19.1 Potential Energy

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Harvey Beasley
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1 Chapte 19: Electic Potential Enegy & Electic Potential Why electic field contains enegy? Is thee an altenative way to undestand electic field? Concepts: Wok done by consevative foce Electic potential enegy Electic potential ti

2 19.1 Potential Enegy The electic foce, like gavity, is a consevative foce: Recall Consevative Foces 1. The wok done on an object by a consevative foce depends only on the object s initial and final position, and not the path taken.. The net wok done by a consevative foce in moving an object aound a closed path is zeo. Let s place a positive point chage q in a unifom electic field and let it move fom point to (no gavity): How much wok is done by the field in moving the chage fom to? y q o * Remembe, W F d xd,wheef d is the component of the constant foce along the diection of the motion. E Hee, F qe, so W qe( y y ) f o qeδy y f Electostatic Potential Enegy (EPE)

3 Thus, W ΔEPE The wok done is equal to the change in electostatic potential enegy! Now let s divide both sides by the chage, q: W ΔEPE q chage. We call this the Electic Potential, V: q Enegy Chage The quantity on the ight is the potential enegy pe unit J C EPE V q Units? [ Volt] [ V ] The electic potential is a scala! Review of Wok: 1. Wok is not a vecto, but it can be eithe positive o negative: Positive Foce is in the same diection as the motion Negative Foce is in the opposite diection as the motion. If positive wok is done on an object, the object speeds up. 3. If negative wok is done on an object, the object slows down.

4 19. Electic Potential Diffeence We can talk about the value of the potential at diffeent points in space: Fo example, what is the diffeence in electostatic potential between two points, and, in an electic field??? q V V E So, Let s say the chage at point is positive: EPE q EPE q Δ V V V W q W q Why is thee a minus sign??? If I elease it, which way will it move? It moves down towad! Since the foce is down and the motion is down, positive wok is done on the chage. Thus, W is positive. This means that (V V ) is negative, o V > V. We say that point is at a highe potential than point.

5 Positive chages, stating fom est, will acceleate fom egions of high potential and move towad egions of low potential. Negative chages, stating fom est, will acceleate fom egions of low potential and move towad egions of high potential. One common object associated with voltages is a battey: Wies Light bulb 1.5 V 1.5 J The battey supplies 1.5 Joules of enegy fo evey coulomb of chage. C 1.5 Volts Notice that the positive chage moves fom attey highe potential () to lowe potential (). The wod volt also appeas in a unit of enegy: Let s acceleate an electon fom est though a potential diffeence of 1 Volt:

6 1 Volt attey Hole The electon gets acceleated fom low potential to high potential It gains kinetic enegy. The enegy gained by an electon when acceleated though a potential diffeence of 1 Volt 1 electon volt 1 ev. **If I acceleated an electon fom est though a potential diffeence of 50, V, then I know immediately that its kinetic enegy is 50, ev. Enegy is usually expessed in Joules: 1 ev J Just like in a gavitational field, in an electic field, potential enegy (PE) can be conveted into kinetic enegy (KE): Example: Let s bing a small positive test chage fom vey fa away in towad a fixed, positive point chage: s I push the chage in close and close, the epulsive foce on it gets bigge and bigge: *Thus, I have to do wok on the chage to move it close. F The wok I do on the chage goes into inceasing its potential enegy!

7 Now elease the chage.. F The chage convets its stoed EPE into KE!!! Remembe: The total mechanical enegy of a system must be conseved E Tot mv Iω mgh kx ot EPE KE Tans KE Rot PE Gav PE Elastic PE Electo nd,.if the wok done by nonconsevative foces is zeo, then: E E Tot f Tot o

8 19.3 The Electic Potential of a Point Chage n electic potential exists aound chages. What is the fom of the potential fo a point chage? Let s place a positive test chage nea a positive fixed point chage: Q q F The electic field ceated by the point chage does wok on the test chage and moves it to the ight What is the wok done by the field in moving the chage fom to? qq F k Well, W F d, and the foce is given by Coulomb s Law: ut, the foce is not constant as the chage moves fom to, since the foce depends d on. Thus, we have to use calculus and integate the foce ove the distance: Result: W k qq k qq

9 Fom ealie, we know that: V V W q k Q k Q Q If we let be eally fa away, i.e., then V k 0. *This sets ou zeo potential ti at tinfinity. it Q V k Thus, we ae left with:. is just some abitay distance fom the point chage, so we dop the subscipt: V k Q This is the electic potential due to a point chage. We now have 3 equations which kind of look simila: qq Q F k E k V k Q Foce between two chages. Electic field of a point chage. Electic potential of a point chage.

