Lab 2-Capacitance v4

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					Electric and magnetic fields lab                                   capacitance and permittivity




Lab 2. Measurement of capacitance and permittivity of vacuum.
   Effect of dielectrics. Measurement of relative permittivity.

Objectives
      To determine the capacitance of a parallel-plate capacitor.
      To study the effect of a dielectric on capacitance.
      To measure permittivity of air and permittivity of acrylic.
      To understand the concept of time constant of an RC circuit, and to test it with a number
       of resistor values.
      To learn how to use a multifunction oscilloscope


Apparatus
      Digital Multimeter or LCR




                                              Figure 1   DMM

      Oscilloscope with function generator: Agilent InfiniiVision MSO-X 2002A Mixed Signal
       Oscilloscope (with built-in pulse generator)




                                   Figure 2     The oscilloscope

      a pair of capacitance plates (40cm x 40cm, 2mm thick aluminium)
      sheets of acrylic, 2mm thick, dimensions 40 x 40 cm
      non-conducting washers (1mm thick), plywood 3mm spacers
      decade resistor box
      resistors (1M5M 10M capacitor)
      breadboard
      Cables: BNC-BNC cable (1), BNC splitter (1), BNC-Clippers cable (2)




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Electric and magnetic fields lab                                capacitance and permittivity




Preparation for the lab:
Read and understand chapter “Capacitance and Dielectric Materials” of Schaum’s Outlines –
Electromagnetics.

Make appropriate notes for all the experiments, including of relevant procedures and
formulae, in your lab book.

Find relative permittivity for acrylic from available sources and make note of it in the lab book.

Prepare your spreadsheet for experiments 1 and 2. That means, prepare a table for entries
of measured values of capacitance for the two cases, for a range of separations d, with entries
for measured values for permittivity, and measurement errors. (For capacitor with air the
separations will vary from 1mm to 5 mm, in 1mm steps. For acrylic, separations are multiples of
2mm: 2, 4, and 6 mm. )

Include unit conversions: The capacitances you measure will be in nF or pF, and the distances
in mm or cm. Think and make notes:
    - Should your graph go through (0,0)?
    - Will the errors be larger for small or large plate separations?

Ensure your spreadsheet has graphs properly prepared (including graphs of residuals); all
quantities labeled with units; all calculations of results and of errors properly incorporated. That
means, plot the function of C versus 1/d for each of the two cases of capacitors: gap filled with
air and gap filled with dielectric.

Write the formulae (including for errors) explicitly in your notebook.

For experiment 3, read up on time constant of an RC circuit, make sure you can define it and
understand it. You will need to demonstrate your understanding to the instructors.

Calculate the capacitance of the parallel plate capacitor (dimensions given under Apparatus
section at the beginning of the lab sheet), for a capacitor filled with acrylic, with separation
d=2mm. Then calculate the time constant for an RC circuit for this capacitor and 3 different
resistor values: 1M, 5M, and 10M. Make a table of calculated values in your lab book
(make sure you have the correct units):

     R      Time constant,         Measured      Capacitance         Amplitude Vc
                            capacitance    measurement         
            Calculated Measured       [units]        error               (measured)
  1M
  5M
  10M


Write an equation for finding capacitance from a measured time constant.




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Electric and magnetic fields lab                                 capacitance and permittivity




Theory required for analysis

                          + + + + +            + + + + +

                   V                                    Field
                                                        E
                          _ _ _ _ _            _ _ _ _ _
                          _                    _

From the diagram, if there is a charge density of  coulombs/m2 on the top plate, an equal and
opposite density will be induced on the bottom plate. By Gauss' Law, an electric field E will be
produced between the plates, where



o is the permittivity of free space (8.854 x 10-12 farad m-1) and is the relative permittivity of
the medium between the plates (1 for air).

Since E is uniform and perpendicular to the plates, there will be a potential difference V
between the plates which is given by

                                           d
                                V  Ed         ,
                                            0
where d is the distance between the plates. The total charge on each plate is

               Q = A ,

where A is the area of the plate; therefore, for a given Q, the potential difference V established
between the plates is given by
                                     Qd
                                V         .
                                     A 0
Hence the capacitance C is

                             Q             1
                        C       0 A  .                                      (1)
                             V             d

This is a measure of the amount of electrical energy that can be stored in the space between
the plates.

Note:
        1) This value depends only on the dimensions and distance of the plates and the
        electrical properties of the medium between them.
        2) The calculation assumes that there are no “edge effects” due to the plates
        having a finite size.




