Electric Field, Potential Energy, & Voltage

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Charlene Lester
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1 Slide 1 / 66 lectic Field, Potential negy, & oltage Wok Slide 2 / 66 Q+ Q+ The foce changes as chages move towads each othe since the foce depends on the distance between the chages. s these two chages get close togethe, it takes moe foce to keep them fom flying apat since they'e epelling each othe. It takes wok to move the chages. W = Foce x distance paallel W = kqq lectic Potential negy Q+ Q+ W = kqq Slide 3 / 66 If we have two chages initially at est, and infinitely fa apat: 0 + W = f 0 + kqq = f f = U = kqq U = kqq Potential negy due to two point chages

2 lectic Potential negy Slide 4 / 66 This is the equation fo the potential enegy due to two point chages being nea each othe. U = kqq negy is NOT a vecto, it is a scala. Thee is no diection, but the sign mattes. If you have two positive chages o two negative chages, thee will be a positive potential enegy. This means it is taking enegy to keep them fom flying apat. If you have a positive chage and a negative chage nea each othe, you will have a negative potential enegy. This means that it takes enegy to keep them fom getting close togethe. +Q 1 +Q 2 Slide 5 / X(m) positive chage Q1 = 5 m is located at x1 = -8 m and a chage Q2 = 2.5 m is located at x2 = 3 m. ompute the potential enegy of the two chages. U = kqq = (9x10 9 Nm 2 / 2 )(5x10-3 )(2.5x10-3 ) 11m U = J negy of Multiple hages Slide 6 / 66 To get the total enegy fo multiple chages, you must fist find the enegy due to each pai of chages. Then, you can add these enegies togethe. Since enegy is a scala, thee is no diection involved. U total = U 1 + U 2 + U

3 lectic Potential o oltage Slide 7 / 66 We know that: Just as we can beak electical foce into two pats: F = q and = kq, 2 we could sepaate potential enegy into two pats: U = q and = kq whee is called the voltage. lectic Potential o oltage Slide 8 / 66 oltage is also called electic potential (NOT to be confused with electic potential enegy). oltage is measued in olts () whee one = J oltage is NOT a vecto, so multiple voltages can be added diectly (sign is impotant!). Unifom lectic Field Slide 9 / 66 The field cancels outside the plates and adds togethe between the plates fo a stong electic field. Unifom means that the stength of the field is the same eveywhee (between the plates).

4 Unifom lectic Field Slide 10 / 66 + Only some equations we have leaned apply to unifom electic fields. Point chages have a non-unifom field stength since the field weakens with distance. F = kqq 2 = kq 2 U = kqq = kq Use ONLY with point chages. quations with the "k" ae point chages ONLY. F = q U = q Use in NY situation. Fo point chages N unifom electic fields Slide 11 / 66 lectic Field & oltage Slide 12 / 66 mg FN Hill a The slope of the plane detemines the acceleation and the net foce on the object. FN mg Slope = 0 F net = 0 no acceleation!

5 lectic Field & oltage Slide 13 / 66 If we look at the enegy of the block on the inclined plane W = f whee W = 0 2 mgδh = ½mv f If v 2 f = v aΔx and v 0 = 0 then v 2 f = 2aΔx mgδh = ½m(2aΔx) mgδh = maδx gδh = aδx a = gδh Δx lectic Field & oltage Slide 14 / 66 simila elationship exists with unifom electic fields and voltage. With the inclined plane, a diffeence in height was esponsible fo acceleation. Hee, a diffeence in electic potential (voltage) is esponsible fo the electic field. f o lectic Field & oltage Slide 15 / 66 The change in voltage is defined as the wok done pe unit chage against the electic field. Theefoe enegy is being put into the system when a positive chage moves in the opposite diection of the electic field (o when a negative chage moves in the same diection of the electic field). f o

