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t P s B n S e U t L , M MISN-0-114 COULOMB'S LA COULOMB'S LA J. Ko acs 1. In tro duction 2. orce et een Tw oin Charges a. Static Tw o-Bo dy Gra vitational orce b. Tw o-Bo dy Electrostatic orce c. The Electrostatic orce (Qualitativ e) d. Coulom b's La e. Basic Electric Charges 3. Sev eral oin Charges a. Cho ose Con enien Co ordinate System b. Apply Coulom b's La to All Charge airs c. otal orce is ector Sum d. Numerical Example Ac kno wledgmen ts Glossary

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ID She et: MISN-0-114 Title: Coulom b's La Author: J. S. Ko acs, Dept. of Ph ysics, Mic h. State Univ. ersion: 10/22/2001 Ev aluation: Stage Length: hr; 20 pages Input Skills 1. cabulary: electric harge, electric force (MISN-0-121). 2. State Newton's second and third la ws (MISN-0-16). 3. Express an arbitrary ector in terms of the unit ectors asso ciated with xed co ordinate system (MISN-0-2). 4. Add or more ectors and determine the magnitude and direc- tion of their resultan (MISN-0-2). Output Skills (Kno wledge) K1. cabulary: harge, coulom b, Coulom force, electrostatic (in ter- action), electrostatic force constan ). K2. State Coulom b's la and iden tify eac of the quan tities in the expression, lab eling eac with its appropriate dimensions. Also dra sk etc sho wing harges and lab eled with the quan tities in Coulom b's la w, sho wing the direction of the forces in olv ed in relation to the signs and ositions of the harges. Output Skills (Problem Solving) S1. Calculate the magnitude and direction of the resultan force ex- erted on giv en oin harge giv en spatial arrangemen of other oin harges. ost-Options 1. \P oin Charge: Field and orce" (MISN-0-115). 2. \Electrostatic oten tial Due to Discrete Charges" (MISN-0-116). THIS IS DEVELOPMENT AL-ST GE PUBLICA TION OF PR OJECT PHYSNET The goal of our pro ject is to assist net ork of educators and scien tists in transferring ph ysics from one erson to another. supp ort man uscript pro cessing and distribution, along with comm unication and information systems. also ork with emplo ers to iden tify basic scien tic skills as ell as ph ysics topics that are needed in science and tec hnology um er of our publications are aimed at assisting users in acquiring suc skills. Our publications are designed: (i) to up dated quic kly in resp onse to eld tests and new scien tic dev elopmen ts; (ii) to used in oth class- ro om and professional settings; (iii) to sho the prerequisite dep enden- cies existing among the arious unks of ph ysics kno wledge and skill, as guide oth to men tal organization and to use of the materials; and (iv) to adapted quic kly to sp ecic user needs ranging from single-skill instruction to complete custom textb oks. New authors, review ers and eld testers are elcome. PR OJECT ST AFF Andrew Sc hnepp ebmaster Eugene Kales Graphics eter Signell Pro ject Director AD VISOR COMMITTEE D. Alan Bromley ale Univ ersit E. Leonard Jossem The Ohio State Univ ersit A. A. Strassen burg S. U. N. Y., Ston Bro ok Views expressed in mo dule are those of the mo dule author(s) and are not necessarily those of other pro ject participan ts. 2001, eter Signell for Pro ject PHYSNET, Ph ysics-Astronom Bldg., Mic h. State Univ., E. Lansing, MI 48824; (517) 355-3784. or our lib eral use olicies see: http://www.physnet.org/home/modules/license.html

