Electric fields and electrical potential lab reprot

timer Asked: Feb 5th, 2019

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Electric Fields and Electric Potential Introduction In this lab you will map out electric potential (V) around electrodes in a water bath. This is done by measuring the voltage drop between one fixed location and a large number of various locations in the water. From these measurements you can also determine the associated electric field. Text Reference: Young and Freedman §§21.4; 21.6; 23.2; 23.4-5. Theory In this experiment you will study the relationship between electric potential V and the electric field E. The electric field vector always points from high potential to low potential, and in this lab you will learn that, roughly speaking, the electric field strength E (the magnitude of the electric field vector E) at any point of interest can be easily calculated from a map of the electric potentials by taking the potential difference between two closely-spaced points on either side of the point of interest and dividing it by the distance between those two closely-spaced points. This description of the relationship between E and V is quite rough; but after completing this lab, you should be able to explain exactly how to calculate the electric field strength (E) and the direction of E from any detailed potential map. The electric field due to highly symmetric charge distributions can be determined using Gauss’s law. For a long, straight wire the electric field can be derived: 𝑬= Where k = * +,-. $%& ' 𝑟̂ (1) and 𝜖0 is the permittivity of free space, 𝜆 is the charge per unit length of the wire, r is the distance from the center of the wire to the field point, and unit vector 𝑟̂ gives the direction of the electric field. If the electric field E between two points a and b is known, then the potential difference between those two points can be calculated as follows: 9 Δ𝑉 = − ∫: 𝑬 ∙ 𝑑𝒍 (2) where the quantity 𝑑𝒍 represent the path taken between the points a and b. This relationship can also be inverted and the components of the electric field can be expressed in terms of derivatives such as <= < 𝑬 ; = − <; (3) Where <; denotes a partial derivative. That is, take the derivative but hold all other variables constant except the one you are differentiating with respect to. To create an electric field in the first part of the experiment you will use parallel plates with the conducting ring in a water tank. The arrangement of conductors in the second part of this lab will be coaxial rings. The inner ring will have an electric field described by equation (1) which when substituted into equation (2) can be integrated to find the potential difference between the inner ring of radius a and a second point at distance of r from the common center of the two rings: ' 𝑉(𝑟) = 2 𝑘 𝜆 𝑙𝑛 DEF (4) Page 1 of 4 It must be noted that the experiment is done with the probes submerged in water so equation (1) and (4) must be modified for the presence of water. Water is a dielectric material (a nonconductor) so we need to replace the permittivity of free space which is part of the constant 𝑘 by 𝑘𝜖0 , where 𝑘 is the dielectric constant. Procedure A separate sheet of the procedure will be posted. Please download it and follow the steps during the experiment. Part 1a – Parallel Conductors Create the experimental set-up shown in the figure below without the conducting ring. We use an AC source to generate the field in the water tank; a DC source would result in a buildup of charge near the electrodes. You will use a digital multimeter (DMM) to measure the RMS voltage drop between a selected fixed location and a number of various locations in the tank. There should be a piece of graph paper attached under the tank from which you can read coordinates. Use the signal generator (f = 200 Hz, amplitude maximum) to generate an AC signal between the bars. Pick one of the two bars as the fixed location for the black lead from your digital multimeter (DMM); this location becomes the SELECTED location at which the potential is zero. Use the red DMM lead to measure the RMS voltage at various locations within the tank. Begin by recording the RMS voltage at the bars (note that this voltage is independent of the placement of the red lead on the bars; try it!). Record your data on a separate copy of graph paper. Your objective is to trace two equipotential lines (see the instruction in class); one from left to right, the other one only in left half of the tank and take advantage of the symmetry of the set up to deduce what the lines look like in the other half of the tank. Using the single lead probe measure the potential of a number of equally spaced points in direction perpendicular to the bars. Record the voltage measurements from the DVM directly on a sheet of paper that is a copy of the investigated system. In GA make a graph of voltage V vs. distance y from grounded bar. What does the slope of the linear fit represent? Part 1b – Parallel Conductors with Metal Ring. Add the conducting ring in the center between the bars. You will then draw the appropriate equipotential lines (lines of constant potential). You may ignore errors for this part. Your objective is to trace two equipotential lines in half of the tank. Each line will have a distinct voltage; you must find enough locations having that voltage to trace the equipotential line throughout your selected half. Indicate your actual data points with small circles, and label each line with its RMS voltage. Be sure to measure the RMS voltage at the center of the tank and at several