Lab 2: The Permittivity of Free Space Edited 10/3/14 by Joe Skitka, Stephen Albright, EAG, DGH, WL, & JCH

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1 Lab : The Permittivity of Free Space Edited 10/3/14 by Joe Skitka, Stephen Albright, EAG, DGH, WL, & JCH Figure 8: The electrostatic balance Objective An attempt will be made to measure the permittivity of free space as accurately as possible. This will be computed from the force acting on an electrostatic balance. The results, including detailed uncertainty analysis, will be written up in a full lab report (see the Lab Report Guide and the Guide to Uncertainty Analysis on the lab wiki). An additional exercise which involves mapping electrostatic fields in two dimensions follows the main component of the lab. A write up is not required for this exercise. Be sure to reserve the last 45 minutes to complete this. Introduction The permittivity of free space, also known as the vacuum permittivity and represented with ε 0, is a constant appearing in Maxwell s equations which, together with the permeability of free space, μ 0, uniquely determines the speed of light, c, and the relative magnitude of electric and magnetic fields in vacuum EM radiation. Although most physical constants are already known to many digits, measuring them to an ever higher degree of precision is of perpetual interest in research because they can be a limiting factor in other experiments and such precision can be helpful in building theoretical models beneath those already known. In this lab, the student is challenged to measure ε o with a small, carefully computed uncertainty using a provided electrostatic balance and associated devices. It should be noted that ε o is not considered a fundamental physical constant; rather, it is a defined quantity and is limited in practice by the precision of definitions of the units in which it is expressed.

2 Experimental Details The vacuum permittivity, ε 0, will be measured using the electrostatic repulsion between two plates held at a specific potential difference by a DC power source. Repulsive forces are determined using a digital scale. Specific information on this equipment is provided below. Some of this information is repeated in the procedure for easy reference. Electrostatic Balance: The electrostatic balance consists of two parallel plates (which are effectively a parallel plate capacitor) of adjustable plate separation that is mounted to a digital scale. When a voltage is applied, a charge accumulates and the plates are repulsed, pushing down on the scale. This apparatus allows for the plate separation and voltage to be adjusted freely. OTE: under no circumstances should you take apart the balance. If you need to measure dimensions, a sample plate is available (ask the TA if it is not in sight). Also, do not allow the plates within 3mm of each other when the power source is on. This will result in a short circuit and/or sparking. Figure 9: The Electrostatic Balance and Experimental Setup Scale: The digital scale (or balance) can be turned on/off and zeroed/tared (same thing) with a thin rod located at each experimental setup. The scale is designed to freeze the display once a reading is stable. This may cause the scale to stop responding. If this happens, turn the voltage down significantly and restore it or bump the scale surface with the thin rod. Digital Depth Gauge: The readout should be self-explanatory. ote that the depth gauge locks; this must be done when making measurements. Also note that all measurements are relative the gauge is not necessarily zero when the plates are touching. This will contribute an extra factor to the uncertainty of any measurement; one for the zero (plates touching) measurement and one for the non-zero measurement. Instructions for measuring and estimating the uncertainty of the zero are given directly in the procedure. Power Source: A direct-current (DC) power supply is used for this lab. Current and voltage knobs can both be used to vary power flowing through the circuit; however, these are not independent parameters. As one of these quantities is adjusted, the other will follow according to ohm s law at a constant resistance. It is recommended you set the current to a fixed value and adjust the voltage as needed. There are both coarse and fine adjustment knobs for each of these parameters. On some of the power sources the potentiometers on the fine adjustment knobs are finicky and may not allow for precise fine tuning.