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MISN-0-114 COULOMB'S LA J. Ko acs 1. In tro duction One of the fundamen tal in teractions in nature is the electromagnetic in teraction. It is the in teraction that is most directly resp onsible for all of the hemical and biological phenomena that ou observ e: the colors of ob jects, the solidness of bric all, the ap or pressure of mercury the com bustibilit of gasoline, etc. Con trasted with this, our eigh on the surface of the earth is consequence of the gra vitational in teraction, while the energy output of the sun and the stars is consequence of the uclear in teraction. The basic prop ert asso ciated with the electromag- netic in teraction is the electric harge that substances carry This can attributed to the harge carried the microscopic constituen ts of matter. The basic consequence of the existence of these harges is that harged particle exerts force on another harged particle. The precise nature of this force is expressed in Coulom b's la w. 2. orce et een Tw oin Charges 2a. Static Tw o-Bo dy Gra vitational orce. Tw stationary masses exert an attractiv force on one another that is prop ortional to their masses and in ersely prop ortional to the square of their separation gra (1) This is utually attractiv force: eac mass attracts the other with this same force, in agreemen with Newton's third la w. Although the magnitudes of the forces on the are equal, their accelerations will not equal if their masses are not equal. or example, the sun and the earth eac exert the same force on the other but the earth's acceleration to ard the sun is 10 times the sun's acceleration to ard the earth. 2b. Tw o-Bo dy Electrostatic orce. In addition to the gra vitational force (and apparen tly indep enden tly of an gra vitational force), parti- cles ma exert force on eac other whose origin is attributed to another See \Newton's La of Univ ersal Gra vitation" (MISN-0-101). Recall Newton's second la w. MISN-0-114 Figure 1. Distances and direc- tions for illustrating the Coulom force of article on article 2. basic prop ert of the particles, electric harge. That is, if in addition to the mass, eac particle has an electric harge, the particles will exert force on one another that is consequence of the existence of their harges. This force is called the electrostatic force. Unlik the gra vita- tional force, whic can only attractiv e, the electrostatic force can either repulsiv or attractiv e. urthermore, for the masses and harges of the elemen tary particles of nature suc as the electrons and protons, the electrostatic force is man orders of magnitude larger than the gra vita- tional force et een the same particles. 2c. The Electrostatic orce (Qualitativ e). Qualitativ ely the force that harged particles exert on eac other via the electrostatic in ter- action can describ ed as: (a) alw ys along the line joining them, (b) either repulsiv or attractiv dep ending up on whether the harges ha the same sign or ha opp osite signs, and (c) dep enden up on the sepa- ration of the particles, with the magnitude of the force decreasing with increasing separation et een them. In fact, giv en the particles' harges and their lo cations, one can calculate the force eac exerts on the other. The mathematical expression that yields the force is called Coulom b's la w. It is kno wn to alid er the remark able range of separation distances from as small as 10 15 meters to as large as 10 meters. The \static" stresses the fact that this force exists ev en if the particles are sta- tionary relativ to the observ er. This is to distinguish it from the magnetic force that arises when harged particles mo relativ to the observ er. This is not to sa that the result is incorrect at distances larger than million meters; that um er is merely the upp er limit to whic exp erimen has eried this in erse-square-la force.

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MISN-0-114 2d. Coulom b's La w. The simplest expression that demonstrates ho the electrostatic force dep ends up on the separation et een the harged particles can written do wn referring to Fig. 1. The magnitude of the electric force exerted on either particle the other is: (2) The co ecien is indep enden of the particle separation, dep ending only on basic in trinsic prop ert of the particles themselv es, their electrical harges. The co ecien can oth ositiv and negativ e, unlik the gra vitational coun terpart whic is alw ys negativ e, indicating oth repulsion and attraction as ossible et een the harged parti- cles. Because the electrostatic force can demonstrated to prop or- tional to the harge on eac of the particles, the pro duct of the individual harges app ears in aking this in to accoun t, Eq. (1) ma written for the force exerted on harged particle harged particle 1: (on particle 2) (3) where and are the harges of particles and 2, resp ectiv ely and is the unit ector oin ting along the radius ector from particle to particle (see Fig. 1). The unit of harge that en ters in this expression, when and are in MKS units, is the \coulom b." The constan quan tit is called \the electric force constan t." Its alue is dene to e: 99 10 (4) The newton and meter are dened elsewhere, while the coulom (sym ol \C") is dened in terms of electrical curren and time. Note that is the force in newtons that iden tical one-coulom harges ould exert on one another if they ere separated distance of one meter, just as is the gra vitational force, in newtons, that one-kilogram masses ould exert on eac other when separated one meter. Similarly the gra vitational force et een particles dep ends up on another in- trinsic prop ert of the particles, their masses. The expression for the gra vitational force et een particles (see MISN-0-101) is similar to Eq. (1), oth of them decreasing as the in erse square of the in ter-particle separation. The distance-indep enden co- ecien ts, ho ev er, are astly dieren t. As an illustration of this, the electrostatic force et een electrons for giv en separation is exactly the same as the electro- static force et een protons. Ho ev er, the gra vitational force et een protons is more than million times the gra vitational force et een electrons for the same separation. See the ables at the end of this ok. MISN-0-114 2e. Basic Electric Charges. Normal atoms and molecules consist of electrons, protons, and neutrons, and harge is strictly additiv when these particles com bine to form comp osite particles or collections of par- ticles. Electrons, protons, and neutrons ha these amoun ts of harge: (1 60217733 00000049) 10 19 +(1 60217733 00000049) 10 19 1) 10 21 will generally require only three digits of accuracy so tak the harge on the neutron as zero and the harges on the proton and the electron as equal but opp osite in sign. or notation, use where: 602 10 19 (5) All harges that are ev er observ ed are ositiv or negativ in teger ul- tiples of the harge on the electron (or proton). Ob viously it tak es large um er of electrons and/or protons to mak coulom or, for that matter, an of the normally observ ed amoun ts of harge that are dealt with in standard electronic apparatus. 3. Sev eral oin Charges 3a. Cho ose Con enien Co ordinate System. When there are more than harges, the in tro duction of suitable co ordinate system with corresp onding set of unit ectors greatly simplies the okk eeping in olv ed in determining the force on an one of the particles. An example with three harges is sho wn in Fig. 2. 3b. Apply Coulom b's La to All Charge airs. Coulom b's la still describ es the electrostatic in teraction et een all pairs of oin harges in the system. There are con tributions to the force on the harge due to harges and in Fig. 2: one that ould exerted on if only the pair and ere presen and the other that ould exerted on if only the pair and ere presen t. These forces are: Due to 13 Actually 10 21 The particles called \quarks" constitute the only exception: they are the con- stituen ts of protons and neutrons and their harges are one-third and o-thirds the electronic harge

