Physics - lecticity and Magnetism Lectue 3 - lectic Field Y&F Chapte Sec. 4 7 Recap & Definition of lectic Field lectic Field Lines Chages in xtenal lectic Fields Field due to a Point Chage Field Lines fo Supepositions of Chages Field of an lectic Dipole lectic Dipole in an xtenal Field: Toque and Potential negy Method fo Finding Field due to Chage Distibutions Infinite Line of Chage Ac of Chage Ring of Chage Disc of Chage and Infinite Sheet Motion of a chaged paicle in an lectic Field - CRT example Copyight R. Janow Fall 6
Basics: Recap: lectic chage Positive and negative flavos. Like chages epel, opposites attact Chage is conseved and quantied. e.6 x -9 Coulombs Odinay matte seeks electical neutality sceening In conductos, chages ae fee to move aound sceening and induction In insulatos, chages ae not fee to move aound but mateials polaie Coulombs Law: foces at a distance enabled by a field F epulsion shown 3d law pai of foces q q F Constant k8.99x 9 Nm /coul Foce on q due to q (magnitude) Supeposition of Foces o Fields F n F net on F,i F, F,3 F,4... i k q q ˆ Copyight R. Janow Fall 6
Scala Field xamples: Vecto Field xamples: Fields Tempeatue - T() Pessue - P() Gavitational Potential enegy U() lectostatic potential V() lectostatic potential enegy U() Velocity - v() Gavitational field/acceleation - g() lectic field () Magnetic field B() Gadients of scala fields Fields explain foces at a distance space alteed by souce g() Gavitational Field vesus lectostatic Field foce/unit mass Fg () Lim ( ) > m m m is a test mass foce/unit chage () Fe () Lim ( ) > q q q is a positive test chage Test masses o chages map the diection and magnitudes of fields Copyight R. Janow Fall 6
() TST CHARG q x Field due to a chage distibution ARBITRARY CHARG DISTRIBUTION (PRODUCS ) F () () Lim ( ) q > q most often F () () q y Test chage q : small and positive does not affect the chage distibution that poduces. A chage distibution ceates a field: Map field by moving q aound and measuing the foce F at each point () is a vecto paallel to F() field exists whethe o not the test chage is pesent vaies in diection and magnitude F() q () F Foce on test chage q at point due to the chage distibution xtenal electic field at point Foce/unit chage SI Units: Newtons / Coulomb late: V/m Copyight R. Janow Fall 6
lectostatic Field xamples Field Location Inside coppe wies in household cicuits Nea a chaged comb Inside a TV pictue tube (CRT) Nea the chaged dum of a photocopie Beakdown voltage acoss an ai gap (acing) -field at the electon s obit in a hydogen atom -field on the suface of a Uanium nucleus Value - N/C 3 N/C 5 N/C 5 N/C 3 6 N/C 5 N/C 3 N/C Magnitude: F/q F q Diection: same as the foce that acts on the positive test chage SI unit: N/C Copyight R. Janow Fall 6
lectic Field due to a point chage Q Coulombs Law test chage q F 4πε Find the field due to point chage Q as a function ove all of space F q F q Q 4πε ˆ ˆ Qq ˆ q Q Magnitude KQ/ is constant on any spheical shell (spheical symmety) Visualie: field lines ae adially out fo Q, in fo - Q Flux though any closed (spheical) shell enclosing Q is the same: Φ A Q.4π /4πε Q/ε Radius cancels The closed (Gaussian) suface intecepts all the field lines leaving Q Copyight R. Janow Fall 6
Use supeposition to calculate net electic field at each point due to a goup of individual chages xample: fo point chages at,.. F F F... F net net Ftot F F F... q q q q... n n n net at i 4πε j q i ij ˆ ij Do the sum above fo evey test point i Copyight R. Janow Fall 6
Visualiation: lectic field lines (Lines of foce) Map diection of an electic field line by moving a positive test chage aound. The tangent to a field line at a point shows the field diection thee. The density of lines cossing a unit aea pependicula to the lines measues the stength of the field. Whee lines ae dense the field is stong. Lines begin on positive chages (o infinity and end on negative chages (o infinity). Field lines cannot coss othe field lines Copyight R. Janow Fall 6
