E-Field+Plotting+manual

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Arizona State University *

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Dec 6, 2023

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1 ELECTRIC FIELD PLOTTING SUMMARY In this experiment, we will map equipotential surfaces in order to plot electric field lines between oppositely charged conductors of various geometries. EQUIPMENT Field Mapping Apparatus & ASU PIRT 3D printed probe, Galvanometer, Pasco 850 Universal Interface, two short patch cables (red/black), two long patch cables (red/black), ruler INTRODUCTION The electric field at a given point is the electric force per unit charge at that point. Contrary to an electric force, which requires the presence of two charges, the electric field represents the possibility for an electrostatic force to be applied to a second charge if it was at that location , based on the magnitude of the single charge and the distance from the charge. It can be written as 𝐸𝐸 �⃗ = 𝐹𝐹 ⃗ 𝑞𝑞 (1) where 𝑞𝑞 is the charge, 𝐹𝐹 ⃗ is the electric force acting on 𝑞𝑞 , and 𝐸𝐸 �⃗ is the electric field that charge 𝑞𝑞 is in. Electric field lines are curves drawn in a region of space to help us visualize invisible electric fields. Electric Potential is electric potential energy per unit charge, 𝑉𝑉 = 𝑈𝑈 𝑞𝑞 (2) where 𝑞𝑞 is the charge, 𝑈𝑈 is the electric potential energy and 𝑉𝑉 is the electric potential. Similar to the relationship between electric field and electric force, the electric potential only requires the presence of one charge, and represents the possibility for there to be electric potential energy if there was a second charge at a given location. Every point in space has it’s own value of electric potential; however, two points that have the same value of electric potential lie on what are called equipotential surfaces . Just like electric field lines help us visualize electric fields, equipotential surfaces can be drawn to help us visualize electric potential. Electric field vector components can be calculated from the change in electric potential between two points in space. For example, in 2-dimensions, using x-y coordinates we can write 𝐸𝐸 𝑥𝑥 = − 𝛥𝛥𝑉𝑉 𝛥𝛥𝛥𝛥 𝑎𝑎𝑎𝑎𝑎𝑎 𝐸𝐸 𝑦𝑦 = − 𝛥𝛥𝑉𝑉 𝛥𝛥𝛥𝛥 (3) where 𝛥𝛥𝛥𝛥 and 𝛥𝛥𝛥𝛥 give the separation between two points along the x and y directions.

2 In this experiment, we will map equipotential surfaces in order to plot electric field lines between oppositely charged conductors of various geometries. In order to do so, we must follow these simple rules: 1) Electric field lines do not intersect. This follows from the definition. At any point in space there is a unique electric field vector and this vector is tangent to the electric field line. Field lines do not intersect, since a vector cannot be tangent to both lines at the intersection point. 2) Electric field is stronger when the field lines are closer to one another. This follows from eqn. (3). If the separation is smaller, electric field magnitude is larger. 3) Electric field lines point from (+) to (-). This follows from the definition of the electric field. 4) Electric fields are always perpendicular to equipotential surfaces. Once again, this follows from eqn. (3). Let us imagine splitting the electric field into two components: One perpendicular to the equipotential surface (y) and the other parallel to the surface (x). We can write the parallel component as 𝐸𝐸 𝑥𝑥 = − 𝛥𝛥𝛥𝛥 𝛥𝛥𝑥𝑥 Since the potential is the same on every point on the equipotential surface by definition, ΔV is zero across the surface. Therefore 𝐸𝐸 𝑥𝑥 always has to be zero, which means the electric field vector has to be perpendicular to the equipotential surface. 5) Electric field lines are always perpendicular to the conductor surface. This follows from the conducting surface itself being an equipotential surface. Also, the electric field is zero inside the conductor. 6) Equipotential surfaces do not intersect. This follows from the definition. Every point on an equipotential surface has the same electric potential. Surfaces do not intersect, since the intersection point cannot have two different potentials. Figure 1: Electric field lines and equipotential surfaces of an electric dipole

3 PROCEDURE 1) In this experiment, we will use two different field plates which are shown in Figure 2. We will start with the dipole (two-point) configuration. Figure 2: Parallel plate and dipole configurations 2) Turn the Field Mapping Apparatus upside down. Unscrew the two screws at the center. Place the dipole field plate with the conducting side facing away from the apparatus, as shown in Figure 3. Put the thumb screws back and tighten them to be snug against the plates. Figure 3: Mounting the field plate on the back of the apparatus

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4 3) Set up the equipment as shown in Figure 4. Add a corresponding figure of the dipole geometry on top using the alignment pegs so that the position of the paper diagram closely matches the position of the metal thumbscrews on the opposite side. Make sure the thumbscrews are tight to ensure solid electrical contact with the conducting sheet. Use the provided binder clips to secure the paper diagram to the top of the board. Inside of the Pasco Capstone software, set the output voltage to 8.0V by selecting the Signal Generator tool on the left side toolbar. Under the “850 Output 1” tab change the waveform to DC then type “8” into the DC Voltage textbox. Finally , click the “On” button. Figure 4: The experiment setup

