ConcepTest: Electric Potential

Transcription

1 ConcepTest: Electric Potential Which point has the largest potential when Q > 0? E Which two points have the same potential? (a) A and C (b) B and E (c) B and D (d) C and D (e) no pair C A Smallest radius Same radius B E Q D PHY2049: Chapter 24 23

2 Multiple Charges: Superposition 3 charges: Find total potential at a point in space Q 3 +Q 1 +Q 1 r 1 x Q 3 r 3 r 2 +Q 2 +Q 2 Q1 Q2 Q Vtot = V1+ V2 + V3 = k + k + k r r r V is a scalar No directions to worry about! But you do have to watch signs! PHY2049: Chapter

3 ConcepTest: Electric Potential What is V at point A? (a) V > 0 (b) V = 0 (c) V < 0 A 60 cm B Closer to + charge 30 cm 40 cm 26 cm 26 cm Q 2 = +50μC Q 1 = 50μC What is V at point B? (a) V > 0 (b) V = 0 (c) V < 0 Equal distance to both charges PHY2049: Chapter 24 25

4 ConcepTest: Electric Potential At which point does V = 0? (a) C (b) A (c) D (d) B (e) all of the above All points equidistant from charges A B C +Q Q D PHY2049: Chapter 24 26

5 ConcepTest: Electric Potential Which configuration gives V = 0 at all points on x axis? +2μC +1μC +2μC +1μC +2μC -2μC x x x -2μC A -1μC -1μC B -2μC -1μC C +1μC (a) A (b) B (c) C (d) All of the above (e) None of the above All points on x axis equidistant from each pair of charges PHY2049: Chapter 24 27

6 ConcepTest: Fields & Potentials Find E and V at the center of the square. (a) E = 0 V = 0 (b) E = 0 V 0 (c) E 0 V 0 (d) E 0 V = 0 (e) E = V regardless of the value -Q +Q -Q +Q PHY2049: Chapter 24 28

7 ConcepTest: Electric Potential You move a positive charge Q from A to B along the path shown. What is the sign of the work done by you? (a) W AB < 0 (b) W AB = 0 (c) W AB > 0 No change in potential since distance from center is the same A B PHY2049: Chapter 24 29

8 Potential of Charge Distribution Generalize superposition to continuous distribution V tot kdq = r Distribution can be any shape Line, surface, volume Express dq in terms of charge density Line or arc: dq = λds or dq = λdx (λ = linear charge density) Surface: dq = σda (σ = surface charge density) Volume: dq = ρdv (ρ = volume charge density) Express r in terms of a problem s natural coordinates x, θ, r, PHY2049: Chapter 24 30

9 Example: Charged Ring Find V at a point z above axis of charged ring of radius R V r z V R kdq 2π k( λrdθ ) = = r z + R 2π kλr kq = = z + R z + R Q λ = 2π R dq = λrdθ 2 2 r = z + R E z For V = = z z R E kqz ( 2 2 z + R ) 3/2 z = kq PHY2049: Chapter z 2

10 Example: Charged Line Find V above midpoint of line of charge Q, length L λ = Q/ L y P 2 2 r = x + y L x dq = λdx V kdq L/2 k λdx = = r y + x ( ) L/2 2 2 = k 2 2 y + L /4 + L/2 λ ln 2 2 y + L /4 L/2 PHY2049: Chapter 24 32

11 Charged Line: Limit of L y Rationalize expression inside ln() For L y 2 ln L 2 y ( ) y + L /4 + L/2 2 2 y + L /4 + L/2 ln = ln y L /4 L/2 y + V = 2kλ ln L y PHY2049: Chapter 24 33

12 Charged Line (cont) Calculate y component of electric field at midpoint 2 2 y + L /4 + L/2 V = kλ ln 2 2 y + L /4 L/2 E y V kλl = = y y y + L 2 2 /4 Agrees with calculation in previous chapter PHY2049: Chapter 24 34

13 Example: Charged Disk Find V at a point z above axis of charged disk of radius R r z V kdq = = r σρ ρ θ ( ) R 2π k d d z + ρ ρ Q σ = π R 2 R (surface charge density) V = R k σ 2πρdρ ( ) z + ρ dq = σda = σ ρdρdθ ( ) ( ) 2 2 V = 2πkσ z + R z PHY2049: Chapter 24 35

