Equipotential Lines

Objective:

In this exercise students will calculate the electric potential in the two-dimensional region around 1, 2, 3, or 4 small, uniformly-charged spheres. The region is divided into a 25´25 grid. The upper left corner of the grid corresponds to x = 0.5 m, y = 0.5 m, and the lower right corner corresponds to x = 24.5 m, y = 24.5 m. The charged spheres can be placed anywhere on the grid. They will be located in the x-y plane. Students will calculate the potential at each grid point and construct a surface and a contour plot of the potential.

The potential at r = (x,y,z) outside a uniformly charged sphere centered at r’ = (x’,y’,z’) is

In the x-y plane we have z=0 and

The constant k has a value of 9´10⁹ in SI units. If we measure q in units of nC = 10⁹C, then kq = 9q Nm²/C.

Procedure:

Open a Microsoft Excel spreadsheet.

Let cells B1 - Z1 and cells A2 - A26 contain the numbers 0.5 - 24.5 in increments of 1 as shown below.

Cells B2 - Z26 are the grid points whose x- and y-coordinates (in units of m) are listed in cells B2 - Z2 and cells A2 - A26.

Into cells A31 - C34 type the x- and y-coordinates (in units of m) of the positions and the magnitudes (in units of nC) of four charges. Start with a +10 nC charge at x = 13m, y = 13m and let all the other charges all have zero magnitude. (Let the x- and y-coordinates always be integers. This avoids “divide by zero” errors, since the grid points have half integer x- and y-coordinates.)

Now find the potential due to the four charges at grid point B2.

Into cell B2 type

=9*$C$31/SQRT((B$1-$A$31)^2+($A2-$B$31)^2)
+9*$C$32/SQRT((B$1-$A$32)^2+($A2-$B$32)^2)
+9*$C$33/SQRT((B$1-$A$33)^2+($A2-$B$33)^2)
+9*$C$34/SQRT((B$1-$A$34)^2+($A2-$B$34)^2)

This is the sum of due to the four charges.

Now copy cell B2 into the other cells of your grid. The grid consists of cells B2 - Z26.

Highlight your grid and choose to construct a chart.

Choose a surface chart of subtype 3-D surface.
You will see a two-dimensional surface plot of the potential outside a uniformly charged sphere.

Construct another chart of subtype contour.

On this chart right-click the legend and choose to format the legend. Choose scale, maximum 100 and major units 5.
(If you are using Excel 2007, select the graph, and from the "Chart Tools" menu choose Layout, Axes, Primary Vertical Axis, More Primary Vertical Axis Options. Now choose maximum 100 and major units 5.)

You now have a contour plot of the potential outside a uniformly charged sphere. The contour lines are equipotential lines. They are spaced in 5V intervals.

(a) Describe you graphs. What do they tell you about the potential outside a uniformly charged sphere? Can you get information about the electric field outside a uniformly charged sphere from these graphs?

Now change the positions and magnitudes of your charges. Use

x	y	q
10	13	10
16	13	10
0	0	0
0	0	0

Just type in the new numbers into the cells A31 -C34 and the spreadsheet and the graphs will update automatically.

(b) Describe your graphs. What do they tell you about the potential of this charge distribution?

Again change the positions and magnitudes of your charges. Use

x	y	q
10	13	10
16	13	-10
0	0	0
0	0	0

Again change the positions and magnitudes of your charges. Use

x	y	q
10	10	10
16	10	-10
10	16	-10
16	16	10

(d) Describe your graphs. What do they tell you about the potential of this charge distribution?

Again change the positions and magnitudes of your charges. Use

x	y	q
10	10	20
16	10	-10
10	16	-10
16	16	20

(e) Describe your graphs. What do they tell you about the potential of this charge distribution?

Pick you own positions and magnitudes for the charges.

To earn extra credit , answers questions (a) through (e). Save your Excel document (your name_exm5.xls) with graphs of the potential of a charge distribution of your choice and attach the Excel document and a Word document with your answers to an e-mail message to mbreinig@utk.edu.