7.2. Coulomb s Law. The Electric Force

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1 Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat simila to gavity Both ae non-contact foces Simila to the foce of gavity, the electic foce becomes weake as the distance,, between the chaged objects inceases (Figue 1) Electic foce becomes stonge as the amount of chage on eithe object inceases, in the same way that the foce of gavity becomes stonge with an incease in mass of eithe object In this section, you will lean moe about how the electic foce depends on chage and distance You will also lean how to solve poblems elated to the electic foce 7 electic foce ( ) a foce with magnitude and diection that acts between two chaged paticles (a) 1 4 F 1 E 4 (b) Figue 1 The electic foce of epulsion between two identical chages deceases as the sepaation inceases The Electic Foce If you dop a tennis ball, the foce of gavity is esponsible fo its fall The tennis ball will take appoimately 1 s to fall fom a height of m How long do you think the tennis ball will take to stop as it hits the gound? It will take a lot less time to stop than it took to fall What foce is esponsible fo making the tennis ball stop? The electic foce of epulsion between the potons in the tennis ball and the potons in the gound stop it This electic foce must be significantly stonge than gavity Conside two chaged objects so tiny that you can model them as point paticles These objects have chages and and ae sepaated by a distance (Figue ) The magnitude of the electic foce between and is epessed by the euation Figue The electic foce between two point chages and is given by Coulomb s law k This euation epesents Coulomb s law Coulomb s aw The foce between two point chages is invesely popotional to the suae of the distance between the chages and diectly popotional to the poduct of the chages The constant k, which is sometimes called Coulomb s constant, has the value k N# m /C Do not confuse this constant k with the sping constant in Hooke s law Most physicists use the same symbol fo both The diection of the electic foce on each of the chages is along the line that connects the two chages, as illustated in Figue fo the case of two like chages As mentioned ealie, this foce is epulsive fo two chages with the same type of chage Stictly speaking, the value of in the euation fo Coulomb s law applies only to two point chages Howeve, it is a good appoimation wheneve the sizes of the paticles ae much smalle than thei distance of sepaation Coulomb s constant (k) the popotionality constant in Coulomb s law; k N# m /C NE 7 Coulomb s aw 37

2 Investigation 71 Coulomb s aw (page 366) Now that you have leaned how to calculate the electic foce between two chages with a given sepaation, pefom Investigation 71 to detemine how electic foce vaies with distance and 1 You will use this infomation with two diffeent chage values to detemine Coulomb s law Coulomb s law has seveal impotant popeties: You have aleady seen that the electic foce is epulsive fo like chages and attactive fo unlike chages Mathematically, this popety esults fom the poduct in the numeato of the Coulomb s law euation The magnitude of the electic foce,, is always positive In physics, it is convenient to use a negative sign to show diection Including the negative sign of a negatively chaged object would imply a diection You neve include the sign of the chage when solving poblems elated to Coulomb s law The electic foce is invesely popotional to the suae of the distance between the two paticles The magnitude of the electic foce is the magnitude of the foce eeted on each of the paticles That is, a foce of magnitude is eeted on chage by chage, and a foce of eual magnitude and opposite diection is eeted on by This paiing of eual but opposite foces is descibed by Newton s thid law, the action eaction pinciple Using the euation fo Coulomb s law, we can show that electic foces can be etemely lage Suppose you have two boes of electons, each with a total chage of T C sepaated by a distance of 10 m Fo simplicity, assume that each bo is so small that it can be modelled as a point paticle The magnitude of the total electic foce is k a N # m N C b C C 110 m This is an etemely lage foce, all fom just two small containes of electons Note that the negative sign fo chage was not included Why, then, does the electic foce not dominate eveyday life? The fact is that it is essentially impossible to obtain a bo containing only electons Recall foalie science studies that a neutal atom contains eual numbes of electons and potons If ou two point-like boes had contained eual numbes of electons and potons, thei total chages, T, would have been zeo, and so would the foce calculated using Coulomb s law Odinay matte consists of eual, o nealy eual, numbes of electons and potons The total chage is theefoe eithe zeo o vey close to zeo At the atomic and molecula scales, howeve, it is common to have the positive and negative chages (nuclei and electons) sepaated by a small distance In this case, the electic foce is not zeo, and these electic foces hold matte togethe Compaing Coulomb s aw and Univesal Gavitation The euation fo Coulomb s law may seem familia to you, even though you have only just leaned it This is because it is simila to the univesal law of gavitation, which you leaned in Chapte 6 Both of these laws descibe the foce between two objects In addition, both depend on cetain popeties of the objects involved Fo gavitation, this popety is mass Fo the electic foce, this popety is electic chage Both mass and chage can be consideed to be concentated at a cental point If you think of the mass o chage as a solid sphee, you can teat this same mass o chage as if it wee a point at the cente of that sphee Anothe similaity between the two laws is that both the gavitational foce and the electic foce decease as the distance between the two inteacting objects inceases If the distance between the objects is geat (compaed to the size of eithe object), the actual size and shape of the objects involved become mathematically ielevant The theoetical mass and chage at the cente point of each object ae then used in calculations 38 Chapte 7 Electic Fields NE

