Download Capacitance graphs support

23 Capacitors
Support
AQA Physics
Capacitance graphs
Specification reference


3.7.4
MS 0.1, 0.3, 0.5, 2.2, 2.3, 2.4, 3.1, 3.8, 3.10, 3.11, 3.12
Introduction
Capacitors are electronic components that store charge and so can be used in time
delay circuits. For example, as you open your front door there is a delay (giving you
a chance to enter the correct code) before the burglar alarm goes off. Time delay
circuits work with a combination of a capacitor and a resistor. The larger the values
of resistance and capacitance in the circuit, the longer the time delay will be.
Figure 1
Capacitors come in different shapes and sizes. They can be two parallel plates with
air or another insulating material called a dielectric between them, or cylindrical
where the two ‘plates’ and the dielectric are effectively rolled up rather like a ‘swiss
roll’ − visualise this as the ‘cake’ being the plates and the ‘filling’ the dielectric and
charges sandwiched between.
Capacitors can be charged and discharged and the graphs produced are
exponential growth or decay.
© Oxford University Press 2016
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
1
23 Capacitors
Support
AQA Physics
Learning outcomes
After completing this worksheet you should be able to:





explain the meaning of capacitance
use the capacitance equation
explain the similarities and differences between graphs showing charging and
discharging of a capacitor and be able to sketch them all
explain the meaning of the time constant and be able to find it from the equations
and from a graph
find the energy stored in a capacitor from the equations and also from a graph.
Background
Capacitance
Units: charge (coulombs, C), capacitance (farads, F)
potential difference (volts, V)
Capacitance is often quoted in microfarads (μF)  10–6 F
Figure 2
Energy stored in capacitor =
1
1
1 Q2
QV = CV 2 =
2
2
2 C
Capacitor discharge
Q  Q0 e
t
– RC
Figure 3
© Oxford University Press 2016
V  V0 e
Figure 4
t
– RC
I  I0 e
t
– RC
Figure 5
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
2
23 Capacitors
Support
AQA Physics


All curves in Figures 1 to 3 have the same shape.
RC is known as the time constant  time taken for charge to drop to e−1
 0.37 of initial value

After a time RC the voltage, charge, and current will all have been reduced to
0.37 of their original value.
As the discharge of a capacitor is exponential you could be asked to plot a
ln graph.
–
t
V  V0 e RC
Taking logs of both sides
ln V0
t
RC
Comparing with y  mx  c
Intercept on ln V axis  ln V0
ln V  ln V 0 -
Figure 6
Gradient  −
1
RC
Charging a capacitor
V  V0 (1– e
t
– RC
Q  Q0 (1– e
)
Figure 7

Figure 8
t
– RC
I  I0 e
)
Figure 9

The Q and V curves in figures 7 and 8 have same shape but I in Figure 9 is
opposite.
As a capacitor is charged, charge is flowing so Q increases but due to
electrostatic repulsion, current I ( charge flowing per second) decreases.
Current  0 when capacitor is fully charged.

I =

t
– RC
Q
= gradient of Q curve.
t
© Oxford University Press 2016
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
3
23 Capacitors
Support
AQA Physics
Worked examples
Example 1: Energy stored  area under V–Q graph
When fully charged by a 20 V DC supply a capacitor carries a charge of 5.0 μC.
Calculate:
a the capacitance
b the energy stored in the capacitor.
Q  5.0 μC
Step 1 Use C 
=
V  20 V
C?
Energy  ?
Q
and substitute in values.
V
5.0 ´10-6
20
 0.25 × 10−6 F or 0.25 μF
Step 2 Use Energy stored 
QV (or you could use

CV 2 if you wish).
1
× 5.0 × 10−6 × 20
2
 5.0 × 10−5 J
OR
The information from the same question could have been represented graphically as
shown below.
Figure 10
In this case,
a
Capacitance 
1
gradient
5.0 ´10-6
20
 0.25 × 10−6 F or 0.25 μF
=
b Energy stored  area under graph
1
× 20 × 5.0 × 10−6
2
 5.0 × 10−5 J

