --- Capacitor

Example 4.5

Figure 4.3 shows a combination of three capacitors where C1=
100 μF, C2 = 22 μF and C3 = 47 μF. A 20 V
supply is connected to the combination.

Determine

(a) the effective capacitance in the circuit,

(b) the charge stored in the capacitor C1,
(c) the potential difference across the

capacitor C2, Figure 4.3

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34Topic --- Capacitor

Solution:

a. Capacitors C2 and C3 are connected in series then C23 is

Therefore the effective capacitance, Ceff is given by

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34Topic --- Capacitor

b. Since the capacitors C1 and C23 are connected in parallel, thus
Hence the charge stored in the capacitor C1 is

c. The total charge stored in the circuit is given by

Thus the charge stored in the capacitor C23 is
The capacitors C2 and C3 are connected in series, thus

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34Topic --- Capacitor

c. Therefore the potential difference across the capacitor C2 is

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34Topic --- Capacitor

C = Q/V Unit:
Q = CV
C V-1 @
Farad

Capacitance - To store
Measurement of the charges
ability of capacitor

to store charges

Capacitor

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34Topic --- Capacitor

Example 4.6

Consider the circuit shown in Figure 4.4, where C1= 50 μF, C2 =
25 μF and V = 25.0 V.

Capacitor C1 is first charged by
closing a switch S1. Switch S1 is
then opened, and then the

charged capacitor is connected

to the uncharged capacitor C2 by

closing a switch S2. Figure 4.4

Calculate the initial charge acquired by C1 and the final charge

on each capacitor.

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34Topic --- Capacitor

Solution:
Switch S1 is closed:
When the capacitor C1 is fully charged, the charge has been
placed on its plate is given by

Switch S2 is closed and S1 is opened:
The capacitors C1 and C2 (uncharged) are connected in parallel
and the equivalent capacitance is

By using the principle of conservation of charge, the total charge
Q on the circuit is given by

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34Topic --- Capacitor

Solution:
Switch S1 is closed:
When the capacitor C1 is fully charged, the charge has been
placed on its plate is given by

Switch S2 is closed and S1 is opened:
The capacitors C1 and C2 (uncharged) are connected in parallel
and the equivalent capacitance is

By using the principle of conservation of charge, the total charge
Q on the circuit is constant, where

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34Topic --- Capacitor

Solution:
The potential difference across each capacitor is the same
(parallel connection) and given by

Therefore the final charge accumulates on the
capacitor C1 :

capacitor C2 :

OR

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34Topic --- Capacitor

EXERCISE 3.1

Given ε0 = 8.85 × 10−12 C2 N−1 m−2
1. Four capacitors are connected as shown in Figure 4.5.

Calculate Figure 4.5

(a) the equivalent capacitance
between points a and b,

(b) the charge on each capacitor if
Vab=15.0 V.

(Physics for scientists and engineers,6th
edition,Serway&Jewett, Q21, p.823)

ANS: 5.96 μF; 89.5 μC on 20 μF, 63.2
μC on 6 μF, 26.3 μC on 15 μF and on 3

μF

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34Topic --- Capacitor

2. Determine the equivalent capacitance between
points a and b for the group of capacitors
connected as shown in Figure
4.6.
Take C1 = 5.00 μF, C2 = 10.0
μF and C3 = 2.00 μF.

(Physics for scientists and engineers,6th
edition,Serway&Jewett, Q27,
p.824)
ANS: 6.04 μF

Figure 4.6

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34Topic --- Capacitor

3. An electronic flash unit for a camera contains a capacitor of
capacitance 850 μF. When the unit is fully charged and ready for
operation, the potential difference across the plates is 330 V.
What is the magnitude of the charge on each plate of the fully
charged capacitor?
(Physics,3rd edition, J.S Walker, Q59, p.692)
ANS: 0.28 C

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34Topic --- Capacitor

4.2 CHARGING &

31 DISCHARGING OF CAPACITORS

(a) State physical meaning of time constant, and use

τ = RC

(a) Sketch and explain the characteristics of Q-t and I-t
graph for charging and discharging of a capacitor

