PHY 2 Experiment Report - NEEDED IN 24 hours

Experiment 2 Electric Potential and Field Mapping

Introduction

In this experiment, you use a voltage probe and a computer data acquisition system to measure the electric potential between two metal electrodes. The electrodes are placed in a tray, which contains a shallow layer of water. The electrodes are connected to a D.C. power supply, which maintains a constant potential difference. The water allows an electric current to flow from the positive electrode to the negative electrode. See Figure 1.

After measuring the electric potential surrounding the electrodes, you will transfer these numbers to an Excel spreadsheet. There you will produce surface plots of the electric potential. For one particular arrangement of electrodes, you will also use a digital multimeter to measure the potential difference between two closely spaced points in the water. This will allow you to calculate the strength of the electric field between these points. These electric field strengths, and the location of the corresponding points, will then be graphed to test Gauss’ Law. All results are displayed graphically, and the data sheets constitute the data and nearly all of the data analysis for the report. Spend time adding labels and color-codes to your data sheets. Though a large amount of numerical data will be recorded and graphed, the results of this experiment are largely qualitative. Therefore, a quantitative error analysis is not required for this experiment’s report.

!

Figure 1. The apparatus set up with two parallel plates.

Concept

Suppose a charged test particle is brought near other electrically charged objects. In the experiment, the test particle is the tip of a metal probe placed in the layer of water and the other charged objects are the electrodes. In this case, the test particle experiences a force of attraction or repulsion depending upon the sign of the electrodes (positive or negative).

One way of depicting the influence of electrically charged objects is by examining the energy a charged test particle will gain or lose when it is moved around in the neighborhood of the main charged objects. One must push against a force of repulsion to move a positively charged test particle toward a positive electrode. This force, multiplied by the distance the test particle is moved is the amount of work required to move the particle. In the experiment, you

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data acquisition: Vernier Lab Pro

AC/DC Adapter

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won't actually feel the repulsive force. It is far too small to experience tactilely. However, the data acquisition probe and the digital multimeter measure the energy (per unit charge) of free charge in the probe tip. This energy per unit charge (Joule/ Coulomb) is called electric potential. The difference in the electric potential at two different locations is called potential difference or the more commonly voltage. Hence 1 Joule/Coulomb = 1 Volt.

! Figure 2. Electric field lines of a dipole located at the origin. Drawn in Grapher as r = cos2θ.

� ! !

Eqn. (1) (2) (3)

The electric field is the electrostatic force exerted on a charged test particle per unit of charge on the test particle. See Eqn. (1). The electrostatic force is the Coulomb’s Law force produced by a nearby charged object. q is the charge of the test particle. The electric field has units of Newtons / Coulomb.

The total electric potential difference is computed by adding all of the individual amounts of work (energy) used to move this charge against the force of the electric field from one point to another. Work done by an electric force is defined as the scalar (or dot) product of the distance the test particle moves and the electrostatic force. See Eqn. (4). By dividing Eqn. (4) by the charge, q, and using Eqns. (1) and (3) we obtain Eqn. (5). When, the field is uniform over a small distance, Eqn. (5) reduces to (6).

! ! ! Eqn. (4) (5) (6)

! E =

! F q

V = W q

ΔV = ΔW q

W = ! F id

! l∫ V =

! E id

! l∫ ΔV =

! E iΔ

! l

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If ! and the electric field are perpendicular to one another, then the dot product of these two vectors is zero. Then there is no change in potential along a path perpendicular to the direction of the electric field. This path is an equipotential since all points along this path have equal potentials. To conclude, field lines are always perpendicular to equipotential lines.

It also stands to reason; the electric field should be strongest where the equipotential lines are most dense. The geometry of the charged object also affects how the electric field varies with distance from the object. Eqn. (6) is a valid approximation if the field doesn’t change over small distances.

Just as a marble or ball rolls down hill due to gravity, an unbound, charged test particle will move to a region of lower electric potential. In this way, an analogy can be drawn between electrostatics and gravitation. This analogy is exact due to the similarities between Newton’s Law of Gravity and Coulomb’s Law. Here, the electric field is analogous to the gravitational field g. Geographic contour lines are lines of constant elevation above mean sea level. In this experiment, equipotentials are lines of constant potential above ground potential, which is defined as V = 0.

Gauss’ Law allows us to derive expressions, which describe the geometry of the electric field for a given distribution of charged particles. Two charge distributions of practical interest are the long, charged line and the charged, infinite sheet. These have practical use since charged metal sheets are used to build capacitors, and long charged lines are the basis for electrical current in wires.

! (7)

If the linear charge density is λ, and for a point a radius r from the line, the electric field is given by Eqn. (7). Consult your textbook to see how this equation is derived from Gauss’ Law. Figure 3 is a two-dimensional slice of this particular three-dimensional electric field. Hence, Eqn. (7) implies the magnitude of the electric field is proportional to 1/r.