© Oxford University Press 2016
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
4
23 Capacitors
Support
AQA Physics
Example 2: Discharging Q and V graphs
In the circuit in Figure 11 a fully charged capacitor has a potential difference of 6.0 V
across it.
The switch is then closed.
Calculate:
a
b
c
d
the time constant
the pd across the capacitor after 20 s
the charge on the capacitor after 20 s.
Show all of this information graphically. (You
will need to find initial charge Q0 and calculate
the values of V and Q when t  time constant.)
Figure 11
a
R  100 kΩ C  500 μF
Step 1 Time constant  R × C
 100 × 103 × 500 × 10−6
 50 s
b V0  6.0 V t  20 s RC  50 s
V decreases exponentially
Step 2 Use V  V0 e
t
– RC
 6.0 e
.
20
– 50
 6.0 e −0.4
 4.0 V
c At t  20 s V  4.0 V C  500 × 10−6 F
Step 3 Use Q  CV.
 500 × 10−6 × 4.0
 2.0 × 10−3 C
d Graphically
Initially charge Q0  CV  500 × 10−6 × 6.0  3.0 C
When t  RC  50 s, V  0.37 V0, and Q  0.37 Q0
Figure 12
© Oxford University Press 2016
Figure 13
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
5
23 Capacitors
Support
AQA Physics
Example 3: ln graph
A 10 μF capacitor is charged to a potential difference of 12 V and then discharged
through a 100 kΩ resistor.
a
Calculate the time taken before the potential difference across the capacitor
drops to 1 V.
b Show this information graphically.
V0  12 V V  1 V R  100 kΩ C  10 μF t  ?
Step 1 Calculate RC  100 × 103 × 10 × 10−6  1 s
Step 2 Use V  Vo e
t
– RC
–t
1  12 e 1
Step 3 Take ln of both sides
ln 1  ln 12 − t
Step 4 ln 1  0
t  ln 12
t  2.5 s (to two significant figures)
Figure 14
Questions
1
a
Calculate the charge stored on the plates of a capacitor of capacitance 8 μF
when it is connected to a 2.0 V battery.
(2 marks)
b Calculate the energy stored in the capacitor.
(2 marks)
c
Show this information graphically.
(2 marks)
© Oxford University Press 2016
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
6
23 Capacitors
Support
AQA Physics
2
A 1000 μF capacitor is charged to 5000 V, disconnected from the power supply
and then allowed to discharge through a 100 kΩ resistor. Calculate the time
taken for the voltage across the capacitor to fall to:
a 200 V
(3 marks)
b 100 V
(2 marks)
c
1V
(2 marks)
3
A 2500 μF capacitor is charged through a 1 kΩ resistor by a 12 V power supply.
Calculate the voltage across the capacitor after 5 s.
(2 marks)
Graphical tasks
A 200 mF capacitor is charged to 10 V and then discharged through a 250 kΩ
resistor.
a Calculate the time constant RC.
b Calculate the pd across the capacitor at intervals of 10 s and complete the
second row of Table 1 using V  V0 e
t
– RC
t/s
0
10
20
V/V
10
8.2
6.7
ln (V / V)
2.303
I /μA
40
Q /mC
2.0
.
30
40
50
60
70
Table 1
© Oxford University Press 2016
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
7
23 Capacitors
Support
AQA Physics
c
d
e
f
g
Draw a V against t graph or produce one using a spreadsheet.
Mark on the graph the value of V when t  RC.
Complete the third row of the table.
Draw a ln V against t graph or produce one using Excel.
Determine the gradient. What does this represent?
V0
and complete the fourth row of the table.
R
i Calculate Q (remember Q0  CV0).
j Draw a Q against t graph or produce one using Excel.
k Determine the gradient at t  10 s. What does this represent?
l Draw a I against t graph or produce one using Excel.
h Calculate I (remember I0 
© Oxford University Press 2016
http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
8