(b) Use for discharging and

for charging
(d) Determine the time constant of an RC circuit

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34Topic --- Capacitor

Charging & Discharging of Capacitor

Charging Discharging

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34Topic --- Capacitor

Charging a Capacitor

Time constant, τ
• Scalar

quantity
• Unit: s
•A

measurement
of how
quickly the
capacitor
charges or
discharges

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34Topic --- Capacitor

Potential Difference (Voltage) Charging
Across Charging Capacitor

• The voltage V across the • τ is defined as the time

capacitor, increase from required for the capacitor to
reach (1− e−1) = 0.63 or 63% of
zero at t = 0 to maximum its maximum voltage

values V0 after a very long
time

and

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34Topic --- Capacitor

Charge on Charging Capacitor Charging

• The charge Q across the • τ is defined as the time
required for the capacitor to
capacitor, increase from reach (1− e−1) = 0.63 or 63% of
its maximum charge
zero at t = 0 to maximum

values Q0 after a very long
time

and

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34Topic --- Capacitor

Current In Resistor Charging

36 • τ is defined as the time
required for the current
• the current drops drops to 1/e = 0.37 or 37% of
exponentially in time constant its initial value(I0)

36 τ

and

Q0: maximum charge
V0: maximum (supply) voltage

I0: maximum current
R: resistance of the resistor
C: capacitance of the capacitor

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34Topic --- Capacitor Discharging

Potential Difference (Voltage)
37Across Discharging Capacitor

Discharging: V-t, Q-t & I-t graph

• the charge Q, the voltage V
and the current I is seen to
decrease exponentially in
time constant τ

• τ is defined as the time
required for the charge on
the capacitor (or voltage
across it or current in the
resistor) decreases to 1/e =
0.37 or 37% of its initial value

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34Topic --- Capacitor Discharging

Discharging a Capacitor Current through resistor

38 The negative sign indicates that as the
capacitor discharges, the current
Charge on capacitor
direction opposite its direction when
the capacitor was being charged

The charge on the capacitor The current through the resistor
decreases exponentially with time decreases exponentially with time

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34Topic --- Capacitor

Example 4.7

In the RC circuit shown in Figure 4.5, the battery has

fully charged the capacitor. At time t = 0 s, a switch S is

thrown from position a to b. The battery voltage V0 is
12.0 V and the capacitance

C = 3.00 μF. The current I is

observed to decrease to 0.45

of its initial value in 60 μs.

Determine

(a) the value of R. Figure 4.5

(b) the time constant, τ

(c) the value of Q, the charge on the capacitor at t = 0.

(d) the value of Q at t = 100 μs

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34Topic --- Capacitor

Solution:
a. By applying the equation of current for discharging process,

Then by taking natural logs on both sides, thus the value of R is

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34Topic --- Capacitor

b. The time constant is given by

c. By using the equation of charge for discharging process and
the time, t = 0 hence

and

d. By using the equation of charge for discharging process and
the time, t = 100 × 10−6 s hence

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34Topic --- Capacitor

EXERCISE 3.2

1. a. A parallel-plate, air-filled capacitor has circular

plates separated by 1.80 mm. The charge per unit area
on each plate has magnitude of 5.60 pC m−2. Calculate

the potential difference between the plates of the

capacitor.

(University physics,11th edition, Young&Freedman, Q24.4, p.934)

b. An electric field of 2.80 × 105 V m−1 is desired

between two parallel plates each of area 21.0

cm2 and separated by 0.250 cm of air. Determine the

charge on each plate. (Physics for scientist & engineers ,3rd

edition, Giancoli, Q14, p.628) ANS: 1.14 mV; 5.20 × 10−9 C

2. When the potential difference between the plates of a

capacitor is increased by 3.25 V, the magnitude of the
charge on each plate increases by 13.5 μC. What is

the capacitance of this capacitor?

(Physics,3rd edition, J.S.Walker, Q86, p.694) ANS: 4.15 μF

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34Topic --- Capacitor

3. A 10.0 μF parallel-plate, air-filled capacitor with circular
plates is connected to a 12.0 V battery. Calculate
a. the charge on each plate.
b. the charge on each plate if their separation were twice
while the capacitor remained connected to the battery.
c. the charge on each plate if the capacitor were
connected to the 12.0 V battery after the radius of each
plate was twice without changing their separation.

(University physics,11th edition, Young&Freedman, Q24.5, p.934)
ANS: 120 μC; 60 μC; 480 μC

4. A capacitor stores 100 pC of charge when it is connected
across a potential difference of 20 V. Calculate
a. the capacitance of the capacitor,
b. the amount of charge to be removed from the
capacitor to reduce its potential difference to 15 V.

ANS: 5.0 pF; 25 pC

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34Topic --- Capacitor

Next Chapter…

CHAPTER 5 :
Magnetism and Electromagnetic

Induction

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