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Electric and magnetic fields lab                                  capacitance and permittivity




Experiment 1. Capacitance of a parallel plate capacitor.
              Permittivity of air 0.
Procedure:

Set up the parallel plate capacitor with the two plates separated by non-conducting washers.
Start with minimum separation. Measure the capacitance using the DMM or the LCR supplied,
and make a table of capacitance as a function of spacing between the plates. (You should use
the table you prepared in advance as part of the lab preparation.) Measure the area A of the
plates.




                     Parallel-plate capacitor. Figure courtesy http://www.jfxiw.com

The capacitance is measured with a digital multimeter or LCR: be sure to put the wires into the
correct socket – ask technician or the TA if you are not sure. (These sockets are designed to
make it easy to plug in commercial capacitors.)


Results

Task 1.1:From your table of measured values, plot a graph of C as a function of 1/d and
calculate the value of o. Compare it with the accepted value for electric permittivity of air.
(NB. For air, is very close to 1).

Task 1.2:Considering Note 2 above, draw conclusions as to whether edge effects are
important in this case; and, if they are, whether they are a function of d.
(Hint: given the values of A and d, you can plot the theoretical expression, equation 1, on the
same axes as your experimental results and compare them).

Task 1.3. Discuss what contributes to measurement errors, and which contribution is likely to
be dominant. Explain your answer.




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Electric and magnetic fields lab                                    capacitance and permittivity




Experiment 2. The relative permittivity of acrylic (plastic glass)
Procedure




              Parallel-plate capacitor with dielectric. Figure courtesy http://www.jfxiw.com




Repeat the previous experiment, separating the plates with varying thicknesses of acrylic (by
placing 1, 2 or 3 sheets between the aluminium plates) as shown in the picture above.

Results

Task 2.1: Plot a graph as before and calculate the relative permittivity of acrylic.Compare
with available data.

Question: What is your measurement error?


Experiment 3. Time constant of an RC circuit
Procedure

You will use a function generator and an oscilloscope to observe the charging and discharging
of the capacitor, to measure its time constant, and to measure the capacitance from the time
constant. You will then alter some circuit parameters and observe the effects on the signal.

                                             Leave a single 2mm acrylic sheet between the plates.
                                             Connect the function generator and the 1M resistor
                                             (on a breadboard) in series with the capacitor using
                                             available BNC cables and T-connector – see the
                                             picture below.

                                             In the picture (left), “A” is the output of the function
                                             generator. It provides the signal to the RC circuit. It is
                                             connected to two leads via a BNC T-connector. One lead
                                             provides signal to the RC circuit, and the other BNC cable
                                             connects the function generator to the Channel 1 of the
                                             oscilloscope, port “B” on the picture. Channel 2 of the
                                             scope is port “C”: it measures the signal across the
                                             capacitor.



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Electric and magnetic fields lab                               capacitance and permittivity



Use one set of leads to observe the signal from the function generator. Use the other set of
leads to connect the oscilloscope in parallel with the capacitor, thus to observe the signal
across the capacitor. Turn on the function generator.
Set the wave generator to “PULSE”, frequency to 100Hz, amplitude to 1V (i.e. 2Vpp), and pulse
width to 4ms.
This frequency should allow you to view how the capacitor charges as the polarity of the signal
is switched. Every time the voltage supply to the capacitor changes polarity, the capacitor will
discharge and re-charge with the opposite polarity.


Results

To keep your work tidy and readable, you will need to present your results in a table or
in clearly defined sections.

Task 3.1. Make note of the signal (voltage) amplitude across the capacitor, Vc. (You will
need to do this for each measurement in this experiment.) Change the frequency up and down,
and observe what happens to the signal. Then set the frequency at which the voltage across
the capacitor can reach its maximum. It should be around 100Hz.

Now measure the time constant from the oscilloscope reading of the voltage signal across
the capacitor. Be prepared to explain to the demonstrator how you have done this.
Calculate the capacitance using the known value of the resistor and the measured time
constant. Compare the calculated and measured values for the given RC combination.

Question 3.1: How big is the error, and what are its causes?

Repeat the above measurements for other two resistor values. To do that, you will need to use
a breadboard.

Task 3.2. Sketch the signal from the function generator and the signal measured across the
capacitor, clearly marking the time and amplitude axes, for one of the resistors.

Task 3.3. Explain the change in amplitude across the capacitor Vc when you change resistors.
Confirm with calculations. You can present that as a table.




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