other locations within the ring. What is the electric field strength inside the ring? c) Now, switch to the two-point probe and check the electric field inside the middle ring and outside of the ring at location close to the edges of the bars. The two-point probe is simply a set of DMM leads clamped together at a fixed small separation distance. The two points of the probe should be on either side of your selected point of interest. To determine the direction of E, you will have to rotate the probe until the DMM reads the largest possible RMS voltage. The RMS electric field strength is this largest RMS voltage divided by the distance between the pins. How does the direction of E at various locations relate to the direction of the nearby equipotential lines? On your tracing of the equipotential lines for this case, make a qualitative drawing the electric field lines; please use different colored ink or pencil so that the two sets of lines are easily distinguishable. Page 2 of 4 Part 2 – Coaxial Rings a) With the signal generator turned off remove the bars and the ring from the tank. Use the largest ring and the smallest ring provided in the tray to set up a pair of oppositely-charged coaxial rings in the water tank; the leads for the signal generator should be connected directly to the rings using alligator clips. Select the inner ring as your location of zero potential. Based on the previous experiment, predict whether the cylinders will themselves be equipotential surfaces. Make measurements to test your prediction and discuss the results. If your prediction is incorrect resolve the inconsistency between your prediction and your data. Trace two equipotential lines (start on X axis at r1 = 5 cm and r2 = 7 cm from common center) (see the instruction in the lab). b) Measure the RMS voltage at five different distances measured from the common axis (all five distances should be larger than the radius of the inner ring and smaller than the radius of the outer ring); for each distance r, measure Vrms along two different directions-positive X and Y. Compute the average value of Vrms for each distance. Enter these measurements in Graphical Analysis, creating the necessary columns to plot Vrms vs. ln(r). Your result should be linear; determine the slope with errors, and state your result for Vrms (r) in your conclusions. Use the value of the slope to determine the charge per unit length 𝜆 on the inner ring and its error. c) Switch to the fixed spacing probe assembly to make measurements of the electric field in between the two circles. Measure the electric field every 1.0 cm along two different directions: X and Y. Record these E values as well as the distances r from the center of the circles (choose the r value at a point exactly in the middle of the probe tips). Represent the measured E values with little vectors on your sketch. Your measurements should start at the location of the inner cylinder. Enter the measured E values and corresponding r values in Graphical Analysis. Make a plot of E vs. 1/r, making use of the calculated column option as needed. Use the slope of the straight-line relationship to make determination of the charge per unit length 𝜆 of the inner ring and its error. d) Compare two experimental values of the charge per unit length 𝜆. Page 3 of 4 e) Using the fixed spacing probe assembly measure the electric field E (value and direction) at two points: i) the cross section point of the first EP at r1 = 5 cm and the vertical line X=18 in the first quadrant. Represent values with little vectors on your sketch. ii) outside of the outer ring. Is this result consistent with Gauss’s law? Please note that with the alternating current the charge per unit length is really varying in a sinusoidal fashion with a period of about 5 ms. Since you are measuring the RMS values of the voltage and E-field, you are really determining the RMS value of the charge per unit length of the inner ring. To make this estimate, you must assume that the two cylinders are infinitely long. This assumption is not absurd because the space between the rings is filled with water, which is a reasonably good conductor, and because we are using AC. For either direction of the alternating current, charges quickly gather on the upper and lower surfaces of the water until the field in the water is purely radial, and radially symmetric, as it would be for infinitely long conductors. Hints and Notes Remember to attach your drawings of the equipotential surfaces for the coaxial rings. Also include the plots of Vrms vs. ln(r) and E vs. 1/r. In addition to the questions asked above you should also address the following: From part 1, what do your measurements of the RMS voltage inside the ring tell you about the E - field inside the ring? From your data in part 2, what will be the shape of the equipotential surfaces? Take additional readings if you are not sure. Is the fact that E is proportional to 1/r consistent with the linear dependence of the electric potential on ln(r)? Explain. Are your calculated values for λ, the charge per unit length, consistent with each other? Is the electric field perpendicular or parallel to the equipotential lines you have drawn? In this experiment you used two measuring probes. The single lead probe was used to measure the RMS voltages and the two-point probe was used to determine the electric field. Explain why the two-point probe is needed to measure the electric field. Questions for self-assessment 1. 2. 3. 4. 5. 6. Is the electric field a scalar or a vector quantity? What is the meaning of the electric field? What is the unit of the electric field? Is the electric potential a scalar or a vector quantity? Is the electric field vector parallel or perpendicular to the equipotential line? What is the relationship between electric potential and electric field? Page 4 of 4
