Charges, Coulomb s Law, and Electric Fields


 Flora Dawson
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1 Q&E 1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded by light ( ) electons. nucleus = (+) chaged potons and (0) chaged neutons m poton m neuton >> m electon (m poton 1800 m electon ) q poton = q electon ("q" is symbol fo chage) Calling potons (+) and electons ( ) is a convention. We could just have easily called electons (+) and potons ( ), but Ben anklin chose the othe, and we e stuck with it. The numbe of potons in the nucleus of an atom is called "Z". Z detemines the element: Z =1 is hydogen, Z = 3 is lithium. act : Unlike chages attact, and like chages epel accoding to Coulomb s Law, which says that the magnitude of the foce between two chages q 1 and q sepaated by a distance is given by = k q q 1 whee k = constant = N m /C. + q 1 q + q 1 + q Likesign chages epel. Unlikesign chages attact. In SI units, the unit of chage is the coulomb (C). magnitude of chage of electon = e = C chage of electon = e, chage of poton = +e (by convention, the symbol e > 0, always) A coulomb is a huge amount of chage: Numbe N of e s in 1 C =? Phys110 Lectue Notes, Dubson Univesity of Coloado at Boulde
2 Q&E  1C 1C Ne = 1C N= = = e C act 3: Electic chage is conseved. The net chage of an isolated system cannot change. It is impossible to ceate o destoy net chage. Except in nuclea o highenegy eactions, you can neve ceate o destoy electons, potons, and othe chaged paticles all we can do is move them aound. In high enegy eactions, we can ceate chaged paticles fom enegy (enegy = mc ), but the paticles ae always ceated o destoyed in pais (+1 and 1) so that the net chage is conseved. Aside: As fa as we know, only 4 things in the univese ae conseved: (1) Enegy () Linea momentum (p = mv) (3) Angula momentum (spin = L = Iω) (4) Chage [Not quite tue: in high enegy physics, thee may be othe quantities, like bayon numbe that ae conseved.] act 4: The chage e is the fundamental unit of chage. You neve find a fee paticle in natue with chage = faction of e. You only find chage = e o intege multiple of e. Statements (1) thu (4) ae expeimental facts. Why ae they tue? Why ae thee kinds of chage, not 3? Why e = C, not C? Why is chage conseved? We don t know! And to some extent, physicists don t cae. It is the pimay goal of physics to descibe how natue behaves; a seconday goal is to explain why it behaves that way. (Many theoists ae looking to explain why, but no luck yet ) Notice that Coulomb s law is simila to Newton s Univesal Law of Gavitation: 18 gav Gmm k q q =, coul = 1 1 Simila, except that thee ae two kinds of chage ( + and ), but only one kind (sign) of mass. Gavity is always attactive, but electical foce can be attactive o epulsive. Recall that foce is a vecto a mathematical object that has a size (magnitude) and a diection. oces add like vectos, not numbes. Example: Net foce on an electon due to two neaby potons, each a distance away, 90 o apat as shown. e 1 o 90 q 1 = +e Net foce = net = 1 + In this paticula case, = = = ke 1. q = +e Phys110 Lectue Notes, Dubson Univesity of Coloado at Boulde
3 Q&E 3 = ( not ) net Recall: = ke net 1 net = 1 Hee we have used the Supeposition Pinciple: the net foce on a chage due to othe neaby chages is the vecto sum of the individual foces: net = , whee 1 = net foce due to chage 1, etc. The Electic field (a new concept) Suounding evey chage (o goup of chages) is a thing, called an electic field E (it is a vecto thing) Definition: The electic field E at a point in empty space is a vecto quantity which can be measued by the following pocedue: place a small test chage q at that point, measue the foce on q due to all othe chages. The electic field at that point is given by E q on q Efield at a point is the foce pe chage on a test chage placed at that point. Note! The Efield exists even if thee is no test chage pesent to measue it. Similaly, a gavitational field suounds the eath, even if thee is no test mass neaby to measue the pull of eath s gavity: m mg m on m gavitational field = = g 1 GMm GM, magnitude g = = = m m (M = eath mass, m = test mass, = distance fom m to Eath s cente) The electic field is not just an mathematical invention; it is eal. We cannot (usually) see it o smell it, but we can feel it. In some situations, you can see an electic field: visible light is a apidly oscillating electic field (moe on that late in the semeste.) What is the Efield aound a point chage Q? (Q = souce chage = souce of Efield, q = test chage o pobe chage ) Phys110 Lectue Notes, Dubson Univesity of Coloado at Boulde
