CAPACITORS

OBJECTIVES:

·         To understand how a parallel plate capacitor works.

·         To determine the dielectric constant for virtual paper used as the dielectric in virtual capacitor.

·         To learn how capacitors connected in series and in parallel behave.

·         To find equivalent capacitance for a complex combination of virtual capacitors.

EQUIPMENT:

Computer with Internet access.

INTRODUCTION AND THEORY:

Very common and important components in modern electrical devices are capacitors. They are what makes computer memory work. They are used with resistors in timing circuits, they occur as filters and coupling elements in every radio and TV set, they can eliminate ripples or spikes in DC voltages, they are used in flash units in photography.

A capacitor is considered a passive electronic element because it does not actively affect electrical currents in the circuit nor produce electrons or energy. Instead, it is used to store electrical charge (and hence electrical energy). A capacitor consists of a pair of conductors separated by a non-conductive material (or region) called dielectric which effectively insulates the two conductors.

The schematic symbol of a capacitor has two parallel vertical lines (representing the conductors) set a small distance apart, and two horizontal connecting wires:

                                                                         

Any two conductors separated by an insulator (including vacuum) form a capacitor. When a voltage (potential difference) is applied across the capacitor, a charge is transferred to and from the conductors and an electric field develops in the dielectric. This field stores energy and produces a mechanical force between the conductors which can be released when needed. When a capacitor is charged the two conductors hold equal in magnitude but opposite in sign amount of charge so the net charge on the capacitor as a whole remains zero.

The capacitor’s ability to hold an electric charge is characterized by capacitance C defined as the ratio of the magnitude of the charge Q on either conductor to the magnitude of the potential difference ∆V between the conductors:

                                                                                                                                                 [Eqn. 1]

The SI unit of capacitance is one farad (1 F). One farad is equal to one coulomb per volt.

                                               1 F = 1 farad = 1 C/V = 1 coulomb/volt

The greater the capacitance C of a capacitor, the more charge Q can be stored on either conductor for a given potential difference ∆V and hence greater the amount of stored energy. One farad is a very large unit and we rarely see capacitors this big. In many applications the most convenient subunits of capacitance are the millifarad (1mF = 10^-3 F), the microfarad (1µF = 10^-6 F), the nanofarad (1nF = 10^-9 F) and the picofarad (1pF = 10^-12 F).

The value of the capacitance depends only on the shapes and sizes of the conductors and on the nature of the insulating material between them.  The simplest way of making a capacitor is to build a unit with two parallel conducting fully overlapping sheets, each with area A, separated by a thin layer of air of thickness d. 

Figure 1. Design of a parallel-plate                   capacitor.

                                             

This arrangement is called a parallel-plate capacitor and its capacitance approximately (when d is small compared to the other dimensions and the field fringing effect around the periphery provides a negligible contribution) equals:

                                                                    C₀ =  ,                                                  [Eqn. 2]

where is the permittivity of free space = 8.854^-12 F/m. Please, note that A represents the area of overlap of the conducting surfaces.

The capacitance of a parallel plate capacitor is directly proportional to the area A of each plate (or in general to the area of overlapping) and inversely proportional to their separation d. Plates with larger area can store more charge. Similar effect takes place for the plates being closer together – according to Coulomb’s law when d is smaller the positive charges on one plate exert a stronger force on the negative charges on the other plate, allowing more charges to be held on the plates for the same applied voltage and this constitutes capacitor with larger C.

In commercial capacitors the layer of air is replaced by dielectric material such as Mylar, rubber, mica, waxed paper, silicone oil etc. If the space between the conducting plates is completely filled by the dielectric, the capacitance increases by the factor , called the dielectric constant (or relative permittivity) of the material which characterizes the reduction in effective electric field between the plates due to the polarization of the dielectric.

Figure 2. Charged parallel plate capacitor: –               air filled (top), –               with dielectric (bottom).

                                                 

 

In general the capacitance of a parallel plate capacitor with a dielectric can be expressed as:

                                                           C =  =  ,                                                                  [Eqn. 3]

where  is now the permittivity of the dielectric = κ₀ .           

Capacitors are manufactured with only certain standard capacitances and working voltages. If a different value is needed for a certain application, one has to use a combination (series, parallel or mixed) of those standard elements connected in such a way to achieve the required equivalent capacitance.

For a series connection, the magnitude of charge on all plates will be the same. Conservation of energy dictates that the total potential difference of the voltage source will be split between capacitors and apportioned to each of them according to the inverse of its capacitance. The entire series acts as a capacitor smaller that any of its components individually.

