Physics 2112 Unit 6: Electric Potential

Transcription

1 Physics 2112 Unit 6: Electric Potential Today s Concept: Electric Potential (Defined in terms of Path Integral of Electric Field) Unit 6, Slide 1

2 Stuff you asked about: I am very confused about the integrals done to find the potential at different points on insulator. "Electric Field Lines" Checkpoint questions 2 and 3. Also confused with the equation for Charged Spherical Insulator, r<a. Can you show work how to get that equation Explanation on gradient and applications. Can't wait for lecture. Still having trouble wrapping my head around what electrical potential is as a concept. The guy in the prlecture got 10x more exciting when he yelled that scripted WOW! need to go over the first and last checkpoint..the hill of electric potential confused me, the potential energy gets higher the closer you get to the charge? I think I fully understood this pre lecture. :) the addition of multiple potential was a bit confusing Didn't seem too bad. But im sure it'll get tough when we start doing calculations. none Unit 5, Slide 2

3 Big Idea Last time we defined the electric potential energy of charge q in an electric field: b U a b F dl qe The only mention of the particle was through its charge q. a b a dl We can obtain a new quantity, the electric potential, which is a PROPERTY OF THE SPACE, as the potential energy per unit charge. V ab U q ab b E dl Note the similarity to the definition of another quantity which is also a PROPERTY OF THE SPACE, the electric field. E F q a Unit 6, Slide 3

4 We never talk about the gravitational potential in We just talked about a gravitational force. Let s say we had. What is the magnitude of the gravitational potential close to the surface of the earth? A) g B) mgh C) gh D) mg Electricity & Magnetism Lecture 2, Slide 4

5 Example 6.2a (Potential from Field) 2 1 E q=+8uc = 6 N/C A +8uC charge is placed in a downwards pointing electric field with a magnitude of 6N/C. An outside force moves the charge up a distance of 0.4m from point 1 to point 2. a) What is the force on this charge? b) How much work was done by the outside force during this move? c) How much work was done by the electric field during this move? d) What is the change in the electrical potential energy of the particle? e) What is the difference in electrical potential between points 1 and 2?

6 Question In the example just worked with a positive charge q and the field pointing down, we found that a) the force on the charge was -4.8 X 10-5 N q b) the work done moving the charge up was 1.92 X 10-5 J c) the change in electrical potential energy (EPE) was X 10-5 J d) the change in the electrical potential was +2.4V 2 1 E + + when we moved the charge upwards 0.4m. Which of these values would not change if I redid the problem, switching q to a negative (-8mC) charge? A. (a) B. (b) C. (c) D. ( d) E. All of them would change

7 Question In the example just worked with a positive charge q and the field pointing down, we found that a) the force on the charge was -4.8 X 10-5 N q b) the work done moving the charge up was 1.92 X 10-5 J c) the change in electrical potential energy (EPE) was X 10-5 J d) the change in the electrical potential was +2.4V 2 1 E + + when we moved the charge upwards 0.4m. Which of these values would not change if we went back to the original +8mC charge, but now the field pointed up? A. (a) B. (b) C. (c) D. ( d) E. All of them would change

8 Question In the example just worked with a positive charge q and the field pointing down, we found that a) the force on the charge was -4.8 X 10-5 N q b) the work done moving the charge up was 1.92 X 10-5 J c) the change in electrical potential energy (EPE) was X 10-5 J d) the change in the electrical potential was +2.4V 2 1 E + + when we moved the charge upwards 0.4m. What it the electrical potential of the original point 1? A. -2.4V B. 0V C. 4.8V D. -4.8V E. We can t tell.