DTAIL NAR A POINT CHARG WAK STRONG NAR A LARG, UNIFORM SHT OF CHARG No conducto - just an infinitely lage chage sheet appoximately constant in the nea field egion (d << L) L d q F TWO QUAL CHARGS (RPL) to to The field has unifom intensity & diection eveywhee except on sheet QUAL AND CHARGS ATTRACT Copyight R. Janow Fall 6
Field lines fo a spheical shell o solid sphee of chage Shell Theoem Conclusions Outside point: Same field as point chage Inside spheical distibution at distance fom cente: fo hollow shell; kq inside / fo solid sphee Copyight R. Janow Fall 6
xample: Find net at a point on the axis of a dipole Use supeposition Symmety net paallel to -axis - d/ and d/ kq kq at O Limitation: > d/ o < - d/ O - at O kq ( d/ ) ( d/ ) d q - q - at O kqd ( d xact / 4) Fo >> d : point O is fa fom cente of dipole qd p πε 3 πε qd DIPOL MOMNT p points fom to q d q [ ] since << at O 3 3 d xecise: Do these fomulas descibe at the point midway between the chages Ans: -4p/πε d 3 Fields cancel as d so falls off as / 3 not / is negative when is negative Does fa field look like point chage? Copyight R. Janow Fall 6
lectic Field 3-: Put the magnitudes of the electic field values at points A, B, and C shown in the figue in deceasing ode..b A) C > B > A B) B > C > A C) A > C > B D) B > A > C ) A > B > C.C.A Copyight R. Janow Fall 6
A Dipole in a Unifom XTRNAL lectic Field Feels toque - Stoes potential enegy (See Sec.7) ASSSUM RIGID DIPOL p qd Dipole Moment Vecto Toque Foce x moment am - q x (d/) sin(θ) - p sin(θ) (CW, into pape as shown) τ px toque at θ o θ π toque p at θ /- π/ RSTORING TORQU: τ( θ) τ(θ) Potential negy U -W U τdθ p sin( θ)dθ pcos( θ) U p OSCILLATOR U fo θ /- π/ U - p fo θ minimum U p fo θ π maximum Copyight R. Janow Fall 6
3-3: In the sketch, a dipole is fee to otate in a unifom extenal electic field. Which configuation has the smallest potential enegy? A B C D 3-4: Which configuation has the lagest potential enegy? Copyight R. Janow Fall 6
Method fo finding the electic field at point P - - given a known continuous chage distibution This pocess is just supeposition. Find an expession fo dq, the point chage within a diffeentially small chunk of the distibution dq ρ σλ dl da dv qi P lim ˆ i 4πε q πε fo a linea distibution fo a suface distibution fo a volume distibution i 4 i dq ˆ. Repesent field contibutions at P due to a point chage dq located anyhwee in the distibution. Use symmety whee possible. q ˆ 4πε d dq 4πε 3. Add up (integate) the contibutions d ove the whole distibution, vaying the displacement and diection as needed. Use symmety whee possible. d P dist (line, suface, o volume integal) ˆ Copyight R. Janow Fall 6
xample: Find electic field on the axis of a chaged od Rod has length L, unifom positive chage pe unit length λ, total chage Q. λ Q/L. Calculate electic field at point P on the axis of the od a distance a fom one end. Field points along x-axis. dq λdx d 4πε dq x 4 πε λdx x Add up contibutions to the field fom all locations of dq along the od (x ε [a, L a]). L a a 4 λ πε dx x 4 λ πε L a a dx x 4 λ πε x L a a 4πε Q L a L a 4πε Q a(l a) Intepet Limiting cases: L > od becomes point chage L << a same, L/a << L >> a a/l <<, Copyight R. Janow Fall 6
lectic field at cente of an ARC of chage L dq Unifom linea chage density λ Q/L dq λds λrdθ R θ θ P d p Integate: θ P,y P on symmety axis at cente of ac Net is along y axis need y only kdq k dq cos( θ) dp ˆ dp,y ĵ R R Angle θ is between θ and θ λ ĵ R 4πε R note : cos( θ)dθ θ θ cos( θ)dθ sin( θ) λ P,y k sin( θ) R ĵ and 4 λ πε sin( θ) ĵ R sin( θ) sin( θ) θ θ In the plane of the ac Fo a semi-cicle, θ π/ Fo a full cicle, θ π λ k ĵ P,y R P, y Copyight R. Janow Fall 6