 = ∆V_Source = ∆V₁ + ∆V₂ + ∆V₃ + ….. = +  +  + …

  =  +  +  + ….                   [Eqn.4]                  

 

                             

                    Figure 3. Two capacitors in series. (a) Schematicillustration. (b) Equivalent circuit diagram.

 

Capacitors are combined in series mostly to achieve a higher working voltage.

 

In a parallel configuration all capacitors have the same applied potential difference. However, their individual charges might not necessarily be equal because the charge is apportioned among them by size (C). Since the total charge stored in such combination is the sum of all the individual charges, the equivalent capacitance equals the sum of the individual capacitances and is always greater than any of the  single capacitance in this arrangment.

 

Q_Total = Q₁ + Q₂ + Q₃ + …. = C₁∆V + C₂∆V + C₃∆V +….. = (C₁ + C₂ + C₃ +…)∆V

 

  C_EQ = C₁ + C₂ + C₃ + …       [Eqn. 5]

       

                                                     

      

 

                Figure 4. Two capacitors in parallel. (a) Schematicillustration. (b) Circuit diagram. (c)Circuit

                                 diagram with equivalent capacitance. 

PART I. Parallel plate capacitor.

Open the PhET Interactive Simulations web page http://phet.colorado.edu/en/simulation/capacitor-lab and download the Capacitor Lab.

The two parallel plates connected to the battery constitute the simplest capacitor. You can change the parameters of this capacitor by placing the cursor on the green arrows and moving it along the indicated directions (up and down to modify the plate separation, sideways to alter the area of plates). 

From the right side tool bar menu select all options except the electric field detector. With the mouse left click grab the red test lead of the voltmeter and place it on the upper plate. Next, grab the black test lead and move it to the lower plate. The voltmeter is now ready to measure the potential difference ∆V applied to the capacitor. The potential difference between the capacitor plates will depend on the position of the vertical slider on the battery which allows for adjustment of the battery output.

1.       “Design” your own air-filled parallel plate capacitor with plate separation d = 7 mm and having the plate area A of 250 mm². Apply approximately 1.00 V to it. If the bar graphs displaying the capacitance, the plate charge and the energy go out of range, click the magnifying glasses icons positioned next to them. Capture the screen and paste it into MS Word file – you will have to attach this screen shot to your lab report.

 

                                

                                      Figure 5.  Custom air filled capacitor – example.

              For your custom capacitor calculate its capacitance (based on its dimensions, ), the

              stored charge ( = and the stored energy (). How do the calculated

              values of C, Q and E compare with the corresponding ones shown in the simulation – calculate

              discrepancy?

 

2.       Investigate the effect of changing the plate separation and the plate surface area (one parameter at a time) on the capacitance, the amount of stored charges and energy.  In your statements quote values extracted from the simulation displays.

 

3.       To your custom capacitor designed in step 1 add a dielectric.

Press the “Dielectric” tab in the simulation window and fully insert between the plates a custom dielectric of dielectric constant 5.0. The dielectric is aligned perfectly with the plates when the offset = 0 mm. Adjust the separation and plate area to match your settings in step 1. Apply the same voltage as you had in step 1 and record the values of capacitance, charge and energy displayed by the simulation. Capture the screen and paste it into MS Word file – you will have to attach this screen shot to your lab report.

 

                                 

                              Figure 6. Custom capacitor with dielectric – example.

 

 

By how much are these values different from the corresponding numbers registered for your custom air-filled capacitor? Do you measurements validate the statement that inserting a dielectric between the plates increases the capacitance by a factor equal to its dielectric constant?

 

4.       Now, using your custom capacitor, you will conduct an experiment to determine the dielectric constant of paper. In this part you have to be extremely careful about the units!

In the Dielectric “Material” window select paper from the drop down menu. As you slide the dielectric out of the capacitor, take readings of the capacitance C and the offset ∆x for 5 positions of the dielectric material in between the plates. In Logger Pro program make a plot of C (y-axis) vs. ∆x (x-axis). Apply linear fit to your set of data and from the intercept of this linear fit find the dielectric constant of the paper along with its error. How does your experimental value of the dielectric constant for paper compare with the known value of 3.5.

Attach the graph from Logger Pro to the lab report.

Hint.

