## Physics Laboratory 5

### 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 to keep a log of your experimental procedures and results.  Complete all the tasks (in blue).  Answer all questions.

### Activity 1

The concept of work

(a)  The work W done on an object by a constant force is defined as W = Fd.  It is equal to the magnitude of the force, multiplied by the distance the object moves in the direction of the force.
The SI unit of work is Nm = J.
An object travels from point A to point B while two constant forces of equal magnitude are exerted on it, as shown in the figure on the right.

• Is the work done on the object by F1 positive, negative, or zero?
• Is the work done on the object by F2 positive, negative, or zero?
• Is the net work done on the object positive, negative, or zero?

(b)  An object travels from point A to point B while two constant forces of unequal magnitude are exerted on it, as shown in the figure on the right.

• Is the work done on the object by F3 positive, negative, or zero?
• Is the work done on the object by F4 positive, negative, or zero?
• Is the net work done on the object positive, negative, or zero?

Work and the electric field

In the diagram on the right the red dot denotes a positive point charge.  Points W, X, Y, and Z and the point charge lie in the same plane.  Points W and Y are equidistant from the charge, as are points Y and Z.
Draw the electric field vectors at points W, X, Y, and Z.
(c)  A particle with charge +qe travels along a straight line from point W to point X.

• Is the work done by the electric field on the particle positive, negative, or zero?
Explain using a sketch that shows the electric force on the particle and the displacement of the particle.

(d)  A particle travels from point X to point Z along the circular arc shown.

• Is the work done by the electric field on the particle positive, negative, or zero?  Explain!
Hint:  Sketch the direction of the force on the particle and the direction of the displacement for several short intervals during the motion.
• Compare the work done by the electric field when the particle travels from point W to point X to that done when the particle travels from point W to point Z along the path shown on the right.

Electric potential difference

A potential energy function is a function of the position of an object.  It can only be defined for conservative forces.  A force is conservative if the work it does on an object depends only on the initial and final position of the object and not on the path.
(e)  Suppose the charge in part (c) increases from +qe to +1.7 qe.

• Is the work done by the electric field as the particle travels from from W to X greater than, less than, or equal to the work done by the electric field on the original particle.  Explain!
• How is the quantity "the work divided by the charge" affected by this change?

(f)  The electric potential difference  ∆VWX between two points W and X is defined to be the negative of the work done by the electric field on a charge q, divided by q, as q travels from W to X.

• Does this quantity depend on the magnitude of the charge that is used?  Explain!
• Does this quantity depend on the sign of the charge that is used?  Explain!

When a net force does work on an object, its kinetic energy changes.
Wnet = ½m(vf2 - vi2) = ∆K.
(g)  A particle of charge |qe| = 2*10-6 C and mass m = 3*10-8 kg is released from rest at point W.  The speed of the particle is measured to be 40 m/s as it passes point X.

• Is qe positive or negative?  Explain!
• What is the change in the kinetic energy of the particle as it travels from point W to point X?
• Find the work done on the particle by the electric field between points W and X.
• What is the electric potential difference ∆VWX = VW - VX between points W and X?

(h) 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.

(i)  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.

(j)  Find some equipotential surfaces for the electric dipole charge configuration shown on the right.

### 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

V(r) =  kq/|r - r'| = kq/((x - x'2) + (y - y'2) + (z - z'2))½.

The constant k has a value of 9*109 in SI units.  If we measure q in units of nC = 109 C, then kq = 9 q 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 V(r) =  kq/|r - r'| = kq/((x - x'2) + (y - y'2) + (z - z'2))½ 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 and grid lines.)
• 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!)