9 Electric Potential from E field Consider the three points A, B, and C located in a region of constant electric field as shown. D Dx What is the sign of V AC V C V A? A) V AC < 0 B) V AC 0 C) V AC > 0 Choose a path (any will do!) D C V E dl A C A D E dl V AC C 0 E dl Ex < 0 D Unit 6, Slide 9

10 CheckPoint: Zero Electric Field Suppose the electric field is zero in a certain region of space. Which of the following statements best describes the electric potential in this region? A. The electric potential is zero everywhere in this region. B. The electric potential is zero at least one point in this region. C. The electric potential is constant everywhere in this region. D. There is not enough information given to distinguish which of the above answers is correct. Remember the definition B VA B E dl A Electricity & Magnetism Lecture 6, Slide 10

11 What we said Suppose the electric field is zero in a certain region of space. Which of the following statements best describes the electric potential in this region? A. The electric potential is zero everywhere in this region. B. The electric potential is zero at least one point in this region. C. The electric potential is constant everywhere in this region. D. There is not enough information given to distinguish which of the above answers is correct. A) the Electric Potential is calculated as the Integral of the E-field, and if the E-field is 0, the area under the curve is also 0 C) The Electric field is the derivative of Electric Potential, thus if the V is constant the derivative is zero E 0 V 0 AB V is constant! Electricity & Magnetism Lecture 6, Slide 11

12 Example 6.2 (V from point charges) +5nc A B x +Q Q 10cm 10cm 10cm -5nc What is the electrical potential at points A and B? Define V = 0 at r = (standard) Unit 6, Slide 12

13 Example 6.3 (V from point charges) C +5nc 10cm -5nc +Q Q x 10cm 20cm What is the electrical potential at point C? Define V = 0 at r = (standard) Unit 6, Slide 13

14 E from V If we can get the potential by integrating the electric field: b Va b a E dl We should be able to get the electric field by differentiating the potential? In Cartesian coordinates: V E x dx V E y dy V E z dz E V Electricity & Magnetism Lecture 6, Slide 14

15 Example 6.4 (E-field above a ring of charge) y x P y What is the electrical potential, V, a distance y above the center of ring of uniform charge Q and radius a? (Assume V = 0 at y = ) a What is the electrical field, E, at that point? Unit 2, Slide 15

16 V = k dq/r Q 2πa 0 l ds/(y 2 +a 2 ) 0.5 =kq /(y 2 +a 2 ) 0.5 Recall E y = kq y/(y 2 +a 2 ) 3/2 DV/Dy = Also E x = 0 why? Electricity & Magnetism Lecture 2, Slide 16

17 A solid insulating sphere of radius a = 5.9 cm is fixed at the origin of a co-ordinate system as shown. The sphere is uniformly charged with a charge density ρ = -395 μc/m 3. Concentric with the sphere is an charged spherical conducting shell of inner radius b = 12.6 cm, and outer radius c = 14.6 cm. The conducting shell has a outer surface density of s c = +1.1uC/m 2 and a inner surface charge density of s b = -1.2uC/m 2. What is V(c), the electric potential at the outer surface of the conducting shell? Define the potential to be zero at infinity. What is V(b), the electric potential at the inner surface of the conducting shell? Define the potential to be zero at infinity. What is V(a), the electric potential at the outer surface of the insulating charge? Define the potential to be zero at infinity. What is V(P), the electric potential at a point 20cm from the center of the charge? Define the potential to be zero at the outer surface of the conducting shell. Unit 6, Slide 17

18 CheckPoint: Spatial Dependence of Potential 1 The electric potential in a certain region is plotted in the following graph At which point is the magnitude of the E-FIELD greatest? A. A B. B C. C D. D Electricity & Magnetism Lecture 6, Slide 18

19 CheckPoint: What we thought. The electric potential in a certain region is plotted in the following graph At which point is the magnitude of the E-FIELD greatest? A. A B. B C. C D. D A) The magnitude of V corresponds with the magnitude of the E-field. B) since the potential is changing the most at point b the electric field will be the greatest at that point How do we get E from V? E V V E x dx Look at slopes! Unit 6, Slide 19

20 CheckPoint: Spatial Dependence of Potential 2 The electric potential in a certain region is plotted in the following graph At which point is the direction of the E-field along the negative x-axis? A. A B. B C. C D. D Electricity & Magnetism Lecture 6, Slide 20

21 CheckPoint: What we thought The electric potential in a certain region is plotted in the following graph At which point is the direction of the E- field along the negative x-axis? A. A B. B C. C D. D B) The potential is decreasing at point B, so the change in potential is negative and therefore the E-field is negative and therefore moving along the negative x axis. C) The E-field is negative dv/dx. Since dv/dx is positive only at point C, this is where E is negative. How do we get E from V? E V V E x dx Look at slopes! Electricity & Magnetism Lecture 6, Slide 21

