Online-Lab-3_Electric-Potential

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Apr 3, 2024

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PHYS 1402 Lab 3: Electric Potential Name: _____________________ Objectives:  To review the mathematical definition of work and potential energy in a conservative force field as the sum of the product of displacement and the force-component in the direction of the displacement: W = Σ F cos θ Δs  To understand the definition of electrical potential, or voltage.  To map and examine the potential distribution in two dimensions for a variety of simple charge configurations.  To learn the relationship between electric field lines and equipotential surfaces. Marketable Skills: This course assesses the following Core Objectives. In this assignment, you will develop the following marketable skills : Critical Thinking  Analyze Issues  Anticipate problems, solutions, and consequences.  Apply knowledge to make decisions  Detect patterns/themes/underlying principles  Interpret data and synthesize information Communication  Summarize information  Use proper technical writing skills Personal Responsibility  Accept responsibility  Exhibit Time Management  Show attention to detail  Learn and grow from mistakes Empirical Quantitative  Communicate results using tables, charts, graphs  Contextualize numeric information/data  Demonstrate logical thinking  Draw inferences from data, use data to formulate conclusions  Use appropriate calculations to solve problems 1

Conservative Forces It takes work to lift an object under the influence of the Earth’s gravitational force. This increases the gravitational potential energy of the object. Lowering the object releases the gravitational potential energy that was stored when it was lifted. When you studied the gravitational force, you applied the term conservative to it because it allows the recovery of all the stored energy. You have now found experimentally that the work required to move an object under the influence of the gravitational force is path independent. This is an important property of any conservative force. Given the mathematical similarity between the Coulomb force and gravitational force laws, it should come as no surprise the experiments confirm that the Coulomb force is also conservative. Again, this means that the work needed to move a charge is independent of the path taken between the points. From the definition of work W = Σ F cos θ Δs = Σ qE cos θ Δs is the work done by the electric field E to move a small test charge q between A and B. Activity 1: Work Done on a Charge Traveling in a Uniform Electric Field 1. Suppose that a small positive test charge q is moved a distance d from point A to point B along a path that is parallel to a uniform electric field of magnitude E. Question 1: What is the work done by the field on the charge? Show your calculation. Question 2: How does the form of this equation compare to the work done on a mass m traveling a distance d parallel to the almost-uniform gravitational force near the surface of the Earth? 2. The charge q is now moved a distance d from point A to point B in a uniform electric field of magnitude E, but this time the path is perpendicular to the field lines. Question-3: What is the work done by the field on the charge? Show your calculation. 2

3. The charge q is now moved a distance d from point A to point B in a uniform electric field of magnitude E. The path lies at a 45  angle to the field lines. Question-4: What is the work done by the field on the charge? Show your calculation. Electrostatic Potential Energy and Potential The change in electrostatic potential energy is defined as the negative of the work done by the electrostatic force in moving a charge from point A to point B using any path consisting of a series of small increments of length  s . Thus, using U as the symbol for potential energy, ∆U elec = U B − U A =− ΣqE cos θ Δ s The electrical potential difference ΔV = V B − V A is defined as the charge in electrical potential energy  U elec per unit charge ΔV = ∆U elec q The electric potential difference has units of Joules/Coulomb (J/C). Since a J/C is defined as one volt , the potential difference is often referred to as voltage. Using the equation above for potential energy change, and the definition of potential difference, write the equation for the potential difference as a function of E and  s . Single Point Charge The simplest charge configuration is a single point charge. As you saw in Lab 2, a positive point charge q produces an electric field that points radially outward in all directions. The potential difference between any two points in space A and B can be found using calculus. The result is that if A is infinitely far away, and B is a finite distance r from a point charge q, then the potential difference, V, is given by the expression V = k e q r In this case, the reference point for the potential difference is at infinity, and the potential difference is simply referred to as “the potential”. Activity-2: Electric Potential of a point charge You will be using PhET simulation Charges and Fields (https://phet.colorado.edu/sims/html/charges-and-fields/latest/charges-and-fields_en.html). This 3

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simulation allows you to place multiple positive and negative point-charges in any configuration and look at the resulting electric field and corresponding electric potential. 1. Open the PhET simulation. 2. You can click and drag positive charges (red) or negative charges (blue) into the main screen. If you select Electric Field in the menu, arrows will appear, showing the direction of the electric field. Faint arrows indicate that the electric field is weaker than at locations where the arrows are brighter (this simulation does not use arrow length as a measure of field magnitude). 3. You can drag a positive charge into the main screen and place the crosshairs of the electric potential (voltage) measurement tool on top of any point for which you would like to see the voltage (for this tutorial, voltage will be taken to be the same as electric potential). To move the electric potential (voltage) measurement tool, drag it by clicking and holding onto the body of the tool, and not the crosshairs. Clicking Plot (the pencil icon) on this device shows an equipotential line. All points along an equipotential line have the same potential. 4. Feel free to play around with the simulation. When you are done, click the Reset button. The electric potential (voltage) at a specific location is equal to the potential energy per unit charge a charged object would have if it were at that location. If the zero point of the voltage is at infinity, the numerical value of the voltage is equal to the numerical value of work done to bring in a unit charge from infinity to that location. 5. Select Values and Grid in the menu and drag one positive charge to the middle of the screen, right on top of two intersecting bold grid lines. 6. Using the voltage meter, you should find that 1 m away from the charge, the voltage is 9 V. Question-5: What is the voltage 2 m away from the charge? What is the voltage 3 m away from the charge? Express your answer numerically in volts to two significant figures. Based on this result, the electric potential (voltage) is inversely proportional to the distance r from the charge: V ∝ 1/r. Recall that the magnitude of the electric field E ∝ 1/r 2 . Equipotential Surfaces Sometimes it is possible to move along a surface in a non-zero electric field without doing any work. Thus, it is possible to remain at the same potential anywhere along such a surface. If an electric charge can travel along a surface without any work being done, the surface is defined as an equipotential surface . Consider the three different charge configurations shown below. Where are the equipotential surfaces? What shapes do they have? The purpose of the following activity is to explore the pattern of potential differences in the space around a charge distribution. This pattern of potential differences can be related to the electric field caused by the charges that lie on the conducting surfaces. 4

