k=9.00*10^9 Nm^2/C^2. Pay attention to unit conversions.
In one experiment the electric field is measured for points atdistances r from a very long, straight line of charge that has acharge per unit length λ. In a second experiment, the electricfield is measured for points at distances from the center of auniformly charged insulating sphere that has total charge Q andradiusR = 8.00 mm. The results of the two measurements are listedin the table below, but you are not told which set of data appliesto which experiment.
r (cm) 1.00 | 1.50 2.00 2.50 3.00 | 3.50 4.00 |
Measurement A |
E (105 N/C) 2.72 | 1.79 1.34 1.07 0.902 | 0.770 0.677 |
Measurement B |
E (105 N/C) 5.45 | 2.42 1.34 0.861 0.605 | 0.443 0.335 |
To solve the mystery, start by creating the graph ln(E) versusln(r) for each data set. Make sure you convert r values to metersbefore calculating ln(r).
Use Excel. The axes should be labeled appropriately, and thegraphs should be titled Measurement A and Measurement B,respectively. Show a linear trendline and its equation for eachgraph. Make sure that the numerical coefficients of the trendlineequation show at least five significant digits, you will need themin your calculations below. You will submit the Excel file showingthe graphs to Blackboard, separate from the test. (5 points)
Now answer the following questions:
a) Use these graphs to determine which data set, A or B, is forthe uniform line of charge and which set is for the uniformlycharged sphere. Explain your reasoning.
b) Use the trendline equation of the graph corresponding to theline of charge to calculate λ. Show neatly your calculations.
Use the trendline equation of the graph corresponding to thecharged sphere to calculate Q. Show neatly your calculations.
Calculate the electric field inside the sphere, at 4.00 mm fromthe center. Show neatly your calculations.
Hint: If the electric field is invers proportional to a power ofr, say E = B/rn with B a constant factor, then ln(E) = ln(B) -n∙ln(r). So, the graph of ln(E) versus ln(r) should be a straightline, for which the slope is -n and the vertical intercept isln(B).
Now apply this idea to the electric field of a line of charge forwhich E = 2kλ/r, so ln(E) = ln(2kλ) - ln(r), and to the electricfield outside of a uniformly charged sphere for which E = kQ/r , soln(E) = ln(kQ) -2ln(r). What is left for you is to compare thesetheoretical equations to the trendline equations of the charts andidentify which one applies best to Measurement A and which one toMeasurement B.
