Electric Field Mapping

• Theory:

  Electric Field and Electric Field Lines:
  The electric field is defined as the electric force a unit charge will experience at a given point in space. By moving the unit charge around, we can create a map of the electric field at various points. To represent that map, we draw electric field lines following the rules:

  • The lines always start from a positive charge and always end at a negative charge.
  • At any point, the density of the lines in the vicinity of that point is proportional to the electric field at that point.
  • At any point, the vector of the electric field is tangential to the electric field lines at that point.

  Electric Dipole:    Two charges with equal magnitude but opposite polarity are called "Electric Dipole".

  The Electric Potential and the Equipotential Lines:
  The difference in the electric potential is defined as the change in the potential energy of a particle with a unit charge as it moves across two points. The unit for the electric potential is Volt and is equivalent to 1 J/1 C. Consequently, if the electric potential at a given location is given by V, then a particle with a charge q at that location will have a potential energy equal to U=qV. In terms of the difference in the electric potential and the electric potential energy,
  
           ΔU = q ΔV.
  
  The equipotential lines are the lines that connect points in space with the same value for the electric potential.
  

  Relation between the Electric Field and the Electric Potential:
  For a conservative force such as the electrostatic force, a potential energy is defined so that
  
           ΔU = - W,
  
  where U is the potential energy in Jouls, and W is the work done by the force. If we consider the force applied on a unit charge particle,which is by definition the electric field, the work done by that force will be related to change in the potential energy of that unit charge, that is the electric potential. In one dimension and for two locations that are infinitely close to each other, x₁ and x₂, the relation between the elelectric field and the electric potential can be written as:
  
           E_x = - (V(x₂) - V(x₁)) / (x₂ - x₁).       ( Eq.1 )
  
  For a three-dimensional case, similar relations must be written for the y and z directions.

• Preliminary Setup:
  
  
  • You need: Graphing Paper or a computer with a spreadsheet program.
  • Go to Charges and Fields on-line simulation
    • Start the simulation by clicking on the "Run Now" button.
    • Select "Show E-Field", "Grid", "Show Numbers", and "Tape measure"


• Activity 1. Map the Equipotential lines of a single positive charge
  • Click and drag a "+1 nC" charge to the center of the viewing window.
  • Click and drag the volt probe, in the lower left corner of the applet, to a position of 4 V. Click on the plot button, so that you will see the 4-V equipotential line.
  • Repeat the previous step to plot the equipotential lines for 4 V, 6 V, 8 V, 10 V, 12 V, 14 V, 16 V, 18 V, and 20 V
  • Results. The diagram of the E-Field lines and the Equipotential lines are your results. You must make a copy either
    • By hand on a graphing paper, or
    • printing it off your computer screen:
      • By pressing simultaneously the ALT and PRTSC (printscreen) keys on your computer, take a snapshot of the active window.
      • Paste it into an Image Editor. (MS Office Word might work, too). Save the file in .jpg or .png format. Note that .bmp files are much bigger than .jpg or .png.
      • Possible image editors that can be used: Paint (on any Windows machine) ca be accessed from the Start Menu -> Accessories. You can also download the free GNU editor GIMP which has available versions for both Windows and Linux. For Macs, one possibility is OmniGraffle.
      • The .jpg or .png files are your data and results.



• Activity 2. Map the Equipotential lines of a single negative charge.
  • Click and drag a "-1 nC" charge to the center of the viewing window.
  • Click and drag the volt probe, in the lower left corner of the applet, to a position of "-4 V". Click on the plot button, so that you will see the "-4 V" equipotential line.
  • Repeat the previous step to plot the equipotential lines for -4 V, -6 V, -8 V, -10 V, -12 V, -14 V, -16 V, -18 V, and -20 V
  • Results. (See Activity 1)
  
  
  
• Activity 3. Map the Equipotential lines of a dipole
  • Click and drag a "+1 nC" charge to the left on the viewing window.
  • Click and drag a "-1 nC" charge to the right on the viewing window.
  • In the lower left corner, type in 0 Volts and click on the "Plot" button.
  • Repeat the previous step to plot the equipotential lines for ± 2 Volts, ± 4 Volts, and ± 6 Volts.
  • Results. (See Activity 1)
  
  
  
• Activity 4. Map the Equipotential lines of two positive charges
  • Click and drag a "+1 nC" charge to the left on the viewing window.
  • Click and drag a "+1 nC" charge to the right on the viewing window.
  • In the lower left corner, type in 5 Volts and click on the "Plot" button.
  • Repeat the previous step to plot the equipotential lines for 6 Volt, 7 Volt, 8 Volt, 9 Volt, 10 Volt, 11 Volt, 12 Volt, 14 Volt, 16 Volt.
  • Results. (See Activity 1)
  
  
  
• Activity 5. Estimate the Electric Field between two equipotential lines
  • Click and drag a "+1 nC" charge to the center of the viewing window.
  • Plot the V₁ and V₂ equipotential lines.
  • Measure the distance from the charge to the V₁ line, r₁.
  • Measure the distance from the charge to the V₂ line, r₂.
  • Calculate the average distance r as:
    
             
  • Calculate the average el. field, E(r) according to Eq.1 as:
    
             
  • Data and results: Repeat the procedure above for all the values in the following table.
    
    

    V1 (V)	V2 (V)	r1 (m)	r2 (m)	r (m)	E (V/m)
    3 V	4 V
    4 V	5 V
    5 V	6 V
    6 V	7 V
    7 V	8 V

    
    
  • Results:
    Using your data, plot E(r) vs. r. What type of curve is it? 
    Using your data, plot E(r) vs. 1/r².
    The graph should be a straight line. Calculate its slope by using linear regression. Use either of the following options:
    • Calculator,
    • Spreadsheet, or
    • Java Appllet from this website. If you choose the latter, clear the data and type in your own data. The slope of the line is given by "m" in the box below the graph.
    Calculated Slope:
    What is the expected slope: 
    Compare the slope you calculated with what it should be. Write down the percent difference 
    Bring your graphs to class.