Activity-3: Electric Field Lines and Equipotentials Another way to study voltage and its relationship to electric field is by producing equipotential lines. Just like every point on a contour line has the same elevation in a topographical map, every point on an equipotential line has the same voltage. You will be using PhET simulation Charges and Fields 1. Click plot on the voltage tool to produce an equipotential line. Produce many equipotential lines by clicking plot as you move the tool around. You should produce a graph that looks similar to the one shown below. Place several E-Field Sensors at a few points on different equipotential lines and look at the relationship between the electric field and the equipotential lines. Question-6: Which statement is true? a) At any point, the electric field is perpendicular to the equipotential line at that point, and it is directed toward lines of higher voltages. b) At any point, the electric field is parallel to the equipotential line at that point. c) At any point, the electric field is perpendicular to the equipotential line at that point, and it is directed toward lines of lower voltages . Equipotential lines are usually shown in a manner similar to topographical contour lines, in which the difference in the value of consecutive lines is constant. Clear the equipotential lines using the Erase button on the voltage tool. Place the first equipotential line 1 m away from the charge. It should have a value of roughly 9 V. Now, produce several additional equipotential lines, increasing and decreasing by an interval of 3 V (e.g., one with 12 V, one with 15 V, and one with 6 V). Don't worry about getting these exact values. You can be off by a few tenths of a volt. Question-7: Which statement best describes the distribution of the equipotential lines? a) The equipotential lines are equally spaced. The distance between each line is the same for all adjacent lines. b) The equipotential lines are closer together in regions where the electric field is stronger. 5

c) The equipotential lines are closer together in regions where the electric field is weaker. Now, remove the positive charge by dragging it back to the basket, and drag one negative charge toward the middle of the screen. Determine how the voltage is different from that of the positive charge. Question-8: How does the voltage differ from that of the positive charge? Now, remove the negative charge, and drag two positive charges, placing them 1 m apart, as shown below. Question-9: What is the voltage at the midpoint of the two charges? a) Exactly twice the voltage produced by only one of the charges at the same point b) Zero c) Greater than zero, but less than twice the voltage produced by only one of the charges at the same point Now, make an electric dipole by replacing one of the positive charges with a negative charge, so the final configuration looks like the figure shown below. Question-10: What is the voltage at the midpoint of the dipole? 6

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Activity 2-2: Drawing Equipotentials Suppose you have a positive test charge, and you move it in space some distance from the charge below. The arrows represent electric field lines. Question-11: Given that the electric field is non-zero, what path could you move a test charge along without doing any work, i.e., for which E cos θ Δ s is always zero? What is the shape of the equipotential surface? (Remember that in general you can move in 3D). Question-12: What is the relationship between the direction of the equipotential lines you have drawn (representing that part of the equipotential surface that lies in the plane of the paper) and the direction of the electric field lines? Explain. Question-13: Draw some equipotential surfaces for the charge configuration shown below that results from two charged metal plates placed parallel to each other. (The field lines drawn represent a uniform electric field, which is correct in the central region of the plates, away from the edged.) What is their shape? 7

Question-14: Draw some equipotential surfaces for the electric dipole charge configuration shown below. An electric dipole has the same magnitude charge but with opposite sign. What is their shape? In PhET simulation, make an electric dipole. Trace out equipotential lines for potential differences of ±2V, ±4V, ±6V, ±8V, and ±10V. Don't worry about getting these exact values. You can be off by a few tenths of a volt. Take a screenshot and upload it. Question-15: Are the equipotential lines what you expected? Did they agree with your predictions? Explain. Click on “Electric Field” in the simulation. Take a screenshot and upload the picture. Then, use an E-Field Sensor to measure the electric field at a few points while looking at the relationship between the electric field and the equipotential lines. Question-16: Which of the following statements is true? a) The electric field strength is greatest where the voltage is the greatest. b) The electric field strength is greatest where the equipotential lines are very close to each other. c) The electric field strength is greatest where the voltage is the smallest. Reference: Sokoloff, David R.. RealTime Physics Active Learning Laboratories Module 3 Electricity & Magnetism, Wiley Higher Ed. Edition. 8