Question

k=9.00*10^9 Nm^2/C^2. Pay attention to unit conversions.

In one experiment the electric field is measured for points at distances r from a very long, straight line of charge that has a charge per unit length λ. In a second experiment, the electric field is measured for points at distances from the center of a uniformly charged insulating sphere that has total charge Q and radiusR = 8.00 mm. The results of the two measurements are listed in the table below, but you are not told which set of data applies to 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) versus ln(r) for each data set. Make sure you convert r values to meters before calculating ln(r).

Use Excel. The axes should be labeled appropriately, and the graphs should be titled Measurement A and Measurement B, respectively. Show a linear trendline and its equation for each graph. Make sure that the numerical coefficients of the trendline equation show at least five significant digits, you will need them in your calculations below. You will submit the Excel file showing the graphs to Blackboard, separate from the test. (5 points)

Now answer the following questions:

1. a) Use these graphs to determine which data set, A or B, is for the uniform line of charge and which set is for the uniformly charged sphere. Explain your reasoning.

2. b) Use the trendline equation of the graph corresponding to the line of charge to calculate λ. Show neatly your calculations.

3. Use the trendline equation of the graph corresponding to the charged sphere to calculate Q. Show neatly your calculations.

4. Calculate the electric field inside the sphere, at 4.00 mm from the center. Show neatly your calculations.

Hint: If the electric field is invers proportional to a power of r, 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 straight line, for which the slope is -n and the vertical intercept is ln(B).
Now apply this idea to the electric field of a line of charge for which E = 2kλ/r, so ln(E) = ln(2kλ) - ln(r), and to the electric field outside of a uniformly charged sphere for which E = kQ/r , so ln(E) = ln(kQ) -2ln(r). What is left for you is to compare these theoretical equations to the trendline equations of the charts and identify which one applies best to Measurement A and which one to Measurement B.

Answer 1

Excel chart is given used to plot the electric field graph is given below . Natural logrithm of distance in metre and natural logrithm of electric fields are tabulated

Plots of natural logrithm of electric field measured for line charge and sphere is plotted as a function of logarithm of distance. Graphs are given below

Part-(a)

Measurement-A gives slope = -1 and Measurement-B gives slope = -2

It is known that electric field of line charge varies as 1/r and electric field of sphere varies as 1/r2 .

Hence Measurement-A is done for line charge and measurement-B is done for charged sphere.

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Part-(b)

For a line charge that has uniform charge density , we have

if we take logarithm , we get

from graph trendline we get intercept 3.619 ,

hence ,

by substituting Coulomb constant , K = 9 x 109 N m2 C-2

we get   = 1.5 x 10-12 C / m

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For charged sphere

from graph we get intercept -7.548

by substituting for columb constant K = 9 x 109 N m2 C-2 , we get charge on the sphere Q as

Q = 5.85 x 10-14 C

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Electric field of sphere at 4 mm from centre can not be determined from experiemntal data because radius of sphere is 8 mm . If we need to find electric field at 4 mm, then this point is inside sphere.

Inside solid sphere electric field varies as distance r as

...........................(1)

where as for experimentally measured data at outside sphere, field varies as

Hence we calculate electric field at 4 mm for sphere using equation (1) and we use the charge value determined by measured data and its graph