6 lectic Field & oltage Slide 16 / 66 To see the exact elationship, look at the enegy of the system. 0 + W = f whee W = 0 2 q 0 = q f + ½mv f 2 q 0 - q f = ½mv f -qδ = ½mv f 2 whee Δ = f - 0 If v f 2 = v aΔx and v 0 = 0 then v f 2 = 2aΔx -qδ = ½m(2aΔx) -qδ = maδx If F = ma, and F = q, then we can substitute ma = q -qδ = qδx -Δ = Δx = -Δ = -Δ Δx d lectic Field & oltage Slide 17 / 66 The equation only applies to unifom electic fields. 1 N = and a = J m 1 N = (J/) m and a J = N m Δ Δ = _ = _ Δx d It follows that the electic field can also be shown in tems of volts pe mete (/m) in addition to Newtons pe oulomb (N/). This can be shown: 1 N (N m/) = m 1 N = 1 N The units ae equivalent. lectic Field & oltage moe intuitive way to undestand the negative sign in the elationship Δ = _ Δx is to conside that just like a mass falls down, fom highe gavitational potential enegy to lowe, a positive chage "falls down" fom highe electic potential () to lowe. Slide 18 / 66 Since the electic field points in the diection of the foce on a hypothetical positive test chage, it must also point fom highe to lowe potential. The negative sign just means that objects feel a foce fom locations with geate potential enegy to locations with lowe potential enegy. This applies to all foms of potential enegy.

7 1 In ode fo a chaged object to expeience an electostatic foce, thee must be a: Slide 19 / 66 a lage electic potential a small electic potential the same electic potential eveywhee a diffeence in electic potential 2 How stong (in /m) is the electic field between two metal plates 25 cm apat if the potential diffeence between them is 100? Slide 20 / /m 600 /m 800 /m 1000 /m 1200 /m 3 n electic field of 3500 N/ is desied between two plates which ae 4.0 mm apat; what voltage should be applied? Slide 21 /

8 ombining Two Ideas... Slide 22 / 66 = - Δ d U = q Δ Δ = -d U = -qd 4 How much wok (in Joules)is done by a unifom 300 N/ electic field on a chage of 6.1 m in acceleating it though a distance of 20 cm? Slide 23 / x 10-2 J 3.66 x 10-2 J 3.81 x 10-2 J 3.12 x 10-2 J 4.93 x 10-2 J F = kqq 2 = kq 2 U = kqq = kq Use ONLY with point chages. quations with the "k" ae point chages ONLY. F = q U = q = - Δ d U = -qd Use in NY situation. Fo point chages N unifom electic fields ONLY fo unifom electic fields Slide 24 / 66

Topogaphic Maps Slide 25 / 66 ach line epesents the same height value. The aea between lines epesents the change between lines. big space between lines, thee is a slow change in height.

quipotential Lines Slide 26 / 66 0 50 230 300 300 230 50 0 These "topogaphy" lines ae called "equipotential lines" when we use them to epesent the electic potential.

9 Topogaphic Maps Slide 25 / 66 ach line epesents the same height value. The aea between lines epesents the change between lines. big space between lines, thee is a slow change in height. Little space between lines means thee is a vey quick change in height. Whee in this pictue is the steepest incline? quipotential Lines Slide 26 / These "topogaphy" lines ae called "equipotential lines" when we use them to epesent the electic potential The close the lines, the faste the change in voltage... the bigge the change in voltage, the lage the electic field. quipotential Lines Slide 27 / The diection of the electic field and foce ae always pependicula to the lines. 2. The electic field lines ae fathe apat when the equipotentials ae fathe apat The electic field goes fom high to low potential (just like a positive chage).

10 5 t point in the diagam, what is the diection of the electic field? Slide 28 / 66 Up own Left Right How much wok is done to a +10μ chage that moves fom point to? Slide 29 / 66 1 x 10 J 1.3 x 10 J 2 x10 J 3.5 x 10 J 1.5 x 10 J Paallel Plate apacitos Slide 30 / 66 The simplest vesion of a capacito is the paallel plate capacito which consists of two metal plates that ae paallel to one anothe and located a distance apat.

11 When a battey is connected to the plates, chage moves between them. vey electon that moves to the negative plate leaves a positive nucleus behind. Paallel Plate apacitos Slide 31 / 66 The plates have equal magnitudes of chages, but one is positive, the othe negative. Paallel Plate apacitos Slide 32 / 66 Only unpaied potons and electons ae epesented hee. Most of the atoms ae neutal since they have equal numbes of potons and electons. Paallel Plate apacitos Slide 33 / 66 awing the lectic Field fom the positive to negative chages eveals that the lectic Field is unifom eveywhee in a capacito's gap. lso, thee is no field outside the gap.