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MISN-0-114 Figure 2. con enien co ordinate system for three oin harges. where 13 is the distance et een the harges and and is unit ector directed along the ositiv -axis. (The force that exerts on is the negativ of the ab e.) Due to 12 Where is unit ector 1) oin ting along the line from to 3c. otal orce is ector Sum. The resultan force on in Fig. is the ector sum of the forces from the other harges: 13 12 (6) com bine these ectors it is necessary to write unit ector in terms of its and -comp onen ts: (cos (sin Help: [S-2] (7) with as dened in Fig. 2. Then the net force is: 13 sin 12 cos 12 (8) MISN-0-114 Figure 3. The resultan force on harge due to the arrangemen of harges sho wn in Fig. 2. 3d. Numerical Example. Numerically with 13 100 12 0500 sin 800, and cos 600, the net force on in Fig. can determined to (46 (21 (9) The magnitude of this force is 51 newtons (10) The direction of the force mak es an angle 65 with the ositiv -axis as sho wn in Fig. 3. This is go example to hec for ourself. Ac kno wledgmen ts Preparation of this mo dule as supp orted in part the National Science oundation, Division of Science Education Dev elopmen and Researc h, through Gran #SED 74-20088 to Mic higan State Univ er- sit Glossary harge the basic prop ert of piece of matter that denes the strength of its in teraction with other matter through electromagnetic forces. coulom the MKS unit of harge, abbreviated \C," dened as the amoun of harge that passes oin in one second due to curren of one amp ere (\amp ere" and \second" are basic SI units). Coulom force an in teraction et een harged particles that ob eys Coulom b's la w, i.e. that dep ends on the pro duct of the harges and in ersely on the square of the particles' separation. 10

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MISN-0-114 electrostatic (in teraction) an electromagnetic in teraction that is indep enden of the motion of harged particles. electrostatic constan also called the electrostatic force constan t, written It is the in trinsic strength of the in teraction et een harged particles and as suc app ears in Coulom b's la and other electric and magnetic expressions. Its alue is dene to e: 10 99 10 Here is the sp eed of ligh t. 11 MISN-0-114 PS-1 PR OBLEM SUPPLEMENT Note: Problem also ccurs in this mo dule's Mo del Exam 1. or the system of three harged particles sho wn in Fig. 2, deriv the expression that giv es the resultan force on harge due to harges and a. Express the resultan in terms of the unit ectors and asso ciated with the co ordinate system sho wn in Fig. 2, getting Eq. (8) b. Giv en the umerical alues sho wn in Fig. 2, ev aluate the alues of the and -comp onen ts of this resultan force, as sho wn in Eq. (9) c. Putting in the dimensions of all of the quan tities that en ter in to the equation used in part (b), ab e, erify that the correct dimensions of the force comp onen ts are the \newton." d. Determine the magnitude and direction of this resultan force on harge getting Eq. (10) and Fig. 3. 2. rite do wn the expression for Coulom b's la w. Put in the units for eac one of the factors in the expression (including the electrostatic constan and erify that ou wind up with force unit. nswer: 11 3. harge 10 is lo cated at the origin of cartesian co ordinate system while harge +4 10 is lo cated at +0 30 meters, 0, 0. a. Calculate the force on due to nswer: b. Calculate the force on due to nswer: 12