3 Although thee ae similaities between electic and gavitational foces, thee ae also impotant diffeences Gavitational foces ae always attactive The diection of electic foces depends on the types of chage: unlike chages attact, and like chages epel Also, the magnitude of the electic foce is much geate than the magnitude of the gavitational foce ove the same distance The Supeposition Pinciple So fa, you have leaned about the electic foce between two point paticles Now conside how to use Coulomb s law with moe than two chages Suppose you have thee paticles One paticle has a chage of The second paticle has a chage of and is a distance fom the fist paticle, as shown in Figue 3 What is the electic foce on a thid chage,, placed midway between these two chages? Solve this poblem by fist using Coulomb s law to calculate the foce eeted by chage on Then use Coulomb s law a second time to calculate the foce eeted by the chage on The total foce on euals the vecto sum of these two sepaate contibutions This combining of two foces is an eample of the supeposition pinciple The supeposition pinciple says that the total foce acting on is the sum, o supeposition, of the individual foces eeted on by all the othe chages in the poblem Remembe that foce is a vecto, so be caeful to add these foces as vectos This means that, fo chages not on a staight line, the solution euies some tigonomety involving tiangles o the use of vecto components When all chages lie in a staight line, the supeposition pinciple simplifies so that you can add o subtact the vaious individual foces to o fom one anothe to obtain the esultant, o net, foce As shown in Figue 3, the sepaation between and is Coulomb s law fo this pai of chages is theefoe FE1 Figue 3 The total foce on euals the sum of the foces fom chages and supeposition pinciple the esultant, o net, vecto acting at a given point euals the sum of the individual vectos fom all souces, each calculated at the given point k Using the coodinate system in Figue 3, this coesponds to a foce on by in the positive -diection because like chages epel So, is the component of the foce along the positive -ais The chages line up along the -ais, so the component of the foce along the y-ais is zeo Now conside the foce on by the chage in a simila way The sepaation of the chages is again, so, using the euation fo Coulomb s law, you get k 1 k Fom Figue 3 you can see that this coesponds to the foce on by in the negative -diection, again because like chages epel This is because the electic foce acts along the line connecting the two chages, which in this case is the -ais Fo this eason, the component of the foce along the y-ais is again zeo So, the foce eeted by on is in the negative -diection, o to the left The total foce on is the sum of the two electic foces, and Be caeful to take diection into account fo the calculation In this case, the sign of each foce indicates its diection F > E net F > E 1 1 F > E 1 k k F > E net k The negative sign means that, fo the positive poduct, the foce eeted on is along the negative -diection NE 7 Coulomb s aw 39