12 apacitos Slide 34 / 66 NY capacito can stoe a cetain amount of chage fo a given voltage. That is called its capacitance,. = Q This is just a FINITION and is tue of all capacitos, not just paallel plate capacitos. Units of apacitance Slide 35 / 66 = Q The unit of capacitance is the faad (F). faad is a oulomb pe olt. faad is huge; so capacitance is given as picofaad (1pf = F), nanofaad (1nf = 10-9 F), micofaad (1µf = 10-6 F), millifaads (1mf = 10-3 F) 7 What is the capacitance of a fully chaged paallel plate capacito that has a chage of 25 µ and a diffeence in potential of 50? Slide 36 / 66 5 x 10-7 F 2 x 10-7 F 4 x 10-7 F 8 x10-7 F 1 x 10-7 F

13 8 fully chaged 50µF paallel place capacito has a potential of 100 acoss it's plates. How much chage is stoed in the capacito? Slide 37 / 66 6 x 10-3 F 4 x 10-3 F 5 x 10-3 F 9 x 10-3 F 10 x 10-3 F Paallel Plate apacitos Slide 38 / 66 The ea of the capacito is just the suface aea of ON PLT, and is epesented by the lette. The distance between the plates is epesented by the lette d. d Paallel Plate apacitos Slide 39 / 66 Fo PRLLL PLT PITORS, the capacity to stoe chage inceases with the aea of the plates and deceases as the plates get fathe apat. # # 1/d

14 The constant of popotionality is called the Paallel Plate apacitos Slide 40 / 66 Pemitivity of Fee Space and has the symbol # o. # o = 8.85 x /N-m 2 So fo PRLLL PLT PITORS: Paallel Plate apacitos Slide 41 / 66 = #o d The lage the ea,, the highe the capacitance. The close togethe the plates get, the highe the capacitance. Paallel Plate apacitos Slide 42 / 66 Imagine you have a fully chaged capacito. If you disconnect the battey and change eithe the aea o distance between the plates, what do you know about the chage on the capacito? The chage emains the same.

15 Paallel Plate apacitos Slide 43 / 66 Imagine you have a fully chaged capacito. If you keep the battey connected and change eithe the aea o distance between the plates, what do you know about the voltage acoss the plates? The voltage emains the same. 9 paallel plate capacito has a capacitance 0. If the aea on the plates is doubled and the distance between the plates dops by one half, what will be the new capacitance? Slide 44 / 66 0 /4 0 / paallel-plate capacito is chaged by connection to a battey and emains connected. What will happen to the chage on the capacito and the voltage acoss it if the aea of the plates inceases and the distance between them deceases? Slide 45 / 66 oth incease oth decease oth emain the same The voltage emains the same and chage inceases The voltage emains the same and the chage deceases

16 11 paallel-plate capacito is chaged by connection to a battey and the battey is disconnected. What will happen to the chage on the capacito and the voltage acoss it if the aea of the plates deceases and the distance between them inceases? Slide 46 / 66 oth incease oth decease oth emain the same The chage emains the same and voltage inceases The chage emains the same and the voltage deceases -field and oltage in P.P. apacitos Slide 47 / 66 fte being chaged the plates have equal and opposite voltage,. Thee is a unifom electic field,, between the plates. +/2 We leaned ealie that with a UNIFORM -FIL that # = -d; this is tue in the case of the paallel plate capacito. -/2 -field and oltage in PP apacitos Slide 48 / 66 The lectic Field is constant eveywhee in the gap. The oltage (also know as the lectic Potential) declines unifomly fom + to - within the gap; it is zeo at the location midway between the plates. +/2 +/4 0 -/4 -/2 It is always pependicula to the - Field.

17 negy Stoage in apacitos Slide 49 / 66 The enegy stoed in NY capacito is given by fomulas most easily deived fom the paallel plate capacito. onside how much wok it would take to move a single electon between two initially unchaged plates. negy Stoage in apacitos Slide 50 / 66 That takes ZRO wok since thee is no diffeence in voltage. Howeve, to move a second electon to the negative plate equies wok to ovecome the epulsion fom the fist one...and to ovecome the attaction of the positive plate. negy Stoage in apacitos Slide 51 / 66 To move the last electon fom the positive plate to the negative plate equies caying it though a voltage diffeence of. +/2 The wok equied to do that is q#... -/2 Hee the chage of a electon is -e, and the diffeence in potential is - (-/2 - /2)... the wok = e.