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MISN-0-114 PS-2 4. Tw balls eac of mass 0500 kilograms are harged and attac hed to strings of length 600 meters hanging from the same oin t. When the system is in equilibrium eac of the balls mak es an angle of 30.0 with the ertical. a. Dra one-b dy force diagram for the ball carrying harge and iden tify the ob ject that exerts eac of those forces on this ball. nswer: 14 b. Do the same for the other ball. nswer: c. Set up co ordinate system at the lo cation of the ball carrying harge with oin ting horizon tally to the righ t, ertically up ard. Express the three forces acting on this ball in terms of these unit ectors and the sym ols in the sk etc (ab e). nswer: What is the resultan (sym olic) force on the ball, based on its acceleration? nswer: d. The alues of and on the left side of the sk etc (ab e) are the same as those on the righ side of the sk etc h. Do es this mean that the alue of the harge on the left side of the sk etc is the same as that on the righ side (i.e., do es )? nswer: 16 e. If 00 10 what is the alue of nswer: 13 MISN-0-114 PS-3 5. the corners of square of side length 030 are lo cated four harges as sho wn in the diagram ab e. is lo cated at the origin of the co ordinates, is on the -axis and on the -axis. 10 10 +2 10 +1 10 a. What force do es the harge exert on nswer: 13 What force do es exert on nswer: b. What force do es exert on nswer: 10 c. What force do es exert on nswer: d. What is the resultan force on nswer: e. Calculate the direction of this net force measured relativ to the ositiv -axis. nswer: 14

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MISN-0-114 PS-4 6. Three oin harges are lo cated at ositions as indicated in this dia- gram: +5 10 C, lo cated at the origin 10 C, lo cated at (0 m) +8 10 C, lo cated at (6.0 m, 8.0 m) Calculate the magnitude and direction of the resultan force on harge nswer: 15 15 MISN-0-114 PS-5 Brief Answ ers 1. same as in answ er except that in this case is the force on due to 2. (1.2 N) 3. (21 N) (21 N) 4. 80 10 5. sin cos sin 6. 83 10 Help: [S-1] 7. 90 clo kwise from the ositiv -direction 8. 9. 10 10. 10 11. See the mo dule text 12. It is not mo ving, so so 13. 21 (21 14. con tact force exerted the string non-con tact electrostatic force exerted harge non-con tact gra vitational force exerted the earth 16

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MISN-0-114 PS-6 15. 31 N, from the origin at an angle that is 51 from the ositiv -axis and 39 from the ositiv -axis. 16. No, but Coulom b's la pro duces force of on that is equal and opp osite to the force of on regardless of the relativ sizes of and This is in agreemen with Newton's third la w. 17 MISN-0-114 AS-1 SPECIAL ASSIST ANCE SUPPLEMENT S-1 (fr om PS, Pr oblem 4d) 05 kg )(9 m/s )(4)( (sin 30 (tan 30 (4 00 10 C)(8 99 10 /C 83 10 Note: distance et een harges (2)(0 )(sin 30 An alternativ e: note the equilateral triangle formed the strings and an imaginary line connecting the balls: all three angles are equal so all three sides are equal. An yw sk etc the situation roughly to scale and see that the alue of the distance is correct. S-2 (fr om TX, 3c) If ou ha trouble with this equation, review MISN-0-1 where suc expressions are discussed in great detail. ou can alw ys hec suc an expression taking: (i) to see that it is indeed of unit length; (ii) to see that the -comp onen is correct; and (iii) to see that the -comp onen is correct. 18

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MISN-0-114 ME-1 MODEL EXAM 99 10 1. See Output Skills K1-K2 in this mo dule's ID She et 2. Three oin harges are lo cated at ositions as indicated in this diagram: +5 10 C, lo cated at the origin 10 C, lo cated at (0 m) +8 10 C, lo cated at (6.0 m, 8.0 m) Calculate the magnitude and direction of the resultan force on harge Brief Answ ers 1. See this mo dule's text 2. See this Problem in this mo dule's Pr oblem Supplement 19 20