3 Calipers: This device is used to make precision measurements of external and internal dimensions of small objects. External dimensions (as with the diameter of the circular plate in this lab) are made by spreading the large jaws and placing them around the object. A measurement can be made by placing them in contact with the surfaces being measured between. Internal dimensions can be made by spreading the small jaws on the opposite side of the large jaws and placing them within the cavity being measured. At the bottom of the caliper handle, a rod can be used to precisely measured depth. With any type of measurement, be careful to ensure that the calipers are measuring orthogonal to the surfaces in question. If the object has measureable surface imperfections, several readings may be taken at different points and averaged. Cables: The black and red banana-plug cables used here will become a staple of this lab series. It is important to note that red and black cables are not a perfect system for indicating the direction of current flow. Typically, a red cable will be connected to the positive terminal of the power source and a black to the negative, but beyond this, there can be no useful guidelines for the general use of solid-colored cables within a circuit. If the cables were ideally made for the purpose of indicating current flow, they would be red at one end and black on the other, not solid colors, so be wary of using the colors to interpret a circuit. Theory The upper plate exerts a force on the lower plate that is equal to the charge on the lower plate times the electric field of the upper plate: F = QE (1) Since the separation of the plates is sufficiently small, the field lines can be approximated by those associated with plates with infinite diameter. Therefore, the field lines between the plates can be thought of as parallel and the following applies: E = σ ε o = Q Aε o () Because the parallel plates also act as a capacitor, the charge on each plate can be found from the potential difference, V, and separation, d, as: Q = VC where C= ε oa d Q= ε ova d (3) Substituting Equations () and (3) into Equation (1) gives: F= ε oa d V (4) Since the force has a linear dependence on the potential squared, the slope of the graph of F vs. V can be used to determine the permittivity of a vacuum. Alternatively, a constant voltage can be maintained while the distance between the plates is varied; a plot of force versus inverse

4 distance squared will yield a constant slope from which the permittivity of a vacuum can also be determined. Procedure 1. To begin, note all of the quantities required to compute the vacuum permittivity in this experiment (refer to the derived expressions in the theory section): the force on the plate, F, the area of the plates, A, the separation between the plates, d, and the applied voltage between the plates, V. Since only one set of plates is used for the whole lab, the area of the plate and its uncertainty can be determined now and set aside until calculations and analysis are done later. The treatment of uncertainty is detailed explicitly here in excessive detail for instructional purposes. The take-home point of all this detail is this is an example of how one might think about various sources of uncertainty to come up with a good estimate. Please read the Guide to Uncertainty Analysis on the lab wiki, or one of the more formal introductions referenced. Use a digital caliper to measure the diameter, D, of the sample plates which are located in the middle of the room. Please keep the plates in the center of the room as students will have to share in order to check multiple plates. To make a good measurement, place the disk on a flat surface and rotate the jaws across the diameter to the point of greatest constriction. Estimate the uncertainty of this value by considering the following contributions to uncertainty: o precision of the measurement devise o your ability to read its value precisely, if analogue o the uncertainty in the zero of the caliper reading. Close the caliper so that the prongs are touching and zero it. Perform the experiment. At the end of the measurements, put the prongs back together and see what it reads. Even if it says zero, the contribution must be at least equal to the precision of the instrument. ote that this uncertainty is bias and will not be reduced by making multiple measurements. o variation in diameter between plates. Because you cannot measure the plate used in the experiment directly, an individual sample plate may not be a reflection of the plate in your apparatus, so you will need to look at the mean and variation between multiple plates. The standard deviation should be used as an estimate for the uncertainty; however, because you do not know the mean in advance, the first measurement does not give you any information on the variation, so use a corrected standard deviation with an n-1 in the definition, which is the Excel command STDEV.S as opposed to STDEV.P. o deviations in its shape from a perfect circle. This can be determined by taking several measurements rotating the plate in between each one. The standard deviation of these measurements can be used as an estimate of the uncertainty for a single measurement. o temporal fluctuations in the reading (unlikely to be noticeable here.) o the diameter s susceptibility to temperature. Because we do not have a temperature controlled environment, it is difficult to account for this, but