lectic field due to a staight LIN of chage Point P on symmety axis, a distance y off the line x y tan( θ) x y θ y [ tan ( θ)] Find dx in tems of θ: dx d [tan( θ)] y y dθ dθ y [ tan ( θ)] dx y [ tan ( θ)] dθ d d sin( θ) θ cos( θ) dq λdx λy [ tan L d P ( θ) y θ x P d P ( θ) dq λdx θ ( θ)] dθ d d y, P unifom linea chage density: λ Q/L point P is at y on symmety axis by symmety, total is along y-axis x-components of d pais cancel solve fo line segment, then let y << L P tan (θ) cancels in numeato and denominato kdq k ˆ d λ cos( θ)dθ ĵ y Integate fom θ to θ k λ θ y, P ĵ θ P, y S Y&F xample. k dq cos( θ ) k λ cos( θ)dθ ĵ sin( θ) y y k λ y Fo y << L (wie looks infinite) θ π/ P ĵ sin( θ P k λ y ) ĵ Copyight R. Janow Fall 6 Falls off as /y ˆ j θ θ Finite length wie Along y diection
lectic field due to a RING of chage at point P on the symmety () axis Unifom linea chage density along cicumfeence: λ Q/πR dq λds chage on ac segment of length ds Rdφ P on symmety axis xy components of cancel Net field is along only, nomal to plane of ing kdq k dq cos( θ) dp ˆ d,p kˆ dq λ ds λ R dφ cos( θ) / R k λrdφ dp, kˆ 3 Integate on aimuthal angle φ fom to π k λr k π dφ integal π P, 3 / [ R ] πrλ Q total chage on disk kq P, kˆ as 3 / [ R ] (see esult fo ac) ds Limit: Fo P fa away use >> R kq P, Ring looks like a point chage if point P is vey fa away! S Y&F xample.9 R dφ dφ xecise: Whee is a maximum? Set d /d Ans: R/sqt() Copyight R. Janow Fall 6
lectic field due to a DISK of chage fo point P on (symmety) axis d See Y&F x. Unifom suface chage density on disc in x-y plane σ Q/πR Disc is a set of ings, each of them d wide in adius P on symmety axis net field only along dq chage on ac segment dφ with adial extent d da d dφ dq σ da σ d cos( θ) /s s k dq cos( θ) kˆ s 4 πε d dφ σ d dφ [ ] 3/ kˆ R x P θ φ s dadφd Integate twice: fist on aimuthal angle φ fom to π which yields a facto of π then on ing adius fom to R πσ 4πε Note Antideivative R d [ ] [ 3 / ] 3 / d d [ k / ] disk σ ε [ ] R / kˆ Copyight R. Janow Fall 6
lectic field due to a DISK of chage, continued xact Solution: disk σ ε [ R ] Nea-Field: << R: P is close to the disk. Disk looks like infinite sheet. σ σ σ fo /R << : kˆ kˆ disk ε / ε R R ( /R) ε disk σ ε kˆ / kˆ [ ] nea field is constant disk appoximates an infinite sheet of chage Fa-Field: R<< : P is fa fom to the disk. Disk looks like a point chage. fo R/ << : disk σ ε [ (R / ) ] / kˆ Recall σ Q / πr Seies xpansion n ( s) ns /! n(n )s /!... appoximate : / conveges quickly fo s << [ (R/) ] σ R σ R Q kˆ kˆ disk ε 4ε disk 4πε kˆ R... Copyight R. Janow Fall 6 Point chage fomula
Infinite (i.e. lage ) unifomly chaged sheet Non-conducto, fixed suface chage density σ Infinite sheet d<<l nea field unifom field σ ε fo infinite, non - conducting chaged sheet L d Method: solve non-conducting disc of chage fo point on -axis then appoximate << R Copyight R. Janow Fall 6
Motion of a Chaged Paticle in a Unifom lectic Field F q ma F net Stationay chages poduce field at location of chage q Acceleation a is paallel o anti-paallel to. Acceleation is F/m not F/q Acceleation is the same eveywhee in unifom field xample: aly CRT tube with electon gun and electostatic deflecto LCTROSTATIC ACCLRATOR PLATS (electon gun contols intensity) FAC OF CRT TUB heated cathode (- pole) boils off electons fom the metal (themionic emission) LCTROSTATIC DFLCTOR PLATS electons ae negative so acceleation a and electic foce F ae in the diection opposite the electic field. Copyight R. Janow Fall 6
Motion of a Chaged Paticle in a Unifom lectic Field F q Fnet ma Kinematics: ballistic tajectoy v x electons ae negative so acceleation a and electic foce F ae in the diection opposite the electic field. Use to find e/m L y x v y is the DFLCTION of the electon as it cosses the field Acceleation has only a constant y component. v x is constant, a x a y v x e m v x yields time of flight t y y L t Measue deflection, find t via kinematics. valuate v y & v x a a y y t t v e m t ( ) v v x y / Copyight R. Janow Fall 6