Electric Potential. Chapter 23. PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman

Transcription

1 Chapter 23 Electric Potential PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Lectures by Reza Khanbabaie

2 Goals for Chapter 23 Reminder about gravitational force, energy and potential: Gravitational Force Gravitational potential energy Gravitational potential (V) find g from V Electrical Force Electrical potential energy Electrical potential (V) find E from V To study and calculate electrical potential energy and electric potential To define and study examples of electric potential To trace regions of equal potential as equipotential surfaces To find the electric field from electric potential

3 Introduction Electrical potential is sometimes modeled as a river. The width of the river defines how much water will be able to flow through its banks. The concept of electric potential will simplify solving the problems like work and energy in mechanics. As gravitational potential energy depends on the height of a mass above the earth s surface, electric potential energy depends on the position of the charged particle in the electric field. The arc welder in the picture is taking advantage of a potential difference between the welding rod and material to be joined. The arc of electrical flow is so hot that the metals and the rod actually melt into one material.

4 Reminder: Work, energy, and the path of motion Work done by a force to move a particle from a to b is: The work done can be expressed as a change in potential energy U: Work done by a conservative force is path independent and reversible.

5 Work, Energy and Electric Potential Energy The total work done on the particle equal to the change in kinetic energy ΔK= K b - K a. If the force is conservative then ΔK= -ΔU = -(U b - U a ) K a + U a =K b + U b = Constant. The work done on charge q 0 by a uniform electric field: W F. d ( q 0 E)( d) q a b 0 Ed +, because force F and displacement d are in the same direction. Like gravitational potential energy F y =-mg, U g = mgy, we can introduce electric potential energy, U E =q 0 Ey for F y = -q 0 E.

6 An electric charge moving in an electric field W W a b a b ΔU a b b a F. dl ( U ( q0 Eyb q0eya ) q E(y y ) 0 q 0 b U E(y U U increases if +q 0 moves in the opposite direction of E, and U decreases if +q 0 moves in the same direction of E. a b a ) b y a ) U decreases if -q 0 moves in the opposite direction of E, and U increases if -q 0 moves in the same direction of E.

7 Example-I: The work done on a test charge by the field of a point charge The concept of electric potential energy can be applied to any electric field (like a point charge s field) not just uniform E. A test charge will move along a straight line extending radially with respect to a like charge q: F r qq r 0 2 Coulomb s law depends only to the end points

8 The work done on a moving test charge-ii What if the test charge does not move on a straight line? But dl cos dr, so the work is dependent only to the radial component of the displacement. 1. The work W done on the q 0 by E depends only on r a and r b. (path independent). 2. If q 0 returns to its starting point a by a different path, W total = 0 (reversible). So the electric force is conservative.

9 Potential energy curves (vs. r) For a conservative force we define electrical potential energy: The reference point is at infinity: At r, U 0. Therefore U represents the work done on the test charge q 0 by the field of q if q 0 moved from r to. Graphically, the potential energy between like charges increases sharply to positive (repulsive) values as the charges become close. For unlike charges the potential energy will be sharply negative as they become close (attractive).

10 Electrical potential and multiple point charges The potential energy of a charge among other charges is the algebraic sum of individual potential energies: So the total potential energy of the collection is: i j & i<j to avoid self interaction and counting the pairs twice. Figure to the right shows this principle is applied to an ion engine for spaceflight. It uses electric forces to eject a stream of Xe + at speeds over 30 km/s.

11 Electric Potential Electric Potential is potential energy per unit charge: U V or U q0v q 0 The SI unit for Electrical Potential is : VOLT (1V= 1 J/C). Potential of a with respect to b, equals the work done by the electric force when a UNIT charge moves from a to b. Potential of a with respect to b, equals the work that must be done to move a UNIT charge slowly from b to a against the electric force. W q U U a b b a ( ) ( Vb Va q q q ) U V The voltage of a battery equals the difference in potential V ab =V a -V b between point a (+) and point b (-). a V b V Voltmeter measures the difference of potential between two points. ab

12 Calculating Electric Potential r p Calculating electric potential from electric field: moving in the direction of E decreasing V moving against the direction of E increasing V

13 Electron-volt and particle accelerator Electron Volt ( a unit of energy): if a charge q = e moves from a point with V b to a point with V a, the change in potential energy is: U a -U b =e (V a -V b )=ev ab if V ab =1 V, then U a -U b =1 ev= J. A particle accelerator can bring a charged particle to motion at velocities great enough to impart millions, even billions, of ev as kinetic energy. Figure to the right shows a particle accelerator at the Fermi Lab in Illinois.

14 Example-II: Electric force and electric potential

15 Example-III: Potential due to two charges

16 Example III: Finding the potential by integration

17 Example IV: Moving through a potential difference

18 Calculating Electric Potential Strategies

19 Example V: Calculation of electrical potential Let us take V=0 at infinity. The potential outside at a point outside the sphere at a distance r from center is the same as a point charge at the center: V 1 4 q r V surface q R

20 Ionization and Corona Discharge The max potential that to which a conductor can be raised before air becomes a conductor (ionized) is limited. Dielectric strength of air: the max electric field E m at which air becomes conductor. V m =RE m, R= radius of the conducting sphere. Corona: For more than V m the surrounding air will become ionized and a current will go to the air. The resulting current and its associated glow are called corona. The radius of the metal at the top of lightning rod must be big enough to hold a substantial amount of opposite charge with respect the excess charge of the air (as happens during thunderstorm). When the rod discharge them to earth it will attract more charge from the air before the excess charge harm the other buildings.

21 Example VI: 0ppositely charged parallel plates

22 Example VII: a charged, conducting cylinder (line) Find potential at r from a linear charge density λ: So we arbitrarily set V b =0 at a distance r 0 : V V 0 ( / 2 )ln( r 2 0 r ln( r 0 ) 0 0 / r)

23 Example VIII: a ring of charge

24 Example IX: a line of charge

25 Equipotential surfaces and field lines (like countor lones in gravitation) Equipotential surfaces are surfaces of equal potential V. If a charge q is moved on such a surface, the electric potential energy U=qV remains constant. They never intersect or touch. The bigger magnitude of E, the closer the equipotential surfaces. Because U is constant, E can not do any work on q. So E must be perpendicular to the displacement of a charge moving on the surface. Therefore field lines and equipotential surfaces are always mutually perpendicular. 90 o

26 Equipotential surfaces of a uniform field What are the equipotential surfaces of a capacitor? Remember: E is not the same on equipotential surfaces.

27 Field lines and a conducting surface When all charges are at rest, the surface of a conductor is always an equipotential surface. To prove above: we prove that when all charges are at rest, E just outside of a conductor is perpendicular to the surface at any point (E = 0).

28 The surface and interior of a conductor Every point in the cavity, with no charge, is at the same potential. Surface A in the figure is an equipotential. Suppose point p in the cavity is at a different potential, then we can construct a different equipotential surface B including p. We can construct a Gaussian surface between two equipotential surfaces. So there will be an E from B toward A and the flux through the Gaussian surface can not be zero and by Gauss s law the charged enclosed cannot be zero. This contradicts our assumption. So the potential at p is the same as cavity wall. So the entire cavity is at the same potential. For this to be true, E must be zero everywhere in the cavity. Gauss s law: σ = 0. Theorem: In an electrostatic situation, if a conductor contains a cavity with no q in it, then there can be no net charge anywhere on the surface of the cavity.

29 Potential Gradient: Finding E from V The relation between potential and electric field is:, and we can write: Therefore: and or Point charge:

30 Example I: Potential and field of a point charge

31 Example I: Potential and field of a ring of charge