5 could be noticeable if you have temperature swings of a couple degrees during the experiment. It turns out that one of these contributions is more than an order of magnitude greater than the others. When combined with the others via the root-mean square, the contribution will be dominated by this contribution, equaling the total uncertainty up to or 3 decimal places. If you can identify the largest contribution, you may neglect the others, eliminating the need to determine the standard deviation(s) and saving a bit of time. You may want to confirm with the TA to see if you ve found the correct contribution. These contributions can be combined by taking their root mean square, or RMS, which is how contributions of uncertainty, theoretically representing standard deviations of unspecified probability distribution functions, combine under the addition of their governing distributions: i (ΔD i ). Practically speaking, a single contribution may dominate, in which case very small contributions may be dropped. It is important to remember that uncertainty is intended to be an upper bound on the error at a fixed confidence level, so be careful to round up when necessary. Calculate the plate s area and practice propagating the uncertainty of the area using a linear expansion around the measurements. Again, the contribution of each independent parameter combines additively with the RMS, only this time taking into account a general functional dependence on the parameters via partial derivatives: ΔΨ = Ψ (Δφ i ) i, (5) φ i where φ i are all of the parameters on which Ψ, the value you are propagating to, depends. This equation can be used to propagate uncertainty in any situation. In this case, Ψ is the area, A, and it is only a function of one parameter, φ 1 = D. When A = ¼ π D is plugged in, this looks like: ΔA = 1 π D ΔD, (6) If you have a question about this, be sure to ask the TA. Application of equation 5 will be required repeatedly throughout the lab component of this course. ote: DO OT TAKE THE APPARATUS APART (schematic of apparatus shown in Figure 9).. The remaining experimental parameters will vary as measurements are made. The apparatus can now be prepared to take measurements. Both the separation between the plates and the electric potential they are held at could be treated as the free variable in this experiment. The electric potential will be varied first, but the experiment will be repeated treating the separation as the free variable to see if one technique is more accurate than the other. Prior to making measurements of the separation of the plates using the depth gauge, it must be zeroed by putting the plates in contact. Make sure the power supply voltage is set to 0 kv. Use the depth gauge to raise the upper plate so that it is not touching the lower plate. If the scale is not reading zero, tare it by pressing tare or zero or powering it on and off with the wand provided.

6 Gently lower the upper plate so it just touches the lower plate - It may be helpful to use the wheel on the back of the depth gauge. When the scale reads between.01g and 5.0g lock the depth gauge, record the scale reading and zero the depth gauge. ote: always remove your hands from the gauge before taking a reading. 3. Use the depth gauge to set the separation between the capacitor plates to ~15 mm. It will remain at this separation while the voltage is varied independently. The exact value is not important. What is important is that you record the value and record an estimate of its uncertainty for later propagation. This will involve contributions similar to the diameter of the plate (e.g. from the precision of the measurement device) in addition to the uncertainty of the zero position, which encapsulates geometric imperfections. Use appendix to estimate this contribution to the uncertainty. Again, these can be combined via their RMS to arrive at a single uncertainty estimate of the separation. 4. Record balance readings for voltages from kv to 5 kv in 500 V increments. The idea here is to smooth out any biases which may occur at a specific voltage range by taking many values at different voltages and combining the data into a single value. Random fluctuations in the readings will also be smoothed out by taking multiple measurements, but this could have been addressed by taking repeated measurements without changing the voltage. Do not allow the plate separation to be less than ~3mm when the voltage is up sparking between the plates may occur. The plates are attracted instead of repelled so the scale will give negative readings. If the force on the scales does not change for 90 seconds, the scales will shut off. When this happens, set the voltage to 0 and turn the scale back on. Your tare should be preserved. The scale is designed to freeze the display once a stable reading has been taken. This may cause the scale to stop responding. When this happens, turn the voltage down significantly then back to the desired voltage or tap the surface of the scale (gently) using the rod. Periodically confirm that the scale tare has not changed. Be sure to estimate the uncertainty of the balance and voltage readings. 5. Use the digital depth gauge to set the separation between the capacitor plates to ~0 mm. If any biases exist at 15mm and not 0mm, this should provide an indication. Take a set of measurements at this separation just as was done at ~15mm. 6. A quick way to combine these multiple data points at different voltages is to use a linear fit and relate the slope to ε o. Plot the force F versus the potential squared V for both separations. Use Appendix 1 to find the slope of the least squares best fit line where x is V and y is F. Find the uncertainty in the slope using the same appendix. 7. Derive the permittivity of a vacuum from the slope, m, of the graph F vs. V.

