Download Lab/Inquiry Handout

Electric Potential Point Charge Inquiry
Objective:
To use approximate the electric potential for a point charge using a spreadsheet.
Theory:
The change in electric potential is equal to the negative of the area under an
“Electric Field versus Position” graph. In the case of a point charge of charge Q at the origin, the
electric field is given by the expression
and the “zero electric potential” reference point is chosen at infinity. Question: Why is the origin
a bad choice for the reference point for a point charge?
The electric potential at some distance r with respect to the reference point at ri = ∞ is the area
under the “Electric Field versus Position” graph from ri = ∞ to r. The problem is that the area is
not a simple geometrical shape whose area formula we know. With Calculus one can obtain the
formula in the textbook exactly. However, we can roughly approximate the area without
Calculus using basic geometry by adding up little rectangles of area whose base width is –Δr and
whose height is E(r), electric field strength at that position as shown below:
300
250
Electric Field Strength
200
150
100
50
0
0
0.2
0.4
0.6
0.8
1
Distance
While this process would be tedious by hand, it is relatively easy to perform using a spreadsheet
such as Excel.
Procedure:
We are going to model the electric potential for a “1/9 nC” point charge. We have chosen this
value to make things easier as kQ = 1. If you don’t like our choice, feel free to choose any other
value for kQ as it will make no difference.
1. Go into Excel or other spreadsheet program
2. In cell A1, type “kQ=” to label cell B1’s contents
3. In cell B1, type +1 (You can type a different value if you want a different charge)
4. Name cell B1 as kQ using your spreadsheet’s cell name command
5. In cell A2, type “delR =” to label cell B2’s contents as our rectangle’s width
6. In cell B2, type 0.01
7. Name cell B2 as delR using your spreadsheet’s cell name command
8. In cell A3, type “r” to indicate that the column is for the radial distance
9. In cell B3, type “E” to indicate that the column is for the electric field
10. In cell C3, type “V” to indicate that the column is for electric potential
11. In cell A4, type +0.1 as our first radial distance
12. In cell B4, type the formula for calculating the electric field for the distance in cell A4
13. In cell A5, add delR to the value of A4 (ie +A4+$delR )
14. Copy the formula from A5 and paste down the column to A1004
15. Copy the formula from B4 and paste down the column to B1004
At this point, you should have a spreadsheet with the magnitude of the electric field for a
point charge as a function of radial distance. Using equation 15.6 in Serway to check your
results in Column B for r=2, r=4, and r=6. If there is a discrepancy, you must fix the error in
the spreadsheet before you continue.
16. In cell C1004, multiply delR by the value in cell B1004.
This is the electric potential at this point with respect to infinity where we have ignored any
previous area under the curve for larger radial distances (hopefully a slight error). The minus
sign for the area in calculating the electric potential has been canceled with the minus sign
for the width of the rectangle ( -1*delR).
17. In cell C1003, add the contents of cell C1004 to the product of delR and the value of cell
B1003.
18. Copy the formula in cell C1003 and past in column C upward to C4.
19. On a separate sheet make a plot of electric potential versus position and electric field
versus position. This is an x-y plot using Column A as x and Column B & Column C as
y’s. Expand the x-axis of your graph from 0 to 2. Your graph should look like Figure 16.5
in Serway otherwise you must fix the errors in your spreadsheet.
Inquiry A:
1. Compare the electrical potential results from your spreadsheet and formula 16.4 in
Serway for r = 0.2, 1, 2, 3, 4, 5, 8, etc. How does your approximation compare with the
Calculus results? Is the formula in the textbook reasonable?
2. Change the value for delR to 0.1. Compare the electrical potential results from your
spreadsheet and formula 16.4 in Serway for r = 0.2, 1, 2, 3, 4, 5, 8, etc. How does your
approximation compare with the Calculus results?
3. Change the value for delR to 1. Compare the electrical potential results from your
spreadsheet and formula 16.4 in Serway for r = 0.2, 1.2, 2.2, 3.2, 4.2, 5.2, 8.2, etc. How
does your approximation compare with the Calculus results?
4. Based upon your results from 1-3, what value of delR will provide the best
approximation results to formula 16.4 in the testbook. Draw a sketch to explain how
some values of delR cause large errors. Hint: Look at the graph on the first page of the
handout.
Prepare your whiteboard for an outbrief!!
Inquiry B:
1. Develop a spreadsheet for approximating the electric potential for a cylinder whose
electric field is
where λ is the charge per unit length on the cylinder and kλ serves the same function as
kQ in the point charge spreadsheet. Choose a delR of 0.1 and perform the following
tasks:
a) Predict the electric potential at r = 0.2, 1, 2, 3, 4, 5, and 8
b) Plot the electric potential versus distance graph. How does this change versus your
point charge results.
c) Compare your results in a) to the Calculus formula:
Where ro is your “zero potential” reference point which we will take to be at the
radial distance equal to the value of cell A1004 + delR. Question: Why is infinity a
bad choice for a reference point in this problem?
Prepare your whiteboard for an outbrief.