Ko acs 1 In tro duction 2 orce et een Tw oin Charges a Static Tw oBo dy Gra vitational orce b Tw oBo dy Electrostatic orce c The Electrostatic orce Qualitativ e d Coulom bs La e Basic Electric Charges 3 Sev eral oin Charges a Cho ose Con enien Co or ID: 23377

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t P s B n S e U t L , M MISN-0-114 COULOMB'S LA COULOMB'S LA J. Ko acs 1. In tro duction 2. orce et een Tw oin Charges a. Static Tw o-Bo dy Gra vitational orce b. Tw o-Bo dy Electrostatic orce c. The Electrostatic orce (Qualitativ e) d. Coulom b's La e. Basic Electric Charges 3. Sev eral oin Charges a. Cho ose Con enien Co ordinate System b. Apply Coulom b's La to All Charge airs c. otal orce is ector Sum d. Numerical Example Ac kno wledgmen ts Glossary

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ID She et: MISN-0-114 Title: Coulom b's La Author: J. S. Ko acs, Dept. of Ph ysics, Mic h. State Univ. ersion: 10/22/2001 Ev aluation: Stage Length: hr; 20 pages Input Skills 1. cabulary: electric harge, electric force (MISN-0-121). 2. State Newton's second and third la ws (MISN-0-16). 3. Express an arbitrary ector in terms of the unit ectors asso ciated with xed co ordinate system (MISN-0-2). 4. Add or more ectors and determine the magnitude and direc- tion of their resultan (MISN-0-2). Output Skills (Kno wledge) K1. cabulary: harge, coulom b, Coulom force, electrostatic (in ter- action), electrostatic force constan ). K2. State Coulom b's la and iden tify eac of the quan tities in the expression, lab eling eac with its appropriate dimensions. Also dra sk etc sho wing harges and lab eled with the quan tities in Coulom b's la w, sho wing the direction of the forces in olv ed in relation to the signs and ositions of the harges. Output Skills (Problem Solving) S1. Calculate the magnitude and direction of the resultan force ex- erted on giv en oin harge giv en spatial arrangemen of other oin harges. ost-Options 1. \P oin Charge: Field and orce" (MISN-0-115). 2. \Electrostatic oten tial Due to Discrete Charges" (MISN-0-116). THIS IS DEVELOPMENT AL-ST GE PUBLICA TION OF PR OJECT PHYSNET The goal of our pro ject is to assist net ork of educators and scien tists in transferring ph ysics from one erson to another. supp ort man uscript pro cessing and distribution, along with comm unication and information systems. also ork with emplo ers to iden tify basic scien tic skills as ell as ph ysics topics that are needed in science and tec hnology um er of our publications are aimed at assisting users in acquiring suc skills. Our publications are designed: (i) to up dated quic kly in resp onse to eld tests and new scien tic dev elopmen ts; (ii) to used in oth class- ro om and professional settings; (iii) to sho the prerequisite dep enden- cies existing among the arious unks of ph ysics kno wledge and skill, as guide oth to men tal organization and to use of the materials; and (iv) to adapted quic kly to sp ecic user needs ranging from single-skill instruction to complete custom textb oks. New authors, review ers and eld testers are elcome. PR OJECT ST AFF Andrew Sc hnepp ebmaster Eugene Kales Graphics eter Signell Pro ject Director AD VISOR COMMITTEE D. Alan Bromley ale Univ ersit E. Leonard Jossem The Ohio State Univ ersit A. A. Strassen burg S. U. N. Y., Ston Bro ok Views expressed in mo dule are those of the mo dule author(s) and are not necessarily those of other pro ject participan ts. 2001, eter Signell for Pro ject PHYSNET, Ph ysics-Astronom Bldg., Mic h. State Univ., E. Lansing, MI 48824; (517) 355-3784. or our lib eral use olicies see: http://www.physnet.org/home/modules/license.html