14 Another approach: treat disk as concentric charged rings Charged Disk (cont) r z V = R k σ 2πρdρ ( ) z + ρ dq ρ R = σ da = σ ( 2πρd ρ ) ( ) 2 2 V = 2πkσ z + R z Follows from 2 A= πρ da= 2πρdρ PHY2049: Chapter 24 36

15 Charged Disk (cont) Calculate z component of electric field ( ) 2 2 V = 2πkσ z + R z E z V z = = 2πkσ 1 z z + R 2 2 When z very small Ez 2πkσ = σ 2ε 0 Just like sheet of charge PHY2049: Chapter 24 37

16 V Dipole Like charges Q V +Q x V kq = + r No equilibrium since E is never 0 kq r 1 2 Where are equilibrium points? V kq = + r kq r 1 2 Equilibrium is at x = 0, since E = 0 E is dv/dx +Q +Q x PHY2049: Chapter 24 38

17 Conductors are Equipotentials No work to move along conductor W = 0 = qδv AB V is constant in conductor But E = 0 inside surface bounded by conductor V C = V A V is constant within enclosed volume A B C PHY2049: Chapter 24 39

18 Conductors in Electrostatic Equilibrium Electric field is zero everywhere inside the conductor if E 0, then charges would move no equilibrium!! Excess charge on isolated conductor is only on surface Mutual repulsion pushes the charges apart Electric field is perpendicular to the surface of a conductor If a parallel component existed, charges would move!! For irregular shaped conductors, charge density is highest near sharp points, i.e. the field strength is greater there PHY2049: Chapter 24 40

19 Spherical Shell Q Inner radius = r 1 Outer radius = r 2 + What is charge on inner shell? Q What is charge on outer shell? +Q What is V vs radius? Constant from 0 < r < r 2 Falls as kq / r for r > r 2 PHY2049: Chapter 24 41

20 ConcepTest: Electric Energy A positively charged rod is held near a neutral conducting sphere. A positively charged particle is moved from A to B (A, B both on sphere). The mechanical work required to cause this motion is (a) positive (b) zero All points on sphere are at same potential (c) negative (d) depends on the path taken from A to B (e) cannot be determined without more information PHY2049: Chapter 24 42

21 ConcepTest: Electric Energy A positively charged rod is held near a neutral conducting sphere. A positively charged particle is moved from A to B (A is on sphere). The mechanical work required to cause this motion is (a) positive Must push against electrostatic force (b) zero (c) negative (d) depends on the path taken from A to B (e) cannot be determined without more information PHY2049: Chapter 24 43

22 ConcepTest: Electric Energy A positively charged rod is held near a neutral conducting sphere. A positively charged particle is moved from A to B (A on sphere). The electrostatic work done on the particle is (a) positive (b) zero (c) negative Electrostatic force is against direction of motion (d) depends on the path taken from A to B (e) cannot be determined without more information PHY2049: Chapter 24 44

23 ConcepTest: Electric Potential A positively charged rod is held near a neutral conducting sphere (A on sphere). The potential change from A to B is: (a) positive Higher potential near + charge (b) zero (c) negative (d) depends on the path taken from A to B (e) cannot be determined without more information PHY2049: Chapter 24 45

24 ConcepTest: Electrostatics Two charged metal spheres are connected by a copper wire. Note that r A > r B. Which quantity must be the same for both spheres? (a) potential at the surface (b) charge on the sphere (c) surface charge density (d) field at the surface (e) more than one of the above. PHY2049: Chapter 24 46

25 ConcepTest: Electrostatics Two charged metal spheres are connected by a copper wire. Note that r A > r B. Compare q A to q B (a) q A > q B Potential is same, so kq A /r A = kq B /r B (b) q A < q B (c) q A = q B (d) Need more information PHY2049: Chapter 24 47

26 ConcepTest: Electric Potential A solid spherical conductor is given a net nonzero charge. The electric potential of the conductor is (a) largest at the center. (b) largest on the surface. (c) largest somewhere between center and surface. (d) constant throughout the volume. PHY2049: Chapter 24 48

27 Review: Electric Potential -4Q +Q What charge will make the potential zero at X? x +4Q Charge =?? Charge =?? What charge will make the potential zero at X? Charge =?? What charge will make the potential zero at X? 2r x -2Q +5Q -Q r x r +3Q PHY2049: Chapter 24 49

28 PHY2049: Chapter 24 50