18 negy Stoage in apacitos Slide 52 / 66 If the wok to move the fist electon is zeo. nd the wok to move the last electon is e. +/2 The the RG wok fo LL electons is e/2. -/2 negy Stoage in apacitos Slide 53 / 66 Then, the wok needed to move a total chage Q fom one plate to the othe is given by +/2 W = Q/2 That enegy is stoed in the electic field within the capacito. -/2 negy Stoage in apacitos Slide 54 / 66 So the enegy stoed in a capacito is given by: U = Q 2 Whee Q is the chage on one plate and is the voltage diffeence between the plates. +/2 -/2

19 negy Stoage in apacitos Slide 55 / 66 Using ou equation fo capacitance (=Q/) and ou equation fo electic potential enegy in a capacito, we can deive thee diffeent esults. solve fo solve fo Q U = Q 2 substitute substitute U = Q2 2 U = 1/ How much enegy is stoed in a fully chaged paallel plate capacito that is stoing 15 n with 3 acoss its plates? Slide 56 / x 10-8 J 1.9 x 10-8 J 2.3 x 10-8 J 2.7 x 10-8 J 3.0 x 10-8 J 13 How much enegy is stoed in a fully chaged 3 mf paallel plate capacito with 2 acoss its plates? Slide 57 / 66 6 x 10-3 J 3 x 10-3 J 5 x 10-3 J 8 x 10-3 J 12 x 10-3 J

20 14 How much enegy is stoed in a fully chaged 16 mf paallel plate capacito with 220 acoss its plates? Slide 58 / J J J J J 15 paallel plate capacito is connected to a battey. The capacito becomes fully chaged and stays connected to the battey. What will happen to the enegy held in a paallel plate capacito if the aea of the plates inceases? Slide 59 / 66 Remains the same Inceased eceased Zeo Moe infomation is equied 16 paallel plate capacito is connected to a battey. The capacito becomes fully chaged and is then disconnected fom the battey. What will happen to the enegy held in a paallel plate capacito if the aea of the plates inceases? Slide 60 / 66 Remains the same Inceased eceased Zeo Moe infomation is equied

21 ielectics and apacitos Slide 61 / 66 apacitance can be inceased by inseting a dielectic (an insulato) into the gap. efoe the plates ae chages the atoms ae unpolaized, the electon is bound to the nucleus and not oiented in any diection. ielectics and apacitos Slide 62 / 66 When the plates chage, the atoms ae polaized and line up to educe the extenal -field. +/2 This educes the electic field, which lowes the voltage fo a given chage (since = d). -/2 - + Since = Q/, this inceases the capacitance. ielectics and apacitos Slide 63 / 66 vey mateial has a dielectic constant, # (kappa), which is given in a table. +/2 Fo a vacuum, = 1; ai is about 1. If a dielectic is pesent, then: = ##o d -/ The lage #, the lage.

22 apacito Poblems Slide 64 / 66 These two equations ae tue fo all capacitos. = Q U = Q 2 This equation is tue fo Paallel Plate apacitos. Unless indicated othewise, # = 1. = ##o d Some combination of these can solve all poblems elated to the oltage, hage, lectic Field and oltage of a apacito. apacito Poblems capacito is fully chaged by a battey. While the battey is connected the distance between the plates is doubled. y what facto is the enegy stoed in the capacito changed? = Q = ##o d U = Q 2 Since the battey is connected, is constant...but Q will change. Slide 65 / 66 So we need an equation fo enegy which doesn't ely on Q since it may change. ombine = Q and U = Q 2 to get: U = 1/2 2 ombine this with = ##o d to get a elationship between enegy ## and distance fo constant voltage: U = 1/2 o 2 d Fom this we see if we double the distance we halve the enegy. Slide 66 / 66

17.1 Electric Potential Energy. Equipotential Lines. PE = energy associated with an arrangement of objects that exert forces on each other

17.1 Electric Potential Energy. Equipotential Lines. PE = energy associated with an arrangement of objects that exert forces on each other Electic Potential Enegy, PE Units: Joules Electic Potential, Units: olts 17.1 Electic Potential Enegy Electic foce is a consevative foce and so we can assign an electic potential enegy (PE) to the system