From Eqn. (5) it is seen that the electric potential is related to the electric field by an integral. By the same token, the electric field is the spatial derivative of the electric potential (multiplied by -1). This has an important interpretation in the gravitational analogy to electric fields. Since a derivative is the slope of a tangent line, the electric field can be visualized as the slope (or gradient) of a potential surface.

Method

The HY3003D power supply is capable of delivering precise, constant currents or constant voltages. The default mode is constant current. However, we usually want a power supply to operate in the constant voltage mode. This requires setting a maximum limit for the current. Since the fuses in the digital multimeters are rated to 400 mA, we conservatively set the current limit to 0.3 Amps. (In some experiments, we use larger currents and then use the 10 Amp jack on the multimeters.)

! E =

1 2πε

0

1 r r̂

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1) With the power supply OFF, turn all current and voltage knobs to their lowest (most counter-clockwise) setting. Also make certain the push-button labeled “AMPS” is pressed IN.

2) Turn the power supply ON before connecting any wires. 3) To set the current limit, connect one wire (a short) between the power supply’s red and

black jacks. The black and green jacks are common due to a connecting metal strap. 4) Turn the COARSE voltage knob clockwise about 1/10th of a revolution. The red light

labeled “CC” should be illuminated. 5) Increase the COARSE and FINE current knobs to 0.3 Amps. This sets the current

limit. 6) Disconnect the wire between the red and black jacks. The “CC” light should go out and

the green indicator light labeled “CV” should illuminate. 7) The power supply is now ready to be used in a constant voltage mode. Use the

COARSE and FINE voltage knobs to apply the desired voltage. 8) Lower all voltage and current knobs to zero when you are finished. 9) Disconnect all wires before turning the supply OFF. 10) When coils of wire are connected to this power supply, adjust the voltage and current

slowly to avoid a back-EMF that might cause damage.

Procedure

Part 1 Potential Between Parallel Charged Plates

1) Turn on the table’s power strip and then the D.C. power supply and the computer. The computer’s ON button is on the back of the iMac in the lower left corner

2) Place a laminated graph paper grid into the yellow tray and then place two aluminum bars on top of the graph paper. Pour a thin layer of tap water into the tray. Use enough to completely surround the electrodes and cover the entire sheet of graph paper but not enough to submerge the electrodes. Return the tray to your station and dry any spills with paper towel.

3) Connect the voltage probe to one of the analog channels labeled CH 1, CH 2, etc. Start the data acquisition program Logger Pro and Microsoft Excel. Ideally, Logger Pro will automatically recognize that a voltage probe is connected. If it does not execute the following pull-down menu commands. Experiment > Set Up Sensor > Show All Interfaces. Then click on the channel the voltage probe is plugged into. As you click and hold down the mouse button, execute these commands: Choose Sensor > Voltage > Voltage (+ / - 10V).

4) Test to see if the data acquisition unit is working correctly by connecting the voltage probe’s black plug to ground and the red one to the power supply’s 5 V output (middle two banana jacks). The live readouts in the screen’s lower left corner should give very close to 5.0 V. If random or nonsensical voltages are displayed try replacing the voltage probe. The solder joints under the electrical tape sometimes break. Disconnect the voltage probe after this test.

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5) Also, execute these commands: Experiment > Data Collection… and set the Mode to Selected Events. This allows you to collect and save voltages. Close the last dialog box and then click the collect button (the triangle in the green rectangle) to begin data acquisition. Save each datum by clicking the light blue circular icon next to the collect button. You can delete the graph window in Logger Pro since it will not be useful in this experiment. Avoid mistakes by collecting one column of data at a time from the plastic tray and then copying and pasting into Excel (see details below).

6) Starting in 2013, we are using power supplies that contain more sophisticated circuitry and require greater care in their use. Students and instructors must consult the method section on the use of the HY3003D power supplies. Connect the power supply and electrodes as shown in Figure 1. In Fig. 1, heavier lines represent black wires, which connect to the power supply’s ground (zero Volts) jack. Lighter lines represent red wires, which connect to the power supply’s red jack. The same convention is used for the wire leading from the data acquisition board and the (ideally red) multimeter probe with its single prong. In this way, the data acquisition system measures and records the potential at the tip of the probe relative to zero Volts on the power supply. The black wire from the measuring device to the power supply’s ground establishes that the power supply’s ground is your reference zero. Note that this convention as to heavy and dark lines is not continued in other figures in this manual.

7) The graph paper in the bottom of the tray is marked off every two centimeters. Measure the potential at every “+” symbol on the graph paper. When the electrodes obscure the marks, touch the probe to the electrode at the desired location. Be sure to collect data near and all the way around the plates.

8) Do not apply more than 5 Volts to the electrodes. The data acquisition board is limited to 5.12 Volts (512 is a power of two). Do not allow the positive and negative electrodes to touch. This causes a spark and overloads the power supply!

9) After recording one column of data, click the red stop button and then click on the “Potential” column heading in Logger Pro to select the entire column of values. Execute the Copy command in the Edit menu and then click on the green Excel icon in the Dock to switch to Excel. With a blank spreadsheet open, use the Paste Special… command in the Edit menu to transfer the potentials from Logger Pro. Click on the Text radio button and click OK. These procedures adjust for a mismatch between the clipboard Logger Pro writes to and the default Excel clipboard.