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MISN-0-114 COULOMB'S LA J. Ko acs 1. In tro duction One of the fundamen tal in teractions in nature is the electromagnetic in teraction. It is the in teraction that is most directly resp onsible for all of the hemical and biological phenomena that ou observ e: the colors of ob jects, the solidness of bric all, the ap or pressure of mercury the com bustibilit of gasoline, etc. Con trasted with this, our eigh on the surface of the earth is consequence of the gra vitational in teraction, while the energy output of the sun and the stars is consequence of the uclear in teraction. The basic prop ert asso ciated with the electromag- netic in teraction is the electric harge that substances carry This can attributed to the harge carried the microscopic constituen ts of matter. The basic consequence of the existence of these harges is that harged particle exerts force on another harged particle. The precise nature of this force is expressed in Coulom b's la w. 2. orce et een Tw oin Charges 2a. Static Tw o-Bo dy Gra vitational orce. Tw stationary masses exert an attractiv force on one another that is prop ortional to their masses and in ersely prop ortional to the square of their separation gra (1) This is utually attractiv force: eac mass attracts the other with this same force, in agreemen with Newton's third la w. Although the magnitudes of the forces on the are equal, their accelerations will not equal if their masses are not equal. or example, the sun and the earth eac exert the same force on the other but the earth's acceleration to ard the sun is 10 times the sun's acceleration to ard the earth. 2b. Tw o-Bo dy Electrostatic orce. In addition to the gra vitational force (and apparen tly indep enden tly of an gra vitational force), parti- cles ma exert force on eac other whose origin is attributed to another See \Newton's La of Univ ersal Gra vitation" (MISN-0-101). Recall Newton's second la w. MISN-0-114 Figure 1. Distances and direc- tions for illustrating the Coulom force of article on article 2. basic prop ert of the particles, electric harge. That is, if in addition to the mass, eac particle has an electric harge, the particles will exert force on one another that is consequence of the existence of their harges. This force is called the electrostatic force. Unlik the gra vita- tional force, whic can only attractiv e, the electrostatic force can either repulsiv or attractiv e. urthermore, for the masses and harges of the elemen tary particles of nature suc as the electrons and protons, the electrostatic force is man orders of magnitude larger than the gra vita- tional force et een the same particles. 2c. The Electrostatic orce (Qualitativ e). Qualitativ ely the force that harged particles exert on eac other via the electrostatic in ter- action can describ ed as: (a) alw ys along the line joining them, (b) either repulsiv or attractiv dep ending up on whether the harges ha the same sign or ha opp osite signs, and (c) dep enden up on the sepa- ration of the particles, with the magnitude of the force decreasing with increasing separation et een them. In fact, giv en the particles' harges and their lo cations, one can calculate the force eac exerts on the other. The mathematical expression that yields the force is called Coulom b's la w. It is kno wn to alid er the remark able range of separation distances from as small as 10 15 meters to as large as 10 meters. The \static" stresses the fact that this force exists ev en if the particles are sta- tionary relativ to the observ er. This is to distinguish it from the magnetic force that arises when harged particles mo relativ to the observ er. This is not to sa that the result is incorrect at distances larger than million meters; that um er is merely the upp er limit to whic exp erimen has eried this in erse-square-la force.

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MISN-0-114 2d. Coulom b's La w. The simplest expression that demonstrates ho the electrostatic force dep ends up on the separation et een the harged particles can written do wn referring to Fig. 1. The magnitude of the electric force exerted on either particle the other is: (2) The co ecien is indep enden of the particle separation, dep ending only on basic in trinsic prop ert of the particles themselv es, their electrical harges. The co ecien can oth ositiv and negativ e, unlik the gra vitational coun terpart whic is alw ys negativ e, indicating oth repulsion and attraction as ossible et een the harged parti- cles. Because the electrostatic force can demonstrated to prop or- tional to the harge on eac of the particles, the pro duct of the individual harges app ears in aking this in to accoun t, Eq. (1) ma written for the force exerted on harged particle harged particle 1: (on particle 2) (3) where and are the harges of particles and 2, resp ectiv ely and is the unit ector oin ting along the radius ector from particle to particle (see Fig. 1). The unit of harge that en ters in this expression, when and are in MKS units, is the \coulom b." The constan quan tit is called \the electric force constan t." Its alue is dene to e: 99 10 (4) The newton and meter are dened elsewhere, while the coulom (sym ol \C") is dened in terms of electrical curren and time. Note that is the force in newtons that iden tical one-coulom harges ould exert on one another if they ere separated distance of one meter, just as is the gra vitational force, in newtons, that one-kilogram masses ould exert on eac other when separated one meter. Similarly the gra vitational force et een particles dep ends up on another in- trinsic prop ert of the particles, their masses. The expression for the gra vitational force et een particles (see MISN-0-101) is similar to Eq. (1), oth of them decreasing as the in erse square of the in ter-particle separation. The distance-indep enden co- ecien ts, ho ev er, are astly dieren t. As an illustration of this, the electrostatic force et een electrons for giv en separation is exactly the same as the electro- static force et een protons. Ho ev er, the gra vitational force et een protons is more than million times the gra vitational force et een electrons for the same separation. See the ables at the end of this ok. MISN-0-114 2e. Basic Electric Charges. Normal atoms and molecules consist of electrons, protons, and neutrons, and harge is strictly additiv when these particles com bine to form comp osite particles or collections of par- ticles. Electrons, protons, and neutrons ha these amoun ts of harge: (1 60217733 00000049) 10 19 +(1 60217733 00000049) 10 19 1) 10 21 will generally require only three digits of accuracy so tak the harge on the neutron as zero and the harges on the proton and the electron as equal but opp osite in sign. or notation, use where: 602 10 19 (5) All harges that are ev er observ ed are ositiv or negativ in teger ul- tiples of the harge on the electron (or proton). Ob viously it tak es large um er of electrons and/or protons to mak coulom or, for that matter, an of the normally observ ed amoun ts of harge that are dealt with in standard electronic apparatus. 3. Sev eral oin Charges 3a. Cho ose Con enien Co ordinate System. When there are more than harges, the in tro duction of suitable co ordinate system with corresp onding set of unit ectors greatly simplies the okk eeping in olv ed in determining the force on an one of the particles. An example with three harges is sho wn in Fig. 2. 3b. Apply Coulom b's La to All Charge airs. Coulom b's la still describ es the electrostatic in teraction et een all pairs of oin harges in the system. There are con tributions to the force on the harge due to harges and in Fig. 2: one that ould exerted on if only the pair and ere presen and the other that ould exerted on if only the pair and ere presen t. These forces are: Due to 13 Actually 10 21 The particles called \quarks" constitute the only exception: they are the con- stituen ts of protons and neutrons and their harges are one-third and o-thirds the electronic harge

