Electrostatic Potential

Transcription

1 Chapte 23 Electostatic Potential PowePoint Lectues fo Univesity Physics, Twelfth Edition Hugh D. Young and Roge A. Feedman Lectues by James Pazun Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley Modified by P. Lam 7_11_2008

2 Topics fo Chapte 23 Electostatic potential enegy Electostatic potential function, V(), geneated by souce chages Connection between electostatic potential, V(), and the electostatic field, E(), geneated by souce chages Intemission Find V() and E() fo continuous chage distibutions Equipotential sufaces Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley

3 Compae gavitational potential enegy with electostatic potential enegy When two masses ae sepaated (such as a baseball and the Eath) by a distance, we say that thee is a gavitational potential enegy associated with this configuation because if we let go the masses, they will move (towad each othe); potential enegy is conveted into kinetic enegy. Similaly, if two chages ae sepaated by a distance, thee is a electostatic potential enegy associated with this configuation, if we let go the chages, they will move (away fom each othe fo like chages; towad each othe fo opposite chages); potential enegy is again conveted into kinetic enegy. m 2 + OR + m 2 q 1 + q 1 - Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley

4 Expession fo electostatic potential enegy of two point chages m OR m 2 q 1 + Gavitational potential enegy : U g = Gm 1m 2 Lowe potential enegy if the masses ae close. (potential enegy is moe negative) Masses tends to move towad lowe potential enegy if they ae fee to move. Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley q 1 - Electostatic potential enegy: U e = kq 1 (q's can be positive o negative) Fo two unlike chages, lowe potential enegy if chages ae close (they like to move close) Fo two like chages, lowe potential enegy if chages ae futhe apat (they like to move away)

5 Relating wok done by electostatic foce to change in electostatic potential enegy f W= F d f l = kq 1 d i + q 1 + OR Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley i 2 = kq 1 kq 1 f = U f U i i ( ) Wok Kinetic Enegy Theoem : ( ) K f K i = W = U f U i K f + U f = K i + U i + q 1 - (Consevation of enegy) (Electostatic foce is a consevative foce F = du d ˆ ) Example : Given : q 1 =+2C, =+1C, i = 3m m 1 =10kg,m 2 = 5kg. Let go of and it moves to infinity. Find the final kinetic enegy of. Find the final speed of. Challengequestion : Let go both q 1 and ; they both move to infinity. Find the final speeds of q 1 and.

6 Electostatic potential function q 1 Electic field concept : q 1 geneates E ( ) eveywhee at location senses E ( ) and expeiences a foce F = E ( ) Electostatic potential (V( )) concept : q 1 geneates V( ) eveywhee at location senses V( ) and the potential enegy of this inteaction is U = V( ) If q 1 is a point chage, then V( ) = kq 1 = kq 1 x 2 + y 2 + z ; 2 the oigin is chosen to be the location of q 1. Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley

7 Connection between electostatic potential and electostatic field q 1 Use point chages as example F = du E = d ˆ F = du/ ( ) d ˆ = dv d ˆ check :V ( ) = kq 1 E dv d ˆ = kq 1 2 If we expess V( ) = V(x,y,z) = then E = V ˆ i + V x ˆ j + V k ˆ y z ˆ kq 1 x 2 + y 2 + z 2 Note :V ( ) is a scala (not a vecto) so, it is easie to calculate than E ( ). Once V( ) is known, E( ) can be calculated fom V( ). Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley

8 Moe than one souce chage Potential enegy of souce chages q 1,,andq 3 U souce = kq 1 + kq 1q 3 + kq 3 (i.e. all possible pai combinations) ch ag es If a test chage q o is added, then the additional potential enegy is U souce test chage inteaction kq = q k + kq Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley 3 q o V souce That is, the electostatic potential geneated by seveal souce chages is the sum of the individual potentials.

9 Moe than one souce chage - example Given : q 1 =+12nC, = 12nC (1)Find the electostatic potential at points a, b, and c (2) Is the potential at point c zeo? Is the electic field at point c zeo? (3) A test chage q o =+1nC tavels fom point b to point c, how much wok is done by the electostatic foce? Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley

10 Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley Intemission

11 Example a line of chage Refe to Example using Figue (1) Calculate V(x,y = 0,z = 0) fo all x. (2) Fom the potential function V(x,y = 0,z = 0), find E x (x, y = 0,z = 0) (3) How would you find E at any point? Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley

12 Example Calculate V(x,y,z) fo a line of chage y dy dq = x 2 + (y y') 2 + z 2 y P=(x,y,z) x (1) Calculate V(x,y) fo all x and y V(x,y,z) = V(x,y,z) = kdq ; dq = dy' a a k dy' x 2 + (y y') 2 + z 2 Once this is calculated, then we can find E (x,y,z) by taking deivatives, E (x, y,z) = V i x ˆ + V ˆ j + V k ˆ y x You ae NOT equied to do the integals, but need to undestand the concept. Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley

13 Calculation of electical potential fom E-field If the E-field is known (say using Gauss s Law), then one find the potential fom E(). E () = - dv d ˆ V () = E (') d ' Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley

14 Example oppositely chaged paallel plates What is the electostatic potential function fo a constant E-field? Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley

15 Potential enegy vs. potential V=7volts V=3volts V=7volts A positive test chage (if let go) would move fom a egion of highe potential to a egion of lowe potential (fundamentally it moves fom high potential enegy to low potential enegy; U = qv) A negative test chage (if let go) would move fom a egion of lowe potential to a egion of highe potential (again, fundamentally it moves fom high potential enegy to low potential enegy; U = - q V) V=3volts Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley

16 Equipotential sufaces and field lines Sufaces of equal potential ae pependicula to the field lines. Spacing between equi-potentials => stength of E-field close spacing => highe E-field stength Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley

17 Field lines and a conducting suface Refe to Figue to illustate the concept of field lines nea a conducting suface. Copyight 2008 Peason Education Inc., publishing as Peason Addison-Wesley