ELECTRIC POTENTIAL FROM A LINE OF CHARGE – 1302Lab2Prob4

You work with an astrophysics research group investigating the origin of high-energy particles in the galaxy. The group has just discovered a large electrically charged nebula with an irregular shape. In order to understand how this nebula affects the motion of charged particles passing nearby you must find the electric potential near the nebula. Because of its complicated shape you plan to use a computer simulation. You must determine if the simulation results match the physics you know. You decide to test the simulation for the case of a uniformly charged rod, since the situation is simple enough for direct calculation. You decide to calculate the electric potential at a point a short distance from the middle of the rod along its perpendicular axis, and also at a point a short distance from the end of the rod along its parallel axis. You will then compare these results with those obtained using the computer simulation, to see if the simulation can be trusted.

Instructions: Before lab, read the required reading from the textbook and the laboratory in its entirety. In your lab notebook, respond to the warm up questions and derive a specific prediction for the outcome of the lab. During lab, compare your warm up responses and prediction in your group. Then, work through the exploration, measurement, analysis, and conclusion sections in sequence, keeping a record of your findings in your lab notebook. It is often useful to use Excel to perform data analysis, rather than doing it by hand. At the end of lab, disseminate any electronic copies of your results to each member of your group.

Read: Tipler & Mosca Chapter 23. Pay particular attention to Sections 23.1, 23.2 & 23.4 and Examples 23-5 & 23-6.

Equipment

The computer program Electrostatics 3D, a protractor and a ruler.

If equipment is missing or broken, submit a problem report by sending an email to . Include the room number and brief description of the problem.

Warm up

1. Make a picture of the situation. Select an arbitrary point of interest along the perpendicular axis of symmetry. Label relevant distances, angles, and constant quantities. Decide on an appropriate coordinate system. Draw a representative infinitesimal charge element dq somewhere along the rod.

2. Write an expression for the infinitesimal (scalar) electric potential dV produced by dq at an arbitrary point of interest along the perpendicular axis of symmetry.

3. Write an integral total electric potential at the point of interest in terms of dq. The total electric potential due to a charge distribution is found by calculating the contribution from each charge element to the total (scalar) potential field, and summing the contributions. When the charge distribution is continuous, it may be mathematically divided into infinitesimal elements dq; then the individual contributions are added together with an integral

4 Evaluate the integral you set up in question 3 to obtain an expression for the electric potential at the point of interest. In order to evaluate such an integral, all terms in the integrand must be either constants or explicit functions of the integration variable. First, choose an appropriate integration variable. Then, rewrite all variable quantities in the integrand (including dq) in terms of the integration variable you have chosen. Determine appropriate limits for the integration variable you have chosen. Use the Pythagorean Theorem, trigonometry, and the linear charge density to write your integrand(s) in a suitable form.

5. Repeat steps 1-4 for an arbitrary point of interest along the parallel axis.

Prediction

Determine the physics task from the problem statement, and then in one or a few sentences, equations, drawings, and/or graphs, make a clear and concise prediction that solves the task.

Exploration

In the folder Physics on the desktop, open Electrostatics 3D and click on the Point Charge button found on the far left side of the toolbar. You can now place a point charge within the workspace. Once placed, a dialog box opens allowing you to enter the magnitude of the point charge, and whether it is positive or negative.

Click the Closed Equipotential Surfaces button and move the cursor within the workspace to where you would like to evaluate the electric potential. Position and values for potential and field will be displayed on the bottom of the workspace as you move the cursor around the work area. Clicking the mouse will cause an equipotential surface to be displayed, and moving the cursor will display new position and potential values for the new location.

You can reveal simulated electric potential values anywhere in the workspace by moving the cursor where you would like to evaluate the electric field.

To place objects at precise points on the screen you will need to keep track of the position data displayed at the bottom of the workspace. You might find it helpful to map out the (x) and (y) positions required in the workspace to simulate the assigned configurations.

Measure the potential at several locations, as well as the distance from the locations to the center of the charged point object.

Now, explore the line charge configuration. From the toolbar or Add menu select Point Charge and create a line of charge by dragging individual positive charges onto the screen to create a long, uniform line of charge. Hints: make sure the charges are evenly distributed. Optimize the overall charge density and placement of the line on the screen in order to be able to obtain good measurements of electric field vectors. Display equipotential surfaces at the locations of interest for this problem and investigate how the electrical potential depends on position.

After you’ve created a line of charge using individual point charges, create a line of charge using the continuous horizontal linear charge option of Electrostatics 3D. From the toolbar or Add menu select Horizontal Linear Charge. Select a charge density that is similar to the charge density of the line of charge you just created from individual point charges. Display equipotential surfaces at the locations of interest for this problem, and investigate how the magnitude of the electrical potential depends on position. Determine a measurement plan.

Measurement

Measure the electric potential at varying locations along each axis of symmetry. Record the data in your notebook.

Analysis

Use the data for the following analysis (perform on Excel):

1. Using your prediction equation, which is based on Coulomb’s law, calculate the expected electric potential in SI units along each axis of symmetry.

2. Compare the calculated potential to that from the computer simulation on a plot for both data sets (2 axes of symmetry). Include uncertainties. Without them, the results are nearly meaningless.

Conclusion

How did your expected result compare to your measured result? Explain any differences. From your results, which general properties of the electric potential does the simulation faithfully reproduce? What is the specific evidence?

Where is the electric potential defined to be zero? Is this consistent with your results?

Compute the derivative with respect to the distance from the rod along each axis of symmetry. How do these compare with the magnitude of the electric fields from the earlier lab Electric Field from a Line of Charge? Is this consistent with what you know about the relationship between electric field and electric potential? Why or why not?