The capacitor with partially inserted dielectric can be looked at as two capacitors connected in parallel:

–          the first one air-filled of plate area= , where  symbolizes the dielectric offset measured from the left edge of plates, and  represents the width of plates but, since the original plates are square in shape,  is equal to  (A is your custom plate area),

–          the second one filled with dielectric material of plate area =  

If both capacitors have the same plate separation d, we can describe their capacitances respectively as:

 

                               =  =           and           =  =             

 

 

Hence, the equivalent capacitance for the two capacitors in parallel after rearranging some terms comes to:

 

                                        +                                 [Eqn. 6]

 

It is apparent that eqn. 6 is a linear function of  (same as with intercept  

b =    .    

Then   , and if we know the uncertainty in the intercept  from the Logger Pro fit, we can propagate it into the error in dielectric constant:   .

 

 

PART II.Capacitors in parallel – sharing charges.

Download the Circuit Construction Kit (AC + DC), Virtual Lab from the PhET website: http://phet.colorado.edu/en/simulation/circuit-construction-kit-ac-virtual-lab .

Construct the circuit shown in the figure below.

By right-click on the circuit elements adjust their values to the following parameters:

V₀ = 9 Volts,  C₁ = 0.1 F,  C₂ = 0.1 F,  C₃ = 0.05 F.

 

To add the voltmeter, select it from the tools menu on the right hand side panel.

 

 

Step 1. Charging capacitor C₁ (initially discharged) → keep switches W₂ and W₃ open; close switch W₁. Connect the voltmeter leads across C₁ and measure voltage V₁. Is it the same as V₀? Calculate the charge Q₁ stored on capacitor C₁, Q₁ = C₁V₁. What is the magnitude of positive charges deposited on C₁? What is the magnitude of negative charges deposited on C₁?   

Step 2. Sharing the charge stored on C₁ with capacitor C₂ (initially discharged) → keep switch W₃ still open; open switch W₁ and close switch W₂. Measure voltage V₂ across capacitors C₁ and C₂ and calculate the amount of charges stored on each of them, Q₁₂= C₁V₂ and Q₂ = C₂V₂ . Is the measured voltage V₂ different then V₁? Why? Knowing the values of capacitance C₁ and C₂ and the initial charge Q₁ stored on C₁, calculate the theoretically expected voltage V_2Theory after capacitor C₂ got connected in parallel with C₁. Does your measured voltage V₂ agree with the calculated theoretical value of V_2Theory?

Step 3. Sharing the charge stored on C₂ (from step 2) with capacitor C₃ (initially discharged) → keep switch W₁ still open; also open switch W₂ and then close switch W₃. In this configuration the charge stored in step 2 on C₂ will be distributed between C₂ and C₃. Following the procedure from step 2 measure the voltage V₃ across C₂ and C₃, calculate the charges deposited on individual capacitors and find analytically the theoretical voltage V_3Theory. Does your measured voltage V₃ agree with the calculated theoretical value of V_3Theory? Which capacitor stores less charges and why?

Step 4.  Distribution of charges between three capacitors in parallel (all capacitors initially charged) → keep switch W₁ open and the switch W₃ closed. Capacitor C₁ should still hold the charge from step 2 (Q₁₂), while capacitors C₂ and C₃ should have the amounts of charges determined in step 3 (Q₂₃ = C₂V₃, Q₃ = C₃V₃). Close the switch W₂. Measure the new voltage V₄ across capacitors C₁, C₂ and C₃. Prove analytically (by clear calculation) that the measured voltage agrees with the theory.

Hint: The total initial charge in this case is the sum of charges Q₁₂ + Q₂₃ + Q₃. This amount of charge is conserved throughout the experiment and should equal the total final charge stored on all three capacitors after closing the switch W₂ and equalizing the voltage to V_4Theory.

 

PART III. Combination of capacitors in series and in parallel – finding an unknown

                equivalent capacitance of three capacitors connected in series using the

                charge sharing method.

 

Construct the circuit shown in the figure below.

 

 

 

By right-click on the circuit elements adjust their values to the following parameters:

V₀ = 9 Volts,  C₁ = 0.1 F,  C₂ = 0.05 F,  C₃ = 0.1 F,  C₄ = 0.2 F.

 

To add the voltmeter, select it from the tools menu on the right hand side panel.

 

Step 1. Charging capacitor C₁ → keep switch W₂ open; close switch W₁. Connect the voltmeter leads across C₁ and measure the voltage V₁. Calculate the charge deposited on C₁.