Template - PHY 132 Lab report 1. Cover Page: • (10 Points) (3 pts) Title of experiment, name of the student, his/her group number, and names of group members. PHY 132 Lab Title Name: Partners: Section: TA: Date: • (7 pts) Abstract: Briefly describe the title. State aim. Basic idea/concept explored. It shouldn’t be any longer than two paragraphs, and can be shorter if you are skilled at being precise. State key results with a conclusion (whether experiment was successful or not.) If there are no numbers here, you are not being specific enough. If there is only a sentence or two stating the results of your measurement you are not really summarizing the report. 2. Experimental Data: • • ,_ Group #: (20 Points) (15 pts) Attach data sheets and graphs (5 pts) Clearly mention the quantities measured during experiment like lengths, weights, etc. NOTE: This is not a place to show calculations. However, you can do some minor calculations like unit conversion. Tables are really nice for organizing data. They are not required though. If you are not using tables, then maybe have a “subsection” for different kinds of data. For example: PART 1 Inner radius = XX cm Outer radius = XX cm PART 2 Inner radius = XX cm Outer radius = XX cm Just make it really easy to identify what you measured, what the measurement was, and …. DON’T FORGET THE UNITS! 3. Data Analysis and Results: (25 Points) This is where you will show all of your calculations. This will generally include many equations. Be sure to include all of the following: • (5 pts) State the equation(s) you are going to use. (example: Newton’s second law, F=ma) and state tits (heir) name(s), while clearly define all variables/symbols/parameters used (e.g., F is the force in Newtons, m is the mass in kilograms, and a is the acceleration in m/s2) This could be done in many ways. You could have a list of all your parameters and their meaning at the beginning of this section; you could define each parameter as they appear in the report, or any other way that ensures that any reader could identify what all of your symbols mean. Explain any derivations. If an equation is given in the lab manual then you can generally just use it, but if you have to combine any equations to get the one you are using or the source of your equation is in any way unclear you should explain where it comes from. • (10 pts) Calculate/analyze the required quantities in the experiment. Some of them might require derivation or plotting as suggested by your TA during the lab. Just because you wrote down the right equation doesn’t mean that you used it correctly, so if you show me the numbers you used I may catch small errors you made and take off fewer points than if you just show an equation and a result that is incorrect. If you have to do the same calculation many different times you can show a single sample calculation and then just list the various results, but still show at least one calculation for each different equation that you use. • (10 pts) There should always be some sort of error calculations as well. That may be calculations of percent error, percent difference, or error propagations, or all of the above. Clearly state results with their errors in an appropriate format in a tabulated or a clear fashion (e.g. Resistance of coil, R = 8.3 +/- 0.1 ohm). Please do your best to organize this section in such a way that it is easy to identify what calculation you are doing. Also, this section does not need to be typed. If it is easier for you to simply write out your calculations and then include that work here, that is okay. Please, please be sure that I will be able to read your work EASILY. 4. Discussion and Conclusion: (25 Points) Here you answer questions about WHY you did the things you did. Show your understanding here. For example, why did we use the slope of your graph to calculate whatever value? For what purpose was a particular step included in the procedure? • • (5 pts) Describe the aim for the experiment with the physical concept explored. Define/describe the physical concept clearly explaining the laws involved if possible with examples. (10 pts) Describe the measurements and the key results with uncertainties and units obtained, and clarify how they help prove the concept explored above. Interpret your graphs and discuss what trends were observed and what the relationship of the variables in your experiment • • was. Also, answer the questions mentioned in the lab manual (at the end) or points asked by the TA to discuss. (5 pts) Discuss whether the results verify the experiment or if they don’t, why? This is an opportunity to regain some of the points lost in case the experiment went wrong and the results do not verify by explaining the possible mistakes and unexpected sources of errors. (5 pts) What are the usual sources of errors? You need to have your signed data sheets and graphs attached to this report. Whether they are included where they belong in the body of the report, or simply attached here at the end. They can even be attached as separate files when you submit your report, but it is recommended you not do that if you can include them somewhere in the report. Remember, the most important part of your report is that it be CLEAR! If your TA can’t read the report, he can’t grade it.

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