4 Tutoial 1 Using Coulomb s aw This Tutoial shows how to use Coulomb s law to solve chage distibution poblems in both one and two dimensions Sample Poblem 1: Applying Coulomb s aw in One Dimension An ealy model of the hydogen atom depicted the electon evolving aound the poton, much like Eath evolving aound the Sun The poton and electon both have mass, so they eet a gavitational foce upon each othe They also have chage, so they eet an electic foce on each othe (a) The distance between the electon and the poton in a hydogen atom is m, the chage of each is C, the mass of the electon is kg, and the mass of the poton is kg Calculate the atio of the electic foce to the gavitational foce F g (b) Detemine the acceleations of the electon caused by both the electic foce and the gavitational foce Solution (a) Given: m; C; kg; m p kg; k Nm /C ; G Nm /kg Reuied: F g Analysis: Use Coulomb s law to calculate, and use the euation fo univesal gavitation to calculate F g F g Gm p and F E k e p, whee e p Fom these esults, calculate F g Solution: k e p a N # m C b C m N 1two eta digits caied The euation fo Coulomb s law is moe accuately epesented by 0F > E 0 k Hee, the vetical bas suounding the electic foce vecto, 0F > E0, epesent the magnitude of the electic foce, and the vetical bas suounding the chages and, and, epesent the absolute values of the electic chages This epesentation is moe accuate because the magnitude of the electic foce is a stictly positive uantity If you just use the euation k, you will get a negative value fo the magnitude of the electic foce when one of the chages is negative The absolute value bas keep the values of the chages positive Howeve, many people find the moe accuate vesion of the euation cumbesome and confusing and pefe to use k This is what we will do when solving poblems Always emembe to dop the negative signs on chages when using this euation The following Tutoial will give you some pactice in solving poblems using Coulomb s law F g Gm p a N # m 11 kg b kg kg m F g N 1two eta digits caied N F g N F g Statement: The electic foce between the electon and the poton of a hydogen atom is times the gavitational foce F g between these same paticles (b) Given: kg; F g N; N Reuied: a E ; a g Analysis: Use the euation fo electic foce to calculate a E using and ikewise, use the euation fo gavitational foce to calculate a g using the values of F g and a E and a g F g Use 1 N 1 kg# m/s Solution: a E kg # m s kg a E m/s 330 Chapte 7 Electic Fields NE

5 a g F g kg # m s kg a g m/s Statement: The acceleation of the electon caused by the electic foce of the poton is m/s, and the acceleation of the electon caused by the gavitational foce is m/s Sample Poblem : Detemining Electostatic Euilibium Two chages, C and C, ae sepaated by a distance,, of 400 m A thid chage, C, is placed somewhee between and, as shown in Figue 4, whee the net foce eeted on 3 by the othe two chages is zeo Detemine the location of 3 3 Figue 4 Given: 400 m; C; C; C; k Nm /C Reuied:, the location of 3 Analysis: Use the supeposition pinciple, and use the euation fo Coulomb s law to calculate the foces eeted on 3 by and eeted on 3 by The foce euations ae k 3 and F E k 3 1 Since the net foce on 3 is zeo, 0, o Solution: k 3 k 3 1 Divide both sides of the euation by the common tems k and 3, and then simplify Substitute the values fo,, and into the euation, noting that is in metes with thee significant digits Solve fo C C C a C C b o Distance is always a positive uantity, so 1 Statement: The location of 3, such that the electic foces fom and cancel, is 100 m, o 100 m to the ight of Sample Poblem 3: Applying Coulomb s aw in Two Dimensions Two point paticles have eual but opposite chages of 1 and The paticles ae aanged as shown in Figue Suppose a chage is placed on the -ais as shown C, C, and the distance between 1 and is 80 m measued vetically along the y-ais Calculate the magnitude and the diection of the net electic foce on Given: C; C; C; k Nm /C Reuied: net electic foce on Analysis: As shown in Figue, thee ae symmetical ight tiangles above and below the -ais, so we can use the Pythagoean theoem to calculate (the distance sepaating 1 and ) Then use to calculate the electic foce between 1 and and that between and Use tigonomety to calculate the hoizontal and vetical components of these two foces to detemine the magnitude and diection of the net electic foce Use Coulomb s law to calculate the foce between the chages 40 m 40 m y 30 m Figue NE 7 Coulomb s aw 331

