It is postulated that the radial electric field of a group of charges falls off

Question

It is postulated that the radial electric field of a group of charges falls off as Er = C/rn, where r is the distance from the center of the group and n is an unknown exponent. To test this hypothesis, you make a field probe consisting of two needle tips spaced Deltax = 1.00mm apart. You orient the needles so that a line between the tips points to the center of the charges, and then use a voltmeter to read the potential difference between the tips. After you take measurements at several distances from the center of the group, your data are as follows: Graphing either ln(DeltaV/Deltax) versus ln(r) or ln(r) versus ln(DeltaV/Deltax) gives a straight line. In the graph shown above, we chose to plot ln(DeltaV/Deltax) on the vertical axis and ln(r) on the horizontal axis. From the equation for the line of best fit given, calculate the constant C. Express your answer using two significant figures.

Explanation / Answer

Given :

ln( delv / delx ) = -2.50ln(r) - 6.23

ln( delv / delx ) + 2.50ln(r) = -6.23

ln( delv / delx ) + ln(r)2.5 = -6.23

ln ( delv * r2.5 / delx) = -6.23

delv * r2.5 / delx = e-6.23

delv / delx = e-6.23 / r2.5

Also, we know that electric field E can be represented as E = delv / delx

Hence,

E = e-6.23 / r2.5

Comparing it to the given form i.e E = C / rn

We can see that n = 2.5 and C = e-6.23 = 0.0019 = 1.9 * 10-3

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