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MISN-0-114 Figure 2. con enien co ordinate system for three oin harges. where 13 is the distance et een the harges and and is unit ector directed along the ositiv -axis. (The force that exerts on is the negativ of the ab e.) Due to 12 Where is unit ector 1) oin ting along the line from to 3c. otal orce is ector Sum. The resultan force on in Fig. is the ector sum of the forces from the other harges: 13 12 (6) com bine these ectors it is necessary to write unit ector in terms of its and -comp onen ts: (cos (sin Help: [S-2] (7) with as dened in Fig. 2. Then the net force is: 13 sin 12 cos 12 (8) MISN-0-114 Figure 3. The resultan force on harge due to the arrangemen of harges sho wn in Fig. 2. 3d. Numerical Example. Numerically with 13 100 12 0500 sin 800, and cos 600, the net force on in Fig. can determined to (46 (21 (9) The magnitude of this force is 51 newtons (10) The direction of the force mak es an angle 65 with the ositiv -axis as sho wn in Fig. 3. This is go example to hec for ourself. Ac kno wledgmen ts Preparation of this mo dule as supp orted in part the National Science oundation, Division of Science Education Dev elopmen and Researc h, through Gran #SED 74-20088 to Mic higan State Univ er- sit Glossary harge the basic prop ert of piece of matter that denes the strength of its in teraction with other matter through electromagnetic forces. coulom the MKS unit of harge, abbreviated \C," dened as the amoun of harge that passes oin in one second due to curren of one amp ere (\amp ere" and \second" are basic SI units). Coulom force an in teraction et een harged particles that ob eys Coulom b's la w, i.e. that dep ends on the pro duct of the harges and in ersely on the square of the particles' separation. 10

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MISN-0-114 electrostatic (in teraction) an electromagnetic in teraction that is indep enden of the motion of harged particles. electrostatic constan also called the electrostatic force constan t, written It is the in trinsic strength of the in teraction et een harged particles and as suc app ears in Coulom b's la and other electric and magnetic expressions. Its alue is dene to e: 10 99 10 Here is the sp eed of ligh t. 11 MISN-0-114 PS-1 PR OBLEM SUPPLEMENT Note: Problem also ccurs in this mo dule's Mo del Exam 1. or the system of three harged particles sho wn in Fig. 2, deriv the expression that giv es the resultan force on harge due to harges and a. Express the resultan in terms of the unit ectors and asso ciated with the co ordinate system sho wn in Fig. 2, getting Eq. (8) b. Giv en the umerical alues sho wn in Fig. 2, ev aluate the alues of the and -comp onen ts of this resultan force, as sho wn in Eq. (9) c. Putting in the dimensions of all of the quan tities that en ter in to the equation used in part (b), ab e, erify that the correct dimensions of the force comp onen ts are the \newton." d. Determine the magnitude and direction of this resultan force on harge getting Eq. (10) and Fig. 3. 2. rite do wn the expression for Coulom b's la w. Put in the units for eac one of the factors in the expression (including the electrostatic constan and erify that ou wind up with force unit. nswer: 11 3. harge 10 is lo cated at the origin of cartesian co ordinate system while harge +4 10 is lo cated at +0 30 meters, 0, 0. a. Calculate the force on due to nswer: b. Calculate the force on due to nswer: 12

