### Electric potentials and fields

In this laboratory you will explore the connection between electric field lines and equipotential surfaces.  Objects with net electric charge attract or repel each other.  If you want to change the position of a charged object relative other charged objects, you, in general, have to do (positive or negative) work.  But sometimes it is possible to move a charged object relative to other charged objects along a surface without doing any work.  The potential energy of the charged object does not change as you move it.  If an electric charge can travel along a surface without the electric field doing any positive or negative work, then the surface is called an equipotential surface

Open a Microsoft Word document and keep a log of your activities.  Answer all the questions in blue font.

Activity 1

 Assume you have a test charge at rest at a distance of 2 cm from the charge on the right.  You want to move it.  What path could you choose, so you would not have to do any work?   What is the shape of the equipotential surface?   (Remember that in general you can move in three dimensions.)  Explain your reasoning. Find some equipotential surfaces for the charge configuration shown on the right, which consists of two charged metal plates placed parallel to each other.  What is the shape of the equipotential surfaces?  Remember you are trying to decide how a test charge could move so that the electric field does no work on it.  Sketch your predictions and explain your reasoning. Find some equipotential surfaces for the electric dipole charge configuration shown on the right.  Sketch your predictions and explain your reasoning.

Activity 2

You now will calculate the electric potential at grid points in the in the x-y plane due to two small, uniformly-charged spheres.  You will fix the position of the charged sphere with charge q2 and vary the position of the other charged sphere with charge q1.  The x-y plane is divided into a 26 x 26 grid.  The upper left corner of the grid corresponds to x = -12.5 m, y = 12.5 m, and the lower right corner corresponds to x = 12.5 m, y = -12.5 m.  The charged spheres can be placed anywhere on the grid in the x-y plane, as well as above or below the x-y plane.  You will calculate the potential at each grid point and construct a surface and a contour plot of the potential.  The contour plots will display the equipotential lines.  The electric field is perpendicular to the equipotential lines, E = -V.  You will draw field lines indicating the direction and relative magnitude of the electric field in the vicinity of the charged spheres and calculate the magnitude of the electric field at selected points.

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

.

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:

Cells B1 - AA1 contain the numbers -12.5 - 12.5 in increments of 1 and cells A2 - A27 contain the numbers 12.5 - (-12.5).
Cells B2 - AA27 are the grid points whose x- and y-coordinates (in units of m) are listed in cells B2-AA2 and cells A2-A27, respectively.
Row 31 holds the total charge on each sphere (in units of nC) and the  x-, y- and z-coordinates (in units of m) of the positions of the centers of the spheres.  We start with q1 = 0 and q2 = +1 nC at x2 = y2 = z2 = 0.  (If we let the x- and y-coordinates always be integers.  we can avoid “divide by zero” errors, since the grid points have half integer x- and y-coordinates.)
You will later be able to change x- and y- coordinates of q1 using scrollbars.  Therefore cells B31 and C31 contain formulas to copy scaled scrollbar values from the cells below.
Now find the potential due to the two charges at grid point B2.
 Into cell B2 type (or copy)=9*\$A\$31/SQRT((B\$1-\$B\$31)^2+(\$A2-\$C\$31)^2+\$D\$31^2) +9*\$F\$31/SQRT((B\$1-\$G\$31)^2+(\$A2-\$H\$31)^2 +\$I\$31^2) This is the sum of  due to the two charges.
Now copy cell B2 into the other cells of your grid.  The grid consists of cells B2 - AA27.
Highlight your grid (cells B2 - AA27) and choose to INSERT a chart.
 Choose a surface chart of subtype 3-D surface (under All Charts, Surface).  Place the chart on Sheet 2.  (Chart tools, Design, Move Chart) You will see a two-dimensional surface plot of the potential outside a small, uniformly charged sphere.  Excel 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 10, minimum -10, and major units 1.
Construct another surface chart of subtype contour.  Place the chart on Sheet 2.
 Excel 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 10, minimum -10, and major units 1.

You now have a surface and a contour plot of the potential outside a uniformly charged sphere placed at the origin.  The contour lines are equipotential lines.  They are spaced in 1 V intervals.  (Select a chart style with enough colors.)

Now insert a scrollbar .  Click the developer tab.
(If this tab is not available,
click the File tab, click Option, and click Customize Ribbon.  Under Customize the Ribbon and under Main Tabs, select the Developer check box.
To add the Developer tab to Excel 2011 on a Mac, select Excel from the menu.  It is between the Apple logo and File in the upper left hand corner of the menu.  From the drop down menu select Preferences.  In the Sharing and Privacy section, select Ribbon.  In the middle of the Ribbon dialog box you will see a box listing Tab or Group title.  Scroll through this list and find Developer.  Check the box and click OK.  For Excel 2016 on a Mac, go to Excel preferences, view, in ribbon show, developer tab.)
Click Insert, Form Controls, and choose the scrollbar, then click in an empty cell, where you want the scrollbar to appear on sheet 2.  Drag the scrollbar to the orientation and size you want.
To set the properties for the scrollbar, right-click it and chose select Format Control.  Choose minimum value 0, maximum value 24, incremental change 1, cell link Sheet1!\$B\$32, and click ok.
Label the scrollbar by typing "x-position" into a cell next to the scrollbar.  This scrollbar will let you move the charge q2 along the x-direction.  In a cell next to your scrollbar label type =Sheet1!B31.  This copies the x-position of charge q1 onto sheet 2, so you can monitor it while you move the charge.
Repeat the procedure above and insert a second scrollbar.  Right-click it and chose select Format Control.  Choose minimum value 0, maximum value 24, incremental change 1, cell link Sheet1!\$C\$32, and click ok.
Label the scrollbar by typing "y-position" into a cell next to the scrollbar.  This scrollbar will let you move the charge q2 along the y-direction.  In a cell next to your scrollbar label type =Sheet1!C31.   This copies the y-position of charge q1 onto sheet 2, so you can monitor it while you move the charge.
Click Insert, Form Controls, and choose the checkbox, then click in an empty cell, where you want the checkbox to appear on sheet 2.  Right click it, select Format Control, checked, cell link Sheet1!\$A\$32.  By clicking the checkbox you can change the sign of q2
Now go to sheet 1 and enter =A32*2-1 into cell A31 to set q1 equal to 1 nC.

Charge q2 is fixed at the origin.  Both charges have the same magnitude.  You can move the charge q1 to different positions with the scrollbars.  You can change the sign of charge q1 with the checkbox..

Describe you graphs and how they change when you move q1.  What do they tell you about the potential outside two uniformly charged spheres?

Produce a contour plot for

 q1 x1 y1 z1 q2 x2 y2 z2 1 -5 -5 0 1 0 0 0

and paste the plot into your word document.

Draw approximately 8 field lines onto the contour plot.  The field lines should accurately reflect the strength and direction of the electric field.  At each point, the field lines must be perpendicular to the equipotential lines.

Pick a point on your graph at which to calculate the magnitude of the electric field.  Let the point lie on a field line.  Label the point A using the textbox tool.
 Estimate the magnitude of the electric field at that point by finding the change in the potential ΔV over some small distance d along the field line.  Let d go along the field line from one side of the chosen point to the other side.  Calculate E = ΔV/d.  (HINT!)