10 19.4 Equipotential Sufaces So the potential fom a point chage is E V k Q S S This means the potential is the same in evey diection aound the point chage at a distance away. In 3D, this foms a spheical shell of adius aound the chage. Thus, the electic potential is the same eveywhee on this spheical suface (S ). It is called an equipotential suface. Equipotential sufaces ae sufaces of constant potential. Let s look at anothe equipotential suface (S ) aound the point chage: We know the electic field lines point eveywhee adially outwad: Notice: The electic field lines ae pependicula to the equipotential sufaces.

11 E Since S is close to the positive chage than S, S is at a highe potential than S. S S Thus, electic field lines point in the diection of deceasing potential, i.e. they point fom high potential to low potential. Wok? The net electic foce does no wok as a chage moves on an equipotential suface. Why? W We defined V V. q ut, if we ae on an equipotential suface, then V V, and W 0. O. In ode fo the chage to feel a foce along an equipotential suface thee must be a In ode fo the chage to feel a foce along an equipotential suface, thee must be a component of the field along the suface, but E is eveywhee pependicula to the equipotential suface.

12 Fields, Potentials, and Motion of Chages Summay Electic field lines stat on positive chages and end on negative ones. Positive chages acceleate fom egions of high potential towad low potential. Negative chages acceleate fom egions of low potential towad high potential. Equipotential sufaces ae sufaces of constant potential. Electic field lines ae pependicula to an equipotential suface. Electic field lines ae pependicula to the suface of a conducto, thus a conducto is an equipotential suface! Electic field lines point fom egions of high potential towad low potential. Theefoe, positive chages move in the same diection as the electic field points, and negative chages move in the opposite diection of the electic field. The electic foce does no wok as a chage moves on an equipotential suface.

13 Clicke Question 194 Which side of space (left o ight) is at a lowe potential? 1. Left side. Right side 51% The electic field points fom left to ight, and electic field lines stat fom egions of high potential, thus the ight side is at a lowe poential. 49% Left side Right side

14 Paallel plate capacito Δs E 9 V 6 V 3 V 0 V The positive plate is at a potential of 9 V and the negative plate is at 0 V. What would the equipotential ti sufaces look like between the plates? They would be a paallel set of planes! Let the plates be sepaated by a distance Δs. [ Change in voltage] ΔV The electic field is then E [ Change in distance ] Δ s This is called the electic field gadient. Thus, the electic field also has units of [ V / m ]

15 19.5 Capacitos Two oppositely chaged conductos sepaated by some small distance. Volts We can chage the plates by connecting them to a battey: The highe the voltage on ou battey, the moe chage we can put on each plate. Q Q Thus, Q V Make this an equality: Q CV C is a new quantity called the Capacitance. Units? Q [ Chage] C V [ Voltage] C V [ Faad] [ F] * faad is a vey lage capacitance. We often use micofaads (μf) and picofaads (pf) F F The lage the capacitance, the moe chage it will hold!

16 Dielectics We can fill the space between the plates with some insulating mateial, say ai, oil, pape, ubbe, plastic, etc. Dielectic This mateial is called a dielectic. E E o So what effect does the dielectic have on the field between the plates? Since the dielectic is an insulato, the chages in it aen t fee to move, but they can sepaate slightly within each atom: Each one of these atoms now poduces a small intenal electic field which points in the opposite diection to the field between the plates: Thus, the net electic field between the plates is educed by the dielectic. The eduction of the field is epesented by the following: E o is the field without the dielectic E is the field with the dielectic κ E o E κ is called the dielectic constant, and it must be geate than 1.

17 κ E o E Since κ is the atio of two electic fields, it s unitless. Mateial κ Vacuum 1 i Wate 80.4 The lage κ is, the moe it educes the field between the plates! E o κ d Let s say the plates have suface aea and ae sepaated by a distance d. V E 1 Eo d ε o q κ V d E κv d σ q ε o ε κ o ut, q CV, o so ε o κ C d

18 Capacitos stoe chage what about enegy? EPE S d EPE Stoed 1 1 qv CV V Ed and C ε ( ) oκ 1 ε oκ, so EPEStoed E d d d Reaange this: EPE 1 1 κε E ( d) κε E o o (Vol) Volume between the plates EPE Vol 1 Enegy Density κε o E Units? Enegy Volume J m 3 *This expession holds tue fo any electic fields, not just fo capacitos!

The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C

The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C Geneal Physics - PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit