Electric Fields. Phys102 General Physics II. Electric Fields

Transcription

1 lectic Fields Phs10 Geneal Phsics II lectic Fields Topics lectic field = Foce pe unit Chage lectic Field Lines and lectic Flu lectic field fom moe than 1 chage lectic Dipoles Motion of point chages in an electic field amples of finding electic fields fom continuous chages The lectic Field Definition of the electic field. Wheneve chages ae pesent and if I bing up anothe chage, it will feel a net Coulomb foce fom all the othes. It is convenient to sa that thee is field thee equal to the foce pe unit positive chage. =F/q 0. The diection of the electic field is along and points in the diection a positive test chage would move. This idea was poposed b Michael Faada in the 1830 s. The idea of the field eplaces the chages as defining the situation. Conside two point chages: The foce pe unit chage is = F/q 0 The Coulomb foce is F= k q 0 / + q 0 and then the electic field at is = k / due to the point chage. The units ae Newton/Coulomb. The electic field has diection and is a vecto. How do we find the diection.? The diection is the diection a unit positive test chage would move. q 0 If wee positive 1

2 Point negative chage lectic Field Lines Like chages (++) Opposite chages (+ -) = k / This is called an electic dipole. lectic Field Lines: a gaphic concept used to daw pictues as an aid to develop intuition about its behavio. Hee ae the dawing ules. -field lines begin on + chages and end on - chages. (o infinit). The ente o leave chage smmeticall. The numbe of lines enteing o leaving a chage is popotional to the chage The densit of lines indicates the stength of at that point. At lage distances fom a sstem of chages, the lines become isotopic and adial as fom a single point chage equal to the net chage of the sstem. No two field lines can coss. Tpical lectic Fields (SI Units) 1 cm awa fom 1 nc of negative chage = kq / = * 10-9 / 10 - =10 5 N /C N.m /C C / m = N/C Fai weathe atmospheic electicit = 100 N/C downwad 100 km high in the ionosphee ath Field due to a poton at the location of the electon in the H atom. The adius of the electon obit is 0.5*10-10 m. = kq / = * 1.6*10-19 / (0.5 *10-10 ) = *10 11 N /C Hdogen atom + - q. 1N / C = Volt/mete

3 ample of field lines fo a unifom distibution of positive chage on one side of a ve lage nonconducting sheet. This is called a unifom electic field. How would the electic field change if both sides wee chaged? How would things change if the sheet wee conducting? Methods of evaluating electic fields Diect evaluation fom Coulombs Law o bute foce method If we know whee the chages ae, we can find fom = kq i /. i This is a vecto equation and can be comple and mess to evaluate and we ma have to esot to a compute. The pinciple of supeposition guaantees the esult. Instead of summing the chage we can imagine a continuous distibution and integate it. This distibution ma be ove a volume, a suface o just a line. = d = kdq i / whee is a unit vecto diected fom chage dq to the field point. dq = ρdv, o dq =σ da, o dq = λ dl lectic field due to a point chage Diect evaluation fom Coulombs Law o bute foce method If we know whee the chages ae, we can find fom = kq i /. i This is a vecto equation and can be comple F 1 q = = q πε 0 0 Supeposition pinciple F 1 q = = q πε 0 0 F = F1+ F + F Fn F F F F... = n = = q 0 q 0 q 0 q 0 q n F 3

4 ample of finding electic field fom two chages We have q 1 =+10 nc at the oigin, q = +15 nc at = m. What is at =3 m and =0? point P P 3 =10 nc q =15 nc Use pinciple of supeposition Find and components of electic field due to both chages and add them up ample continued Recall =kq/ and k= N.m /C Field due to = N.m /C 10 X10-9 C/(3m) = 11 N/C in the diection. = 11 N/C = 0 Field due to q = N.m /C 15 X10-9 C/(5m) = 6 N/C at some angle φ Resolve into and components = sin φ = 6 3/5 =18/5 = 3.6 Ν/C = cos φ = 6 ( )/5 = /5 =.8 Ν/C φ 5 3 φ =10 nc q =15 nc Now add all components = = 1.6 N/C = -.8 N/C Magnitude = + = -.8 N/C 3 =10 nc Magnitude of electic field = + φ 1 ( ) ( ) = = 1.6 N/C q =15 nc = = 15. N / C ample continued Using unit vecto notation we can also wite the electic field vecto as: =.8 i+ 1.6 j ample of two identical chages on the ais. What is the field on the ais? q =15 nc =* sin φ = 6 3/5 = 7. N/C Ε =0 3 θ 5 φ q =15 nc = N.m /C 15 X10-9 C/(5m) = 6 N/C ample of two opposite chages on the ais. What is the filed on the ais? Ε q = -15 nc Ε =0 θ 3 5 φ q =15 nc =* cos φ = 6 /5 = N/C φ 1 = atan / = atan (1.6/-.8)= 7.8 deg

5 equal chages smmeticall spaced along a line. What is the field at point P? ( and = 0) lectic field due to an lectic Dipole q q 3 q 1 3 θ 1 P θ θ 3 θ φ Diect evaluation fom Coulombs Law o bute foce method If we know whee the chages ae, we can find fom F 1 q = = q πε 0 0 = i cos θi / i i= 1 k q = icos θi / i i= 1 k q Find electic field due to a line of unifom + chage of length L with linea chage densit equal to λ d = k dq / d d d = d cos θ d θ - + -L/ 0 dq = λd dq L/ d = k λ d cos θ / = k λ q cos θ / fo a point chage L / = kλ cosθ d / L / θ0 / cos θ0 = kλ θ dθ = kλ sinθ 0 L / = d = 0 L / d/ =dθ/ = tanθ d = sec θ dθ = sec θ = sec θ L / sinθ 0 = + L / What is the electic field fom an infinitel long wie with linea chage densit of +100 nc/m at a point 10 awa fom it. What do the lines of flu look like? = kλ sinθ 0. =10 cm = *1010 Nm *100*10 9 C /m sin90 = *10 N /C 0.1m Tpical field fo the electostatic smoke cleane 5

6 Back to computing lectic Fields lectic field due to a line of unifom chage lectic field due to an ac of a cicle of unifom chage. lectic field due to a ing of unifom chage lectic field of a unifom chaged disk Net we will go on to anothe simple method to calculate electic fields that woks fo highl smmetic situations using Gauss s Law. L / = d = 0 L / d = k dq cos θ / d = k λ ds cos θ / Field due to ac of chage L / L / θ0 cos / / cos = kλ dθ θ = kλ dθ θ = kλ sinθ 0 What is the field at the cente of a cicle of chage? Ans. 0 θ0 s= θ ds= dθ Find the electic field on the ais of a unifoml chaged ing with linea chage densit λ = Q/πR. z = dcosθ dq λds d = k = k kλcosθ ds z = π π ds = Rdθ = R dθ = πr 0 0 kλcosθ s = Rθ z = R π lectic field due to a chaged disk dq = λds kqz z = ( z + R ) 3/ =z +R cos θ =z/ =0 at z=0 =0 at z=infinit =ma at z=0.7r 6

7 Measuing the elementa chage lectic field gadient When a dipole is an electic field that vaies with position, then the magnitude of the electic foce will be diffeent fo the two chages. The dipole can be pemanent like NaCl o wate o induced as seen in the hanging pith ball. Induced dipoles ae alwas attacted to the egion of highe field. plains wh wood is attacted to the teflon od and how a smoke emove o micowave oven woks. lectostatic smoke pecipitato model Negativel chaged cental wie has electic field that vaies as a 1/ (stong electic field gadient). Field induces a dipole moment on the smoke paticles. The positive end gets attacted moe to the wie In the meantime a coona dischage is ceated. This just means that induced dipole moments in the ai molecules cause them to be attacted towads the wie whee the eceive an electon and get epelled poducing a cloud of ions aound the wie. When the smoke paticle hits the wie it eceives an electon and then is epelled to the side of the can whee it sticks. Howeve, it onl has to ente the cloud of ions befoe it is epelled. It would also wok if the polait of the wie is evesed lectic Dipoles: A pai of equal and opposite chages q sepaated b a displacement d is called an electic dipole. It has an electic dipole moment p=qd qd. + +q d p - -q II ~ kqd/ 3 when is lage compaed to d p=qd = the electic dipole moment P II ~kp/ 3 Note invese cube law II = kq/ Note invese squae law fo a single chage. 7

8 Wate (H O) is a molecule that has a pemanent dipole moment. GIven p = C m And q = -10 e and q = +10e What is d? d = p / 10e = C m / 10* C = m Ve small distance but still is esponsible fo the conductivit of wate. When a dipole is an electic field, the dipole moment wants to otate to line up with the electic field. It epeiences a toque. H O in a Unifom lectic Field Thee eist a toque on the wate molecule To otate it so that p lines up with. Toque about the com = τ F sin θ + F(d-)sin θ = Fdsin θ = qdsin θ = psin θ = p τ = p Potential neg U = -W = -pcosθ= p. Ε Is a minimum when p aligns with Leads to how micowave ovens heat up food Motion of point chages in electic fields When a point chage such as an electon is placed in an electic field, it is acceleated accoding to Newton s Law: a = F/m = q/m fo unifom electic fields a = F/m = mg/m = g fo unifom gavitational fields If the field is unifom, we now have a pojectile motion poblem- constant acceleation in one diection. So we have paabolic motion just as in hitting a baseball, etc ecept the magnitudes of velocities and acceleations ae diffeent. Replace g b q/m in all equations; Fo eample, In =1/at we get =1/(q/m)t ample: An electon is pojected pependiculal to a downwad electic field of = 000 N/C with a hoizontal velocit v=10 6 m/s. How much is the electon deflected afte taveling 1 cm. V e Since velocit in diection does not change, t=d/v =10 - /10 6 = 10-6 sec, so the distance the electon falls upwad is =1/at = 0.5*e/m*t = 0.5*1.6*10-19 **10 3 /10-30 *(10-8 ) = 0.016m d 8

9 Wam-up set Kelvin Wate Dop Geneato Am. J. Phs. 68,108(000) 1. [153709] Can thee be an electic field at a point whee thee is no chage? Can a chage epeience a foce due to its own field? Please wite a one o two sentence answe fo each question.. [153707] An insulato is a mateial that... thee ae coect is not penetated b electic fields none of these cannot ca an electic chage cannot feel an electical foce 3. [153708] Which of the following is tue of a pefect conducto? Thee can be no electic chage on the suface. Thee cannot be an electic field inside. Thee cannot be an ecess electic chage inside. Thee cannot be an electic chages inside. Two of the choices ae coect 9