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MISN-0-114 PS-2 4. Tw balls eac of mass 0500 kilograms are harged and attac hed to strings of length 600 meters hanging from the same oin t. When the system is in equilibrium eac of the balls mak es an angle of 30.0 with the ertical. a. Dra one-b dy force diagram for the ball carrying harge and iden tify the ob ject that exerts eac of those forces on this ball. nswer: 14 b. Do the same for the other ball. nswer: c. Set up co ordinate system at the lo cation of the ball carrying harge with oin ting horizon tally to the righ t, ertically up ard. Express the three forces acting on this ball in terms of these unit ectors and the sym ols in the sk etc (ab e). nswer: What is the resultan (sym olic) force on the ball, based on its acceleration? nswer: d. The alues of and on the left side of the sk etc (ab e) are the same as those on the righ side of the sk etc h. Do es this mean that the alue of the harge on the left side of the sk etc is the same as that on the righ side (i.e., do es )? nswer: 16 e. If 00 10 what is the alue of nswer: 13 MISN-0-114 PS-3 5. the corners of square of side length 030 are lo cated four harges as sho wn in the diagram ab e. is lo cated at the origin of the co ordinates, is on the -axis and on the -axis. 10 10 +2 10 +1 10 a. What force do es the harge exert on nswer: 13 What force do es exert on nswer: b. What force do es exert on nswer: 10 c. What force do es exert on nswer: d. What is the resultan force on nswer: e. Calculate the direction of this net force measured relativ to the ositiv -axis. nswer: 14

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MISN-0-114 PS-4 6. Three oin harges are lo cated at ositions as indicated in this dia- gram: +5 10 C, lo cated at the origin 10 C, lo cated at (0 m) +8 10 C, lo cated at (6.0 m, 8.0 m) Calculate the magnitude and direction of the resultan force on harge nswer: 15 15 MISN-0-114 PS-5 Brief Answ ers 1. same as in answ er except that in this case is the force on due to 2. (1.2 N) 3. (21 N) (21 N) 4. 80 10 5. sin cos sin 6. 83 10 Help: [S-1] 7. 90 clo kwise from the ositiv -direction 8. 9. 10 10. 10 11. See the mo dule text 12. It is not mo ving, so so 13. 21 (21 14. con tact force exerted the string non-con tact electrostatic force exerted harge non-con tact gra vitational force exerted the earth 16

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MISN-0-114 PS-6 15. 31 N, from the origin at an angle that is 51 from the ositiv -axis and 39 from the ositiv -axis. 16. No, but Coulom b's la pro duces force of on that is equal and opp osite to the force of on regardless of the relativ sizes of and This is in agreemen with Newton's third la w. 17 MISN-0-114 AS-1 SPECIAL ASSIST ANCE SUPPLEMENT S-1 (fr om PS, Pr oblem 4d) 05 kg )(9 m/s )(4)( (sin 30 (tan 30 (4 00 10 C)(8 99 10 /C 83 10 Note: distance et een harges (2)(0 )(sin 30 An alternativ e: note the equilateral triangle formed the strings and an imaginary line connecting the balls: all three angles are equal so all three sides are equal. An yw sk etc the situation roughly to scale and see that the alue of the distance is correct. S-2 (fr om TX, 3c) If ou ha trouble with this equation, review MISN-0-1 where suc expressions are discussed in great detail. ou can alw ys hec suc an expression taking: (i) to see that it is indeed of unit length; (ii) to see that the -comp onen is correct; and (iii) to see that the -comp onen is correct. 18

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MISN-0-114 ME-1 MODEL EXAM 99 10 1. See Output Skills K1-K2 in this mo dule's ID She et 2. Three oin harges are lo cated at ositions as indicated in this diagram: +5 10 C, lo cated at the origin 10 C, lo cated at (0 m) +8 10 C, lo cated at (6.0 m, 8.0 m) Calculate the magnitude and direction of the resultan force on harge Brief Answ ers 1. See this mo dule's text 2. See this Problem in this mo dule's Pr oblem Supplement 19 20

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