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 x 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 = 109 C, then kq = 9q Nm2/C.

Procedure:

bulletOpen a Microsoft Excel spreadsheet.
bulletLet 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.

bulletInto 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 = 13 m, y = 13 m 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.)

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

bulletNow copy cell B2 into the other cells of your grid.  The grid consists of cells B2 - Z26.
bullet  Highlight your grid and choose to construct a chart. 
bulletChoose a surface chart of subtype 3-D surface.
You will see a two-dimensional surface plot of the potential outside a uniformly charged sphere.
bulletConstruct another chart of subtype contour. 
bulletExcel 2010:  Select the graph, and from the "Chart Tools" menu choose Format, Vertical (Value) Axis, Axis Options. 
Excel 2013 or later:  Select the graph, click the + sign next to the graph, Axes, More Options, Axis Options, Vertical Value Axis, three bar (rightmost) icon.
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.   (Select a chart style with enough colors.)

(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?

bulletNow 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?

bulletAgain change the positions and magnitudes of your charges.  Use

x

y

q

10

13

10

16

13

-10

0

0

0

0

0

0

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

bulletAgain 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?

bulletAgain 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?

bulletPick you own positions and magnitudes for the charges.

To earn extra credit add your name and e-mail address to your spreadsheet.  In full sentences answer the questions (in blue) posed above.

Save your Excel document (your name_exm5.xlsx), go to Blackboard, Assignments, Extra Credit 5, and attach your document.