6 Solution: Fist, calculate the magnitudes of the electic foces along the hoizontal and vetical diections Both of the foces along the -ais act on the left side of the chage, so the net hoizontal foce is the sum of the two hoizontal foce components Use the Pythagoean theoem to calculate, the distance between 1 and, and that between and "140 m m 0 m Calculate the foce of 1 on, k a N # m 9 b C C C 10 m N 1two eta digits caied The foce between a positive chage and anothe positive chage is epulsion, so this foce is diected towad along the line connecting 1 and Calculate the foce of on, Since the magnitude of is the same as that of 1, the magnitude of is the same as that of N 1two eta digits caied The foce between a negative chage and a positive chage is attaction, so this foce is diected away fom along the line connecting and Now calculate the -component of, 1 et u be the angle that makes with the -ais 1 cos u 1 cos u N a 30 m 0 m b N 1two eta digits caied is diected towad, so 1 is diected towad along the positive -diection By symmety, the -component of,, must have the same magnitude as the -component of and is diected away fom along the negative -diection N 1two eta digits caied Detemine the vecto sum of the two hoizontal foce components F > E net F > E 1 1 F > E N N F > E net 0 N Thee is no net foce on along the -diection Now calculate the y-component of, y1 y1 sin u y1 sin u N a 40 m 0 m b y N 1two eta digits caied is diected towad fom 1, so y1 is diected towad downwad Since, the y-component of, y, has the same magnitude as the y-component of, y1 y y1 y N 1two eta digits caied is diected away fom fom, so y is also diected downwad Detemine the vecto sum of the two vetical foce components Choose upwad as positive, so downwad is negative F > Ey net F > Ey 1 1 F > Ey N N F > Ey net N Statement: The net electic foce acting on is N [down] Pactice 1 Detemine the magnitude of the electic foce between two chages of C and C that ae sepaated by a distance of 00 m T/I [ans: N] Two paticles of chage and (with positive) ae located as shown in Figue 6 A thid chage 3 is placed on the -ais to the left of The total electic foce on 3 is zeo The chages ae sepaated by 1000 m Detemine whee 3 must be located K/U T/I [ans: 414 m [left of ]] 1000 m Figue 6 3 Thee point chages ae placed at the following points on the -ais: 10 mc at 0, 30 mc at 400 cm, and 0 mc at 100 cm Detemine the foce on the 30 mc chage T/I [ans: 0 N towad the negative -diection, o 0 N [left]] 33 Chapte 7 Electic Fields NE

7 7 Review Summay Accoding to Coulomb s law, the foce between two point chages is diectly popotional to the poduct of the chages and invesely popotional to the suae of the distance between the chages, given as k Coulomb s constant, k, is eual to N m /C Coulomb s law applies to point chages and to chages that can be concentated euivalently in points located at the cente, when the sizes of the chages ae much smalle than thei distance of sepaation Thee ae similaities between the electic foce and the gavitational foce The supeposition pinciple states that the total foce acting on a chage is the vecto sum, o supeposition, of the individual foces eeted on by all the othe chages in the poblem Questions 1 Two chaged objects have a epulsive foce of 0080 N The distance sepaating the two objects is tipled Detemine the new foce T/I Two chaged objects have an attactive foce of 0080 N Suppose that the chage of one of the objects is tipled and the distance sepaating the objects is tipled Calculate the new foce T/I 3 Detemine the magnitude of the electic foce between two electons sepaated by a distance of 010 nm (appoimately the diamete of an atom) K/U T/I 4 Two point chages ae sepaated by a distance Detemine the facto by which the electic foce between them changes when the sepaation is educed by a facto of 10 K/U T/I Detemine the distance of sepaation euied fo two 100 µc chages to be positioned so that the epulsive foce between them is euivalent to the weight (on Eath) of a 100 kg mass T/I 6 Conside an electon and a poton sepaated by a distance of 10 nm K/U T/I (a) Calculate the magnitude of the gavitational foce between them (b) Calculate the magnitude of the electic foce between them (c) Eplain how the atio of these gavitational and electic foces would change if the distance wee inceased to 10 m 7 Paticles of chage and 3 ae placed on the -ais at 40 and 0, espectively A thid paticle of chage is placed on the -ais, and the total electic foce on this paticle is zeo Detemine the position of the paticle K/U T/I 8 Two chages of C and C ae placed at a sepaation of 10 cm Detemine whee a thid chage should be placed on the line connecting the two chages so that it epeiences no net foce due to these two chages K/U T/I A 9 Thee chages with C ae located as shown in Figue 7, with cm Detemine the magnitude and diection of the total electic foce on each paticle listed below T/I Figue 7 y (a) the chage at the bottom (b) the chage on the ight (c) an electon placed at the oigin 10 Two pith balls, each with a mass of 00 g, ae attached to non-conducting theads and suspended fom the same point on the ceiling Each thead has a length of 100 m The balls ae then given an identical chage, which causes them to sepaate At the point that the electic and gavitational foces balance, the theads ae sepaated by an angle of 300 Calculate the chage on each pith ball K/U A NE 7 Coulomb s aw 333