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Physics 30: Forces and Fields

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Determining Coulomb's Constant

Purpose:

To determine Coulomb’s constant using a force vs. separation graph for two like charges.

Background:

Electrostatic Force:

Any two charged objects will exert an electrostatic force on each other.  The formula for this force is:

The separation between the centres of both charges is r, and k is Coulomb’s constant.

Curve Straightening:

When the force is plotted as a function of the separation distance between the two charges the resulting graph is not a straight line. In this exercise the curve must be turned into a straight line so that its slope can be found and Coulomb’s constant can be determined. 

The reason that an electrostatic force vs. separation distance graph is not linear is because force is a function of the reciprocal of the square of the separation distance.  To straighten this line, we must make the equation for electrostatic force a linear one of the form y = mx + b.  In essence, the 1/r2 must be replaced with a non reciprocal, non-squared value.  To do this we can make a variable "u" equal to 1/r2.  Writing this mathematically, we say:

Let u = 1/r2

Rewriting the equation for electrostatic force using "u" produces a linear relationship:

, which can be expressed as, so that it more closely resembles the straight-line relationship of the form y = mx + b.  If we write both formulas, one overtop the other, it is easier to see how the two equations are similar and how the variables match up:

 represents the "y" values, kq1q2 is the slope,  and u represents the "x" values.  In this case, the "b" value (y-intercept) equals zero.

Most importantly, the equation for the electrostatic force has no reciprocal or squared values.

A new data table will need to be created that will be used to plot electrostatic force as a function of u.

Instructions:

1. Download and open the accompanying Excel spreadsheet.

Use an appropriate application or graphing calculator to plot the values of the electrostatic force as a function of charge separation (using table 1). The graph that you plot should not be a straight line.
2.

The second table represents the centripetal force as a function of "u." Now fill in the values of "u" (reciprocal of the square of the separation distance) in table 2. To do this:

  • Keep the values of centripetal force from table 1.  Copy them across to table 2.
  • Square, then take the reciprocal of each value of r and place it in the "u" column. 
3. Create a graph from the data in table 2 using an appropriate application or graphing calculator. The centripetal force should be plotted as a function of "u." This new graph should be linear.

Analysis:

1.

The second graph that plots centripetal force vs. u is a straight line.  Determine the slope of the line in the second graph.
2.

In this case the slope value represents kq1q2 (see the "Curve Straightening" notes above).  Using the value of the slope determined in step 3 and the charges of q1 and q2 (from the Excel spreadsheet), determine Coulomb's constant.
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