4 Q&E 4 souce chage Q test chage +q E on q 1 kqq kq ˆ q q on q = = = ˆ ( ˆ ponounced"hat" is the unit vecto pointing away fom the oigin, whee Q is. hat has no dimensions). Magnitude of the E field due to a point chage Q: E = k Q ˆ If the souce chage Q is positive, then the Efield points away fom Q, in the diection of hat. If the souce chage Q is negative then the E field points towad Q in the diection opposite hat. This follows diectly fom the definition E = /q. o instance, if both Q and q ae positive then the foce points away fom Q and so does E. If Q is negative and q is positive, then both and E point towad Q. What if the test chage q is changed fom positive to negative? Then the diection of the foce and the sign of q both flip, which leaves the diection of E unchanged. The size and diection of the Efield is independent of the test chage. The test chage is just an imaginay atifice which we use to measue something which is aleady thee. The Efield aound a positive chage points always fom the chage, and deceases in magnitude with 1 distance as E. We can epesent the Efield at vaious points in space by dawing a little dot at those points and dawing an aow coming out of that dot. The aow epesents the Efield at the dot point. Think of the Efield aow as "packed into the point". The Efield aow is not something "eaching fom beginning to end of aow". The Efield at a point in space exists at that point. Phys110 Lectue Notes, Dubson Univesity of Coloado at Boulde
5 Q&E 5 Notice that E as 0. The electic field diveges nea a point chage. on q Again, what if the test chage q in E = is negative? Efield still points away fom positive souce q chage Q, since both changes diection and q switches sign, which leaves the vecto E unchanged. The Efield points away fom positive chages. It points towad negative chages. We can think of the inteaction between chages in two diffeent ways: Action at a distance vs. ields Action at a distance : Coulomb s Law suggests that two chages exet a foce on each othe though empty space, instantaneously. But Coulomb s law is only valid fo stationay chages. If chage 1 moves, it takes some time fo chage to sense the change. The moe moden fieldview is: Chage 1 ceates an Efield aound it. Chage feels that field. If Chage 1 moves, it takes some time fo the suounding Efield to change, so it takes some time fo chage to eact. The total Efield due to a collection of chages is the vecto sum of the Efields due to the individual chages: Q1 Etotal = E1 + E + E3 + = Ei, whee E1 = E1 = k, E =.., etc i Why? Supeposition Pinciple says that if we place a small test chage q nea othe chages Q 1, Q, Q 3,, then the net foce on q is total 1 total = = + + Etotal = E1 + E + q q q Phys110 Lectue Notes, Dubson Univesity of Coloado at Boulde
6 Q&E 6 Example: Electic fields (qualitative) ou point chages, labeled 1 though 4, all with the same magnitude q, ae placed aound the oigin as shown. Chage is negative, the est ae positive. What is the diection of the Efield at the oigin? y 1 E =? 3 4 x The total Efield at oigin is the vecto sum E tot = E 1 + E + E 3 + E 4 (I'll use bold type to indicate vecto.) 1 Notice that E 3 and E 4 cancel since they have equal magnitude, but opposite diections. The total Efield points ight. E 4 E E E E 1 E 1 E tot What would be the diection of the foce on an electon (chage q = e) placed at the oigin? Since E = /q, we have = qe. If q is negative, the diection of the foce on q is opposite the on q diection of the Efield. So the foce is to the left. e E on q = qe In this equation, the Efield is due to all the othe chages, not the field due to the chage q itself. Example Electic fields (quantitative) Two chage Q 1 = +e and Q = 3e ae placed as shown. What the xcomponent of the electic field at the oigin? y Q = 3e E = E + E E = E + E tot 1 tot,x 1x x Q 1 = +e E x =? x Q1 e Q 3 E = 1 k k, E k k = = = e ( ) (Have used the fact that the distance fom the oigin to Q is ) Phys110 Lectue Notes, Dubson Univesity of Coloado at Boulde
7 Q&E 7 y Q = 3e E 1x = E 1, since E 1 is along the xaxis. E = E cos θ = E cos E o x Q 1 = +e E θ E 1 x e 3e Etot,x = E1x + Ex = k k e e = [ (3/ )] k = 3.06 k Efield due to Continuous distibution of chage Image a continuous distibution of chage with the chage spead out smoothly ove the volume of some object. What is the electic field at some point p due to this volume of chage? A vey small (infinitesimal) volume of the the object has an infinitesimal chage dq. "dq" means a "little bit of chage" This little bit of chage dq ceates an infinitesimal electic field de. The total electic field E at p due to all the bits of chage is E = de. dq p de Example: A semiinfinite line of chage with chage pe length = λ, units [λ] = C/m. What is the E field at a distance d fom the end of the line, as shown? E =? x dx d x chage dq = λ dx dx = "little bit of x", dq = "little bit of chage" Coulomb's Law gives us the magnitude of the field de due to the chage dq, a distance x away: de kdq kdq k λ dx = = = = " little bit of E due to little bit of chage dq" x x Since all diections hee ae along the xaxis, we ae only inteested in de x and can just dop the subscipt x. So instead of woking with the 3D integal E = de, we wok with the 1D integal Phys110 Lectue Notes, Dubson Univesity of Coloado at Boulde
8 Q&E 8 E k λ dx kλ kλ k = de = = = = x x d dλ. d 0 d chage Check units: [λ] = chage/length, so [k λ / d] has units of [k] = kq length. Units check! In D poblems whee the Efield has components along x and y, beak the poblem up into x and y components. E = de E = E xˆ + E yˆ = de xˆ + de x y x Conductos vs. Insulatos Most mateials can be classified as conducto o insulato. A conducto is also called a metal; an insulato also called avdielectic. Metals (Cu, Al, Au, Ag, e ) conduct electicity. In metals, some of the electons (conduction electons) can move feely thu the metal. If thee is an Efield, the conduction electons move in esponse to the foce = q E, and so a flow of chage, o cuent, occus. The inne coe electons ae bound stongly to thei nuclei, but the oute coe, conduction electons ae unbound, fee to move among the nuclei. Metals usually have 1 o conduction electons pe atom. (In chemist talk: valence = means conduction electons pe atom.) Insulatos (plastic, wood, ceamic, sulfu) do not conduct electicity. In insulatos, all the electons ae stongly attached to thei nuclei, and do not move (much), even if thee is an Efield exeting a foce on them. Metals ae shiny, insulatos ae dull. The appeaance is a consequence of the mobility of the electons. Insulatos can have an induced chage due to induced dipole moments. All atoms, some molecules, have no pemanent dipole moment, but acquie an induced moment when an extenal Efield is applied. y ŷ neutal atom E = 0 polaized atom in extenal field E q +q d Phys110 Lectue Notes, Dubson Univesity of Coloado at Boulde
9 Q&E 9 Recall fom chemisty, that a dipole moment is associated with a pai of equal and opposite chages (+q and q) sepaated by a distance d. The dipole moment p is a vecto quantity defined as p = qd, whee the vecto d points fom to +. Some molecules, like H O, have a pemanent dipole moment. In an extenal Efield, the moments align. = Induced Chage H O molecule = dipole moment moments align in Efield A chaged object (+Q, say) bought nea a neutal object induces a chage sepaation in the neutal object The equal and opposite chages on the two side of the object ae called induced chage. Anothe way to descibe this situation is to say that the Efield fom the chage Q induces polaization chage. Notice that an induced chage always esults in a net attaction to the souce of the Efield. The positivelychage +Q is attacting the negativelychaged nea side of the object and epelling the positivelychaged fa side. But the attaction to the neaby side is geate than the epulsion fom the moe distant side, so the net foce is attactive. neutal object (coke can o block of wood) +Q bought nea neutal object Q bought nea neutal object Phys110 Lectue Notes, Dubson Univesity of Coloado at Boulde
10 Q&E 10 Electic ield Line Diagams The electic field can be epesented by a field line diagam. Instead of tying to show the Efield at a whole bunch of individual points (left diagam), we indicate the field eveywhee with a field line diagam (ight diagam) ield line diagam fo (+) point chage. Rules fo field line diagams: 1) ield lines begin on positive chages, end on negative chages, o go off to infinity. Chage q Chage q ) The numbe of field lines coming fom o going to a chage is popotional to the magnitude of the chage: Moe field lines = bigge chage 3) The diection of the Efield at a point is the diection tangent to the field line at that point. 4) The magnitude of the Efield at a point is popotional to the density of field lines at that point. To be pecise, the magnitude of the Efield is popotional to the numbe of field pe aea pependicula to the field diection. Moe densely packed field lines = highe magnitude Efield. Phys110 Lectue Notes, Dubson Univesity of Coloado at Boulde
11 Q&E 11 Conductos in Electostatic Equilibium "Electostatic equilibium" means that all chages ae stationay; so the net foce on evey chage must be zeo (othewise the chage would be acceleating). 3 inteesting facts about metals (conductos) in electostatic equilibium: The electic field in the inteio of a metal must be zeo. Any net chage on the conducto esides only on the suface of the conducto. The electic field must be pependicula to the suface of the conducto. net chage on suface only E = 0 inside Efield suface A metal in electostatic equilibium The Efield must be zeo in the inteio, othewise the conduction electons in the metal would feel a foce = q E = e E and would move in esponse. Electons in motion would mean we ae not in electostatic equilibium. Any net chage esides only on the suface because any net chage in the inteio would ceate an Efield in the inteio which would cause the electons to move. The electons would keep moving until all chages have aanged themselves so that both the total Efield and net chage is zeo eveywhee inside the metal. In the next chapte, we will see a igoous poof of this using Gauss's Law. The Efield must be pependicula to the suface (in electostatic equilibium), othewise the component of the Efield along the suface would push electons along the suface causing movement of chages (and we would not be in equilibium). E E x On the suface of metal, if E x was not zeo, thee would be a foce x = q E x = e E x on electons in the metal pushing them along the suface. metal Phys110 Lectue Notes, Dubson Univesity of Coloado at Boulde
12 Q&E 1 Anothe cuious fact about electic fields: the Efield due to an a vey lage plane (sheet) of chage is constant in both diection and magnitude (as long as we ae "close" to the plane). We will pove this late, using Gauss' Law. E = constant E = constant unifom plane of chage (seen edgeon) One last impotant fact about chages and Efields: The Efield anywhee is always due to all the chages eveywhee. To get the total Efield, must always add up all the Efields due to all chages eveywhee: E tot = E 1 + E + E 3 + You cannot destoy o "block" the Efield due to a chage, but you can ceate a second Efield which cancels the fist Efield. Phys110 Lectue Notes, Dubson Univesity of Coloado at Boulde
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