7 m= ε oa d (7) 8. Once again, use a linear expansion, equation (5), to find the error in the measured permittivity: which ends up looking like: ΔΨ = Ψ i (Δφ i, (8) φ i ) Δε o = ( d 4md A ) + ( m d A ) + ( A md A ) (9) 9. ext, the plate separation will be varied independently. Set the potential to 3.0 kv and measure the force for several different separations between 10 mm and 0 mm. ote: Do not allow the plate separation to be less than ~3mm when the voltage is up sparking between the plates may occur. 10. Plot F vs. 1 d for all separations. We recommend you use Kaleidagraph, available on the lab computers or as a free trial download. See Appendix 3. Find the slope of the least squares best fit line where x is 1 d and y is F. Record this slope. Derive the uncertainty in the slope (again, see Appendix 1.) 11. Derive the permittivity of a vacuum from the slope of the graph, F vs. 1 d m = ε o AV (10) 1. Calculate the uncertainty in the permittivity using the techniques demonstrated in step 8. For reference, the accepted value is ε o = m -3 kg -1 s C.

8 Additional Exercise: Mapping Electrostatic Fields Edited 8/14/14 by Joe Skitka, Stephen Albright, EAG, DGH, WL, & JCH Figure 1: Equipotentials (red dotted lines) and electric field lines (black vectors) for dipole system Objective The purpose of this exercise is to visualize static electric fields and corresponding equipotential surfaces. While most electric fields one encounters are 3-dimensional, to maximize comprehension and simplify the necessary procedure, approximately -dimensional fields are used instead. This is accomplished by confining the electric field to a conducting sheet. Although charges flow through the conducting sheet when an electric potential is applied across it, the resulting field is static and a good approximation of that of a -dimensional vacuum to the extent that the conductor is an infinite sheet and uniform. OTE: This exercise does not require a written assignment. Rather, each group will check out with the TA before or at the end of the lab to answer questions and demonstrate an understanding of the concepts involved. Introduction A conductor used to establish electrical contact with a nonmetallic part of a circuit is called an electrode. Charged electrodes generate an electric field in the space around them. The size and shape of this field is dependent on the shape of the electrodes and their geometric configuration relative to each other.

9 This experiment explores these fields using copper electrodes painted onto graphite-impregnated conducting paper. By connecting the electrodes to a power supply, an electric field is produced in the plane of the paper. A small current flows through the paper generating a potential gradient across the paper. The potential difference (voltage) between any two points on the paper can be measured by touching the paper with the metal probes of an electrometer - this device draws virtually zero current so it does not affect the measurements. If these two probes are secured together at a fixed separation, they can be used to measure the direction and magnitude of the electric field vector. By these means, the relationship between the equipotential curves and the electric field vectors can be investigated for electrodes of various shapes and configurations. on-conducting, colored pencils will be used to map the equipotential curves and field lines on the paper. Theory While electric fields are innately 3-dimensional, the electric field maps created in this experiment are approximately -dimensional. D fields and geometries can be thought of as flat systems extending to infinity in both directions along a third dimension. In this case the system is the graphite sheet confined to the horizontal plane. The electrodes would then be infinite vertical line charges in a vacuum. If this geometry could be realized, there would be no need for graphite conducting paper and the field could be measured in air (an approximate vacuum). Using the conducting graphite sheet allows this ideal setup to be conveniently approximated. Electric field lines indicate the direction a positive charge would accelerate if placed on the line (see Figure 1). Equipotentials are a region in space where all points have the same potential. Like contour lines on a map which trace lines of equal altitude, equipotential curves trace points of equal voltage (in 3-D, these would be equipotential surfaces). For a 3D system, equipotentials are D surfaces. In D systems (as in this exercise), equipotentials are 1D lines. Movement along an equipotential surface requires no work. Can you convince yourself why? Procedure 13. Secure the paper to the corkboard with pushpins. 14. To ensure a good electrical connection with the paper, screw a coffee hook into each of the electrodes (paint spots) until the base of the hook presses against the paint on the paper (see Figures.a and.b). Attach the power supply to the coffee hooks using red and black cables for the positive and negative/ground terminals respectively. These connections will established the desired electric field. Figure.a: Correct Figure.b: Incorrect