5 The eight resistors on the back side of the apparatus divide the total voltage into ΔV=1.0V steps. This means we will be able to measure seven equipotential surfaces with voltages 1V, 2V, 3V, 4V, 5V, 6V and 7V with our setup. 4) Connect the galvanometer to E4 on the apparatus. Make sure the metal conducting tip of the probe goes under the apparatus, touching the conducting sheet, and the part with a hole on the tip goes over the provided paper. 5) Find the points where the galvanometer reads zero and mark them through the hole with a pencil. Choosing points that are about 1cm apart should be enough to see the pattern. The galvanometer is directly reading the current, or flow of charge; however, we can use the measurement of current to infer the electric potential difference between E4 and the tip of the probe. Since E4 is at 4 V, moving the probe to any point that causes no deflection (zero current) means that point must lie on the 4 V equipotential surface. Connect the dots smoothly to draw the equipotential surface. 6) Repeat this process for E3, E5, E2, E6, and E1 & E7 if you can. For each one, make sure to take enough measurements to fill the page. Refer to the figures on the next page to make sure you are measuring in the right places. 7) Draw the electric field lines following the rules listed in the introduction . Indicate the direction of the electric fields with arrows.

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7 8) Turn the power off. Now turn the apparatus over and swap out the conducting sheet for the parallel plate geometry. Turn the power on and repeat your measurements for the new geometry. 9) On your equipotential map use the rules established on page 2 to draw in at least 7 electric field lines. Make sure at least one E-field line is centered between the positive and negative terminals. From there, spread out and fill the page with 3 field lines above and 3 field lines below your first centered line. 10) For both of your E-field drawings we will calculate the E field at two different points. To do this, follow these steps: a) In the middle of the paper along the centered E-field line, draw a point directly in the middle of the E4 and E5 equipotential lines. Using the ruler as a straight-edge, draw a line that connects the center dot you made to the E4 and E5 potential lines, taking the shortest path. Measure this distance and record it in your worksheet. b) Use the ruler to measure 6 cm above or below from your first point and mark another point directly in the middle of the E4 and E5 equipotential lines. Draw a line using the same rules as before that passes through this point. Measure the distance of this line and record it in your worksheet. Note: The electric field in the area around your second point is not actually a straight line—it curves in a way that follows the rules outlined on page 2. However, we can approximate the distance as a straight line using a ruler. If you are interested, discuss with your TA what you think you would need to do if you had to get a more accurate distance measurement. c) Calculate the magnitude of the electric fields at your centered and distant points. Call them 𝐸𝐸 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 and 𝐸𝐸 𝑓𝑓𝑐𝑐𝑓𝑓𝑐𝑐𝑓𝑓𝑐𝑐 . Reference Eq. (3), but only calculate the magnitude (no need to separate x and y components). Show your calculations in a convenient place on your worksheet. A final result without calculations will not receive credit. Remember to include units. 11) Tidy up your lab station. Restart the computer, unplug all patch cables, flip the apparatus upside down, unscrew the thumb screws, remove the conducting sheet, and screw the thumb screws back in so they don’t end up on the floor. 12) Read this entire paragraph. Each student must take a picture of the E-field drawings before leaving the lab. If you don’t have access to a camera, discuss with your TA and/or lab partners to make arrangements. Each student is responsible for turning in their complete report, including copies of the drawings. All post-lab questions should be answered independently and in your own words. Discussion of the concepts with your lab partners is encouraged, but copying is not allowed. For full credit on the drawings, at least five equipotential lines (but preferably seven, if you can find all of them) and at least seven electric field lines with arrows must be present and clearly marked. Make sure the drawings cover a majority of the page.

8 E L E C T R I C F I E L D P L O T T I N G – L A B R E P O R T Name: Partners: TA: Cleaning up & Signing out (3): Data/Plots (40): Analysis (15): Post Lab Qs (30): Lab Report Total (88): ANALYSIS 1) Two-Point charges: 𝐸𝐸 1 , center = ……………… (unit: _______ ) 𝐸𝐸 1 , fringe = ……………… (unit: _______ ) 2) Parallel plates: 𝐸𝐸 2 , center = ……………… (unit: _______ ) 𝐸𝐸 2 , fringe = ……………… (unit: _______ ) 3) Ratio 𝑅𝑅 1 = 𝐸𝐸 1 , fringe 𝐸𝐸 1 , center = ……………… 𝑅𝑅 2 = 𝐸𝐸 2 , fringe 𝐸𝐸 2 , center = ………………

9 POST LAB QUESTIONS 1) For the dipole configuration: a) Are the equipotential surfaces equally separated across a majority of the page? b) Is the electric field uniform across a majority of the page? What makes it uniform/non-uniform? 2) For the parallel-plates configuration: a) Are the equipotential surfaces equally separated across a majority of the page? b) Is the electric field uniform across a majority of the page? What makes it uniform/non-uniform? 3) a) Is 𝑅𝑅 2 greater than 𝑅𝑅 1 ? b) Based on the ratios 𝑅𝑅 2 and 𝑅𝑅 1 , which geometry generates a more uniform electric field? Briefly explain your answer, and make sure to mention the distribution of charge and how it affects the equipotential surfaces. Remember to consider the direction of causality (what causes what?). Don’t forget to answer the questions on the next page!

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10 4) Why do electric field lines not intersect? Explain briefly. A small drawing would be helpful. 5) Why do equipotential surfaces not intersect? Explain briefly. A small drawing would be helpful. 6) Why are electric field lines always perpendicular to equipotential surfaces? Explain your argument mathematically (just showing math without explaining your understanding of it is not sufficient).

E-Field+Plotting+manual

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