10) Within Excel, arrange the numbers in columns and rows, just as the electrodes are arranged in the tray. This will create a one-to-one map of what is in the tray. Now is a good time to save the spreadsheet to the hard disk. Please locate the file on the desktop and delete it at the end of the lab period. Once finished, you will have created a scalar field of electric potential values. Insert column and row headings above and to the left of the array of voltages. Use integers that represent the distance (in cm) along the grid paper.

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11) Plot a surface graph of the electric potential in Excel.

a) Select the array of data including the column and row headings. Then click the Insert menu and select Chart > Surface. Here, the z-coordinate represents the electric potential; x and y represent the spatial coordinates in the tray.

b) Use the Add Chart Element button on the far left of the Chart Design ribbon to enter a graph title and axes labels. You must select the axis before adding the label.

c) Print a copy of the graph for each lab partner. Questions: What does theory predict for the shape of the potential surface between two parallel plates? Qualitatively, does your data agree with theory? How can you use the graph to find the electric field strength between the plates? Explain in your discussion section. Use Eqn. 6 to calculate this electric field from your graph.

Part 2 Potential Around a Charged Line

1) Disconnect and remove the parallel plates. Build the arrangement of electrodes in Fig. 3. Connect the wires to the ring and rod using alligator clips (the ring connects to the black, ground terminal on the power supply). Place the rod in the water at the center of the ring. Connect the black wire from the data acquisition board to the power supply’s ground.

! Figure 3. Two-dimensional slice of a long, charged line

2) Again, use the data acquisition equipment to measure the potential at each + symbol in the tray. Include the potential on the ring and the rod. Produce a surface plot of the data. Insert titles and axes labels and delete the legend.

3) Produce a second surface plot of the data and then execute Chart Design, then Change Chart Type from 3-D Surface to Contour. Again, add a title and axes labels.

4) Print copies of your data table and graphs for each lab partner.

5) Question: What is the mathematical shape of the electric potential surface for Part 2? To answer this question, assume Eqn. (7) is correct and use Eqn. (5) to derive an expression for the radial dependence of the electric potential around a charged line. Include this derivation in the data analysis section of your report. As stated in

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Appendix B, the report is more readable if you write equations by hand instead of using text (such as x-squared or x^2). This also eliminates the time necessary to use an equation editor.

Part 3 Field Around a Charged Line

1) Disconnect the computer data acquisition probe and set it aside. Also quit Logger Pro. Obtain a digital multimeter and a two-pronged probe from the front table. Connect a red wire from the meter’s (+) jack to the red jack of the two-prong probe. Connect a black wire between the meter’s COM jack and the black jack of the two-prong probe. In this way, the meter measures the electric potential at the tip of the red probe relative to the tip of the black probe (no longer relative to the power supply’s ground). See Fig. 4.

! Fig. 4. Set-up for Part 3.

2) Using Vernier calipers, measure and record the distance, Δl, between the two prongs of the probe. Raise the voltage on the power supply to between 10 and 20 Volts. If you don’t recall how to read the Vernier Caliper, see Appendix H.

3) Move the probe so both prongs touch the ring and then the rod. Next, probe a few spots in the water. Twist the two-prong probe in the water, about the vertical axis and examine the voltage on the meter.

4) Find the orientation of the two prongs inside the ring, which produces the smallest (near zero) voltage reading. Record this value and make a sketch of the orientation of the probes relative to the electrodes. Draw a short line between the two probe points in your sketch. Question: What electrical quantity does this line represent? Find the orientation of the two prongs, which produces the largest, positive voltage reading. Draw this line on the sketch and record the voltage. Questions: How does this orientation relate to the direction of the first line? What electrical quantity does this line represent?

5) Use the two-prong probe to measure ΔV at seven points inside of the ring, but at radii of 2, 3, and 4 … centimeters out from the center of the ring. Locate the midpoint between the two prongs at these radii. Record the radii and the voltages in a table, then use Eqn. (6) to calculate the electric field.

6) Measure the electric field outside the ring. Questions: Does this agree with your knowledge of electrostatics? Why does the electric field have this value?

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7) Plot a graph of electric field strength versus 1 / r to verify the functional dependence seen in Eqn. 7. In Excel, choose a scatter chart type and then fit the points with a straight line. Check the appropriate box so the R-squared value is displayed. Print a copy of the data and graph for you and your partner.

Questions for the Discussion

1) What two slightly different quantities do Eqns. (2) and (3) refer to?

2) Write a short paragraph describing the relationship between equipotential lines and electric field lines as well as the relationship between field vector diagrams and field line diagrams.

3) What are the sources of experimental error, and which was largest? What categories do these errors fall into? Error can be intrinsic to the quantity itself or found in the measuring procedure or the tool used. Intrinsic means the quantity being measured has some variability built into it by nature. This error is unavoidable no matter how precise the measuring tool. Error in measurement usually refers to the precision of the tool you are using.

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