10 15. An RCA plug will act as a movable voltage probe on the graphite paper. To measure the electric potential, the voltage must be measured from the negative (ground) electrode, so a second, fixed probe will be attached to that part of the circuit. In practice, this need not be directly attached to the electrode and can instead be attached directly to the ground terminal of the power source, which is at the same potential as the negative electrode because it is connected to it with a conducting cable of much less resistance than the paper. To set this up: Connect the black banana lead of the long thin BC/shielded Banana cable to the COM terminal of the multimeter Connect the red banana lead to the V, Ω, µa terminal of the multimeter. Plug the BC to RCA adapter in the other end of this cable. (see Figures 3.a and 3.b). ATTETIO: Connect the negative terminal of the power supply to the COM terminal of the voltmeter with a banana cable. Set the multimeter to DC Volts and the Range to 0.00V Figure 3.a: BC to shielded Banana cable Figure 3.b: BC to RCA adapter 16. Set the power supply to 5 volts (see Figures 4.a and 4.b). Figure 4.a Figure 4.b 17. Map out at least three equipotential curves on the graphite paper. Readings of the electric potential can be taken by touching the protruding lead of the RCA adapter (positive probe) to the paper at the position of interest. To precisely define the gauge and sign of this reading, if the above instructions were followed exactly, this

11 measurement is the electric potential of the chosen point on the paper minus the electric potential of the ground terminal. Does the sign of the readings make sense? To find an equipotential line, start by making a measurement somewhere between the two electrodes (coffee hooks). Record the voltage and mark the position with a colored pencil. Find several other points on the sheet which are at the same potential, marking each of them. When there are enough individual points, connect the dots. It is recommended that you start at a variety of positions between the electrodes, closer to the positive electrode than the negative, as the fields should be symmetrical. Try at least one rather close to the positive electrode so that the curve forms a closed loop. 18. Measuring the electric field, E, can most easily be accomplished indirectly from the electric potential. The electric field (a vector field) is the negative gradient of the electric potential (a scalar field). The gradient can be approximated by maximizing the voltage difference between two very close points on the graphite paper as a function of direction. This will require two probes separated by a constant distance. To set this up: Remove the banana cable connecting the negative terminal of the power supply to the COM terminal of the voltmeter. Replace the BC to RCA adapter with the BC to mini banana adapter (see Figure 5). 19. Draw -3 electric field lines: Touch the positive probe to a point. Start on an equipotential curve, but you may need to check points elsewhere to connect the lines. Keep the positive probe in place and trace a small circle around it with the negative probe whilst keeping it in contact with the paper. At the orientation of maximum positive voltage, using a different color from the equipotential lines, draw an arrow on the paper from the positive (+) to the negative (-) probe. This arrow represents the direction of E. Sketch in the continuous electric field line by connecting a few arrows. o need to be overly precise here this is meant to be qualitative. Figure 5: BC to mini banana adapter

12 Figure 6: Configuration. 0. Switch to the other provided graphite sheet. Draw equipotential curves and electric field vectors for a point charge in the presence of a conducting plane (figure 6). Here, the plane is just a horizontal line; however, it can be thought of as extending infinitely in the vertical direction to form a surface. Refer back to steps 13 through 19 for instruction as needed. 1. [Optional] The observed fields of these -dimensional configurations can be compared with simulated fields. To do this, go to the Electrostatics Simulation Applet: Set up the first scenarios (Figures 4.a and 6) on the simulation to check if you correctly measured the equipotentials and electric field vectors. (Use Dipole Charge for the first scenario, Charge + Plane for second).. Check in with the TA before proceeding to the main part of the lab. Questions 1. Compare and contrast the two electric fields and equipotentials mapped. What symmetries do you observe? Why does the second configuration (with the conducting plane ) look different?. What might the field map of figure 4.a look like if there were a square cut out between the electrodes?

