Problem 3: It is postulated that the radial electric field of a group of charges falls off as E, = C/r", where Distance, r (cm) Potential difference, AV (mV) C is a constant, r is the distance from the center of the 2.0 34.7 group, and n is an unknown exponent. To test this 4.0 6.6 hypothesis, you make a field probe costisting of two 6.0 2.1 needle tips spaced Ar = 1.0mm apart. You orient the 8.0 1.2 needles so that a line between the tips points to the 10.0 0.6 center of the charges, 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 given in the table. Use an appropriate graph of the data to determine the constants C and n. a) In this experiment you are measuring the change of potential AV over small displacement Ar. In the limit Ar - 0, AV/Ar would give you the derivative of V with respect to r, which is the radial component of the electric field, E, (strictly speaking, with the minus sign). Take the natural logarithm of both sides of Ar = mm to show that In( A, ) can be represented as a linear function of In(r), In( A, ) = a In(r) + b. What are the constants a and b in terms of C and n?' b) Place the measurements In( ) as a function of In(r) on the graph below. W N O In ( AV ) -1 In(r) 1 Note that here we use AV = & instead of E, = -Ap = S because the minus sign is already absorbed into the potential difference values given in the table. The potential in this problem is decreasing with the distance, which means that the value of potential at the tip of the probe that is farther from the center will be smaller than the potential of the closer tip. c) If Er is indeed equal to C/r", your measurements should lie on a straight line (otherwise, you would have to conclude that the field must be described by a different function). Can you connect the dots with a straight line? Determine the constants C and n from this line (to do that, it is helpful to remember that a line y = ax + b intersects y axis at the point y = b and x axis at the point x = -b/a)