Step 2. Finding the unknown equivalent capacitance of C₂, C₃ and C₄ (all initially discharged) connected in series by sharing charge from capacitor C₁ (from step 1) → open switch W₁, then close the switch W₂. Measure the new voltage V₁₂ across C₁. The three series capacitors C₂, C₃, C₄ can be replaced by one capacitor C_EQ of such value that its effect on the circuit would be equivalent. Note that the equivalent capacitor C_EQ would be now connected in parallel with C₁. Following the reasoning from PART II Step 2 find the capacitance C_EQ. How does this experimental value of C_EQ compare to the theoretical one calculated from the following equation:

                                                               =  +  +  

 

or                                                            C_EQ =

 

According to what you have learnt about capacitors connected in series what charge should be stored on each of the three capacitors C₂, C₃, C₄? Measure the voltage across each of the three series capacitors and calculate the corresponding charges Q₂, Q₃, Q₄ deposited on them. Do the experimentally determined charges Q₂, Q₃, Q₄ agree with the theoretical prediction?

* Include answers to all questions in lab report

 

 

Capacitors  –  Post Lab Checklist.

 

      PART I.

·           Screen capture of the air-filled virtual capacitor designed in PhET.

·           Screen capture of the virtual capacitor with dielectric designed in PhET.

·           For TWO plate separations and TWO plate surface areas for the air-filled capacitor and just one set of parameters for the capacitor with a dielectric, calculations of the following:

ü  Capacitance (5 values total),

ü  Stored charge (5 values total),

ü  Stored energy (5 values total).

 

·           In LoggerPro – graph of C vs. Δx for a paper-filled capacitor, with linear fit.

 

·           Calculations of the experimental dielectric constant κ for the paper along with the uncertainty in the measurement of κ.

 

·           Calculation of the discrepancy (expressed in %) between your experimental κ and the accepted value of 3.5.

 

PART II.

 

·           All calculations as detailed in steps 1 – 4 of the lab write-up.

 

PART III.

 

·           Step by step calculations of the experimental equivalent capacitance. Calculation of the theoretical equivalent capacitance. Calculation of the discrepancy (expressed in %) between the experimental and the theoretical equivalent capacitance.

·           Calculations of the charges stored on each capacitor.

 

 

·         ALL PARTS – Address all discussion questions posted in the lab write-up!

 

 

 

LAB REPORT CRITERIA

 

The formal Lab Report is written from the third person; in the passive form, in the past tense. It includes the following parts:

Expression of the experimental results is an integral part of science. The lab report should have the following format:

·          Cover page – course name , title of the experiment, your name (prominent), section number, TA’s name, date of experiment, an abstract. An abstract (two paragraphs long) is the place where you briefly summarize the experiment and cite your main experimental results along with any associated errors and units. Write the abstract after all the other sections are completed.

The main body of the report will contain the following sections, each of which must be clearly labeled:

·         Objectives  – in one or two sentences describe the purpose of the lab. What physical quantities are you measuring? What physical principles/laws are you investigating?

·         Procedure – this section should contain a brief description of the main steps and the significant details of the experiment.

·         Experimental data – your data should be tabulated neatly in this section. Your tables should have clear headings and contain units. All the clearly labeled plots (Figure 1, etc.) produced during lab must be attached to the report. The scales on the figures should be chosen appropriately so that the data to be presented will cover most part of the graph paper.

·         Results  – you are required to show sample calculation of the quantities you are looking for including formulas and all derived equations used in your calculations. Provide all intermediate quantities. Show the calculation of the uncertainties using the rules of the error propagation. You may choose to type these calculations, but neatly hand write will be acceptable. Please label this page Sample Calculations and box your results. Your data sheets that contain measurements generated during the lab are not the results of the lab.

·         Discussion and analysis – here you analyze the data, briefly summarize the basic idea of the experiment, and describe the measurements you made. State the key results with uncertainties and units. Interpret your graphs and discuss what trends were observed, what was the relationship of the variables in your experiment. An important part of any experimental result is a quantification of error in the result.  Describe what you learned from your results. The answers to any questions posed to you in the lab packet should be answered here.

·         Conclusion – Did you meet the stated objective of the lab? You will need to supply reasoning in your answers to these questions.

.

All data sheets and computer printouts generated during the lab have to be labeled Fig.1, Fig. 2, and included at the end of the lab report.

Important Note

·         All data sheets and computer printouts generated during the lab have to be labeled Fig.1, Fig. 2, and included at the end of the lab report.

·         The Post-Lab Checklist does not need to be attached when the lab report is submitted.