Download Construction of capacitor and inductor

Construction of capacitor and inductor
Level: Secondary 6
Project aims:
The main purposes of this project are to let students:
1. Familiarize with the working principles of capacitor and inductor.
2.
Construct workable inductors and capacitors using simple components.
Learning objectives:
1.
2.
Manipulate materials, apparatus and tools for construction.
Apply knowledge and principles of physics to solve problems.
Task 1:
Students have to construct a capacitor and an inductor by using household materials
and/or elementary electrical components.
Task 2:
Students are asked to modify the capacitor and inductor made in Task 1 to achieve
capacitance and inductance to 50nF and 1mH (without ferrite) or 10mH (with ferrite)
respectively.
Apparatus/Materials:
Any appropriate apparatus and materials proposed by students are welcomed. Typical
examples include coil, aluminum foil, connecting wires, multi-meters* (for measuring
capacitance, inductance and resistance) and so on. Ferrite iron or soft iron bar can be
inserted into the solenoid to increase the inductance. Students can try any suitable
insulating materials like paper or plastic wrapping sheet as the dielectric of capacitor.
* If the school do not have such multi-meter students may have to use other method to
find the inductance and capacitance in that sense. Reference on this is shown in the
Appendix.
Pre-knowledge:
Capacitor
The capacitance of parallel plates capacitors are
C
 0 A
d
   [1]
where  is dielectric constant,   is permittivity constant equals to
8.85  10 12 Fm-1, A is overlapping area of two plates and d is separation
between two plates.
Inductor
Solenoid inductance is roughly given by
N B
   [2]
I
where N B  (nl )( BA) and n is the number of turns per unit length, l is
L
the length of the solenoid, B is the magnetic field inside the solenoid and A
is the cross-sectional area. Besides, magnetic field B of solenoid is given
by B  nI , where  is permeability and I is current. The relative
permeability is   k 0 where k is relative permeability,  0 is
permeability constant equal to 4  107 Hm-1. Finally, the inductance could
be expressed as L  n 2 lA .
Questions:
1.
2.
3.
4.
Using the same amount of material, why is the capacitance in a rolled up
cylindrical capacitor higher than that of parallel plate? Argue without using the
formula of capacitance of cylinder.
A company wants to produce the capacitor with highest capacitance and lowest
cost. Can you help them to design the shape and material that should be used?
Inside the coil, soft iron bar or ferrite can be put into it to increase the inductance.
Try other material and argue if they contain iron.
Can you mathematically show that what shape of the coil can produce the
maximum amount of inductance given the same length of wire? Justify your
answer by wrapping a few coils.
Extension:
Based on your constructed capacitor and make use of a multimeter that can measure
capacitance to make a digital balance.
If a multimeter can measure the capacitance of the capacitor, students can make use of
this multimeter to make a digital balance. The capacitor similar to Figure 5 is
sensitive to pressure. Adding mass on the book will change the capacitance and it can
be shown in the digital balance. By putting some known mass on the book and make a
table, students can plot a graph with capacitance against mass. Putting the value of
capacitance in the graph and extrapolate the value of the mass, the digital balance is
ready to use.
Appendix:
Measuring capacitance / inductance by measuring time constant
R
CRO
Signal
Generator
C
Figure 9: Circuit of measuring capacitance without meter
Figure 9 shows a simple RC circuit. The CRO measure the voltage across the
capacitor. Signal generator can charge capacitor or switch to discharge capacitor. The
equation which describes the potential difference across the capacitor VC (t ) in the
circuit of Figure 9 is of the form:
VC (t )  (Vo  V f )e

t
TC
 V f    [1]
Using square wave signal with 8-volt peak-to-peak
voltage, the signal generator can charge the capacitor
and then discharge it. The charging cycle and the
discharging cycle could be observed, as shown in
Figure 10. Now assuming the power supply is +4V,
then Vo would be -4V and Vf = 4V for charging cycle,
and vice versa for discharging cycle.
Voltage
where Vo is initial voltage, Vf is final voltage and Tc is time constant, Tc = RC.
Discharging
Cycle
Charging
Cycle
Time
Figure 10: Charge and
The resistance should be confirmed by voltmeter and
ammeter.
discharge cycle
(Question: Why shall the resistance be measured by voltmeter and ammeter but not
directly through a ohmmeter?)
The resistance should be confirmed by voltmeter and ammeter so that the internal
resistance of the whole circuit can be found.
Compute the time constant
times, as shown in Figure 11. Moving the last term to
the left of the equal sign and dividing these two
measurements yields:
e
( t 2  t1 )
TC

IC
t1 , v(t1)
Voltage
Equation [1] can be used to solve for the time constant
TC by measuring the voltage VC (t ) at two different
t 2 , v(t 2 )
FC
vC (t1 )  V f
Time
vC (t 2 )  V f
Figure 11: Discharge cycle
In this case, solving for the time constant TC is accomplished by taking the natural
logarithm of both sides and rearranging to produce the result:
TC 
ln(
t 2  t1
v(t1 )  V f
v(t 2 )  V f
)
Students may also use charging cycle, but to rearrange the terms of equation [1]
carefully. The time constant for this circuit is given by the equation TC  RC . Students
may use this equation to compute the capacitance, C.
The method of finding inductance is the same by replacing capacitor with inductor.
The equation of time constant is TL  L / R .
Measuring capacitance and inductance by measuring the phase difference
Another method to measure the capacitance and inductance is by measure the phase
difference.
In a circuit diagram similar to Figure 9, we can measure the voltage across the resistor
and across the whole signal generator. The signals can be seen in a CRO like Figure
12. A phasor diagram is constructed in Figure 13.
2
0
6
VC
E
4
VR
2
Ф
VC
0
-2
-4
VR
E
0
2
Figure 12: CRO Image
Figure 13: Phasor Diagram
In Figure 12 and 13, the potential difference across resistor VR, and e.m.f. E are
labeled with red and blue respectively. As a reference potential difference across
capacitor VC is also included and labeled with dotted green line. The e.m.f. of the
circuit is equal to the vector sum of potential difference of capacitor and resistor:
E  VC  VR    [2]
The phase difference  between resistor and e.m.f. is
tan  
VC
1

   [3]
VR 2fCR
The phase difference can be found by measuring the distance between the peak of VR
and E. The phase difference is the ratio this distance and that of one period. By
measuring the phase difference, the capacitance C can be found by equation [3].