13 APPEDIX 1 Least Squares Fitting Although there are many programs which will do least squares fitting, it is instructive to know how the software is working. Here we give the exact formulas for the linear equation y = Bx + A (see Reference 1, Chapter 8 for more details). For data points (xi, yi) the least squares fitting algorithm gives the following formulas for A and B: xi yi xi xi y i i i i A xi xi i i xi xi yi y i i i B xi xi i i If the different values are assumed to follow a Gaussian distribution around the mean, the uncertainties ΔA and ΔB can be computed without estimating the uncertainty of a given measurement, which is what you can do here. These are given by: i i ΔA = 1 (y x i A Bx i ) i i x i i ( i x i ) i ΔB = 1 (y i A Bx i ) x i i ( i x i ) i This technique will not capture biases or varying uncertainties for each data point. A general, more rigorous approach to combining all of this data would be using an uncertainty-weighted average.

14 APPEDIX Finding the Error in the Separation In step, the separation between the plates was zeroed when the scale read between.01 and 5.0 grams. Ideally the scale would read zero while the plates are just barely touching - a very difficult setting at attain with this apparatus. To find the impact of this error, with the voltage off: 1. Gently lower the upper plate onto the lower plate until the scale reads grams.. Lock the depth gauge and remove your hands. Record the scale and depth gauge readings. 3. Gently raise the upper plate until the scale reads several grams less than it did. 4. Lock the depth gauge and remove your hands. Record the scale and depth gauge readings. 5. Divide the difference in depth by the difference in mass. 6. Multiply this ratio by the mass recorded when the depth gauge was zeroed. This is the contribution to the uncertainty in the separation of the plates from the zero measurement. APPEDIX 3 Kaleidagraph Kaleidagraph is a super-quick solution to data analysis, but it is also more versatile than Word (at least for the untrained user.) The biggest advantage, besides its efficiency, is that it will provide uncertainty in fit parameters, both linear and non-linear. Kaleidagraph is available on the lab computers, but is generally not installed on other campus computers. The software should be available for download from but if not, then a free trial may be downloaded from Here are a few key things you need to know to utilize the software: Don t label columns in the top-most cell with text. Rather there are specific title boxes, initially labeled A, B, C, etc. If you do enter text in a box, the column will format for text entry and will throw an error when you try to plot. (To fix this, go to column formatting under the data menu. Format the column for float.) Simply enter columns of data for the independent and dependent variables you wish to plot. To manipulate data, press the up/down wedges/arrows at the top right of the page to expand the column labels for reference. A formula entry window should be floating around the screen somewhere. Columns can be edited with simple commands. For

15 instance, if you put your Voltage data in column c1 and wish to use voltage squared, simply type: c = c1^ in the formula entry. Press Run and you will see the voltage squared has its own column. You may wish to input a column of uncertainties for later use as error bars. To plot: select the Gallery dropdown menu. Select the Linear menu. Select a Scatter plot. Choose one column for your independent variable, and one or more for your dependent variable. To fit: select the Curve Fit dropdown menu. The presets like linear may be used for a quick fit, but they will not give you uncertainty on the fitting parameters. To do this, select General. If you d like a linear fit, fit1 should be set to this be default. Upon clicking it, a box should come up. Select Define to double check the form of the fit. It should read: m1 + m * M0; m1 = 1; m = 1 Here, {m1, m, m3 } are the fit parameters and M0 is the independent variable. Clearly this is just a linear fit. If you, for instance, wanted to make this a quadratic fit instead, you would write: m1 + m * M0 + m3 * M0^; m1 = 1; m = 1; m3 = 1 The values on the right are initial guesses and must be provided for each parameter. When you are done, select OK. Check the box of the dependent variable you would like to fit. Then select OK. The uncertainty in the fit parameter is listed in the Error column in the plot To generate error bars, right click on the plot and select Error bars. Error bars can be added as a percent of the value (in either variable) or from a different column of data. REFERECES 1.) John R. Taylor, An Introduction to Error Analysis nd ed. University Science Books