Electric Field TutorialName: Date:Overview of tutorial:The goal of this worksheet is to help you develop a better understanding of the electric field due to many charges and due to a bar of charge. 1.Charge Densitybegin this discussion, well review a related topic: charge density.: To a.You are told that a2cm bar of charge (located fromx=1cm tox=3cm) has a charge densityρ= 1/18µC/cm i.Show all of your work.Calculate the total charge on the bar. ii.We often read the density as “1/18 micro Coulombspercentimeter”. How would you say this using common English words without the unusual word “per”? b.Imagine that same bar (same length and location) now has a charge density given the formula The variablexgives the distance from the origin of the coordinate system. i.What is the charge density at the left end of the bar? ii.What is the charge density at the right end of the bar? iii.Show all your work. Keep the unitsCalculate the total charge on the bar. in your work.

Electric Field Tutorial

iv.How would tell a fellow studentwhyyou calculated the total charge as you did in the last problem?

v.Another student in your class describes calculating the total charge this wayI can approximate the: I imagine breaking the bar into small chunks. charge for each chunk by assuming the density is constant for the chunk, and then I add up charge for each chunk to get the total charge. This becomes more accurate as the chunks get smaller. In the limit that the chunks are infinitely thin, the sum becomes an integral and my answer is exact. 1.ExplainDo you agree with this students method of calculation? your reasoning.

2.Change her words into a sketch and an equation (usingρandΔx) and compare her equation to yours. Do the equations agree?

3.If you disagree with her, what do you disagree with?

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Electric Field Tutorial

2.QUICK review ofElectric force:Coulomb's law gives the magnitude of theforcebetween two point chargesq1andq2a distancerapart: Give quick qualitative answers to the next questions. i.DoesDoes the force increase or decrease when the charges get bigger? this agree with the equation? ii.Does the force increase or decrease when the distance between them gets bigger? Does this agree with the equation? iii.Given the two point charges (of opposite signs), draw theforce vectorto indicateon each charge the relative magnitude and direction of the force due to the other charge.

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Electric Field Tutorial

3.Electric field:Now we move to the main topic of the worksheet: the electric field. We will begin with a review of the electric field. Then we will calculate the electric field at the origin due to a fixed charge. We will do this three times, with the charge spread out over a larger and larger distance. The calculations on this page should be quick. a.Review:it is the The electric field is closely related to the force: force per chargedue to a given set of charges. For a single point chargeq, the magnitude of the electric field at a pointPa distancergiven by the following formula.away is i.Often we use the electric field to calculate the electric force. If the electric field at pointPis 9 N perµC in thexpoint-direction, and we place a charge of 10µC at pointP, what is the force it feels due to the electric field? Hint: what do the units on E tell you about how to calculate the force? ii.In this worksheet we willElectric field (like force) is a vector quantity. make the vector part of the work easy by looking at electric fields that are only in thexa sketch to convince yourself that if a-direction. Draw charge is on thex-axis, and pointPis on thex-axis, then the electric field is entirely in thex-direction at pointP. iii.One other piece of calculating the electric field is finding the distancer between pointPand the chargeq. We have made this easy as well for this worksheet. If pointPis at the origin and the point charge is located at (x,0)what isras a function ofx? Use the general formula for the distance between two points to find out:

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Electric Field Tutorial

b.Calculations: in the next few pages we will calculate theExdue to the same total charge, but spread out in different ways. For this worksheet, we will always find the electric field at the origin. i.CalculateExat pointPat the origin due to a point charge of 1/9µC 2 2 located atx= 2cm andy=0a sketch. Takecm. Draw k=90N cm /µC . Show all your work and keep the units in. Give your answer in decimal form so that we can compare answers on the next page. ii.How would you calculate the electric field at pointPdue to two charges on thex-axis? Hint: how do we find the total force on an object when more that one force acts on that object? iii.Use the method you just outlined to calculate the magnitudeExat pointPat the origin due to two point charges of 1/18 micro Coulombs each located atx= 5/3cm andx=7/3 cm (both at y=0a sketch.cm). Draw 2 2 Takek=90N cm /µC . Show all your work and keep the units in. Give your answer in decimal form. Page 5 of 9 8/16/07 D. Meredith, University of New Hampshire

Electric Field Tutorial

iv.Even though it would beverytedious, could you calculate the total electric field due to 100 point charges evenly spaced between 1 and 3 cm (y=0cm for all) if you were given the value of each charge and its location? If not, what other information would you need t do the calculation?

v.Below is a table for summarizing the value ofEx. due to 1/9µC as it is spread more smoothly over the region between 1 and 3 cm. We have filled in the values for 4, 10, 100 and 1000 points, put your values from the previous page for one and two points in the appropriate places.

Number of point charges 1 2 4 10 100 1000 10,000

Size of each charge (µC)

1/9 1/18 1/36 1/90 1/900 1/9000 1/90,000

Total charge (µC)

1/9 1/9 1/9 1/9

Total Ex(N/µC)

2.9 3.14 3.31 3.33

vi.Look at the values ofEx; 1.Are the values of E increasing or decreasing? 2.Is the rate of change increasing, decreasing, or staying the same? 3.Based on this pattern (calculations are not necessary; dont take more than a minute on this !) put in your best guess(2 digits only!)forEx. due to 10,000.

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Electric Field Tutorial

c.Electric field for a bar of charge. Up until now we have consideredExat the origin due to a finite number of point charges with a total charge of 1/9µC. But imagine now that we have a bar of charge, where there are infinitely many charges spread evenly over the bar. The bar of charge has a length 2 cm with charge density 1/18µC/cm and is located between 1 cm < x < 3 cm on thex-axis (y=0).

i.ApproximateExat the origin due to the charge between 1 cm< x <1.1 cm by taking the distance to pointPto be constant over this interval and equal to 1 cm.

ii.Write the formula that you used to findExabove in terms of k,ρ(charge density), x (the distance of the chunk from the origin) andΔx (width of chunk).

iii.Given your answer to previous question, describe in words and/or pictures how can you accurately calculate the electric field due to this bar of charge where there are infinitely many charges. Hint: what ideas in this worksheet can you use to figure out how to do this calculation?

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Electric Field Tutorial

iv.Carry out the calculation you just described. v.If you havent already done so, check that the units are correct. vi.If so, is it?Should your answer be consistent with the data in the table? vii.Another student in your class states:To find the electric field due to the bar of charge, because the distance is changing over the bar, I have to integrate the electric field. I integrate with respect to x because x is what is changing. My integral isDo you agree or disagree with this student? Explain. Be sure to compare your equation with hers, and be sure to check the units! Page 8 of 9 8/16/07 D. Meredith, University of New Hampshire

Electric Field Tutorial

Notes for instructors: 1.Be sure that students understand the non constant density on page one. Some students interpret the x value as the length of the bar. 2.On question 1biii 1 some students do not believe that the limit will be exact. You might want to be prepared how to help them see that it is exact. This is a difficult enough issue that it may take more time for some students. 3.We have tried to indicate on pages 3 and 4 that these pages should be quick. Be prepared to urge students along if they are taking it too seriously. 4.Page 7 is optional but there are typically a few students who are curious. 5.This took 1.5 hours for most groups. 6.Question 1bii (how did you know to integrate) – here there is no right or wrong answer. 7.In principle, this work sheet is self-checking; there are many internal consistency checks. If students are off track, ask them if their answer is consistent with other answers.

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