22 WARNING!!!! We re about to do some integrals. Good news: They re going to be really, really simple. Sometimes so simple we can do them conceptually. Unit 6, Slide 22

23 Example 6.5 (V near line of charge) An infinitely long solid insulating cylinder of radius a = 4.1 cm is positioned with its symmetry axis along the z-axis as shown. The cylinder is uniformly charged with a charge density ρ = 27.0 μc/m 3. Concentric with the cylinder is a cylindrical conducting shell of inner radius b = 19.9 cm, and outer radius c = 21.9 cm. The conducting shell has a linear charge density λ = -0.36μC/m. What is V(P) V(R), the potential difference between points P and R? Point P is located at (x,y) = (46.0 cm, 46.0 cm). Point R is located at (x,y) = (0 cm, 46.0 cm). C Unit 2, Slide 23

24 a 4 3 a 2 a 1 cross-section Example 6.6 (V for charges) +Q Point charge q at center of concentric conducting spherical shells of radii a 1, a 2, a 3, and a 4. The inner shell is uncharged, but the outer shell carries charge Q. +q metal What is V as a function of r? metal Plan: Two Step process: - Step #1: Use Gauss Law to calculate E everywhere - Step #2: Integrate E to get V surface E da Q enc o B VA B A E dl Electricity & Magnetism Lecture 6, Slide 24

25 cross-section Question a 4 3 +Q a 2 a 1 Why? Gauss law: +q metal metal E da E4 r 2 r Q enclosed 0 Q + q 0 r > a 4 : What is E(r)? 1 Q A) 0 B) 2 C) 4 ( r a 0 4) 1 Q q D) 2 4 r E) Q + q 2 0 ( r a4) 1 Q + q 2 4 r Electricity & Magnetism Lecture 6, Slide 25 0

26 Calculation: Quantitative Analysis cross-section a 4 3 a 2 a 1 +Q a 4 < r : 1 Q + q 2 4 r 0 +q metal metal r a 3 < r < a 4 : E 0 a 2 < r < a 3 : a 1 < r < a 2 : E 0 1 q 4 r r < a 1 : E 2 0 Electricity & Magnetism Lecture 6, Slide 26

27 cross-section Question a 4 3 +Q a 2 a 1 What is V for a 3 < r < a 4? +q metal metal r A) B) C) V 0 V V Q + q a 4 Q + q a 3 V ( a ) D) V Q + r q Electricity & Magnetism Lecture 6, Slide 27

28 cross-section Question a 4 3 +Q a 2 a 1 +q metal r metal V Electricity & Magnetism Lecture 6, Slide 28

29 Example 6.7 (Insulator) insulator +Q A spherical insulator has a total charge +Q and a radius of a. a What is V as a function of r? Plan: Two Step process: - Step #1: Use Gauss Law to calculate E everywhere - Step #2: Integrate E to get V surface E da Q enc o B VA B A E dl Electricity & Magnetism Lecture 6, Slide 29

30 +Q insulator a E r

31 Equipotentials In previous example, all these points had same V, same electrical potential Line is called equipotenial or a line of equipotential. Unit 2, Slide 31

32 Topographic Map Lines on topo map are lines of equal gravitational potential Closer the lines are, the steeper the hill Gravity does no work when you walk along a brown line.

33 Equipotential for Point Charge Equipotentials produced by a point charge V 0 W 0 E dl 0 elec Equipotentials always perpendicular to field lines. SPACING of the equipotentials indicates STRENGTH of the E field. Electricity & Magnetism Lecture 6, Slide 33

34 Question Which of the equipotential diagrams shown to the left best describes the spherical insulator in the described below? (The colored circle in the center represents the insulator.) Unit 6, Slide 34

35 CheckPoint: Electric Field Lines 1 The field-line representation of the E-field in a certain region in space is shown below. The dashed lines represent equipotential lines. At which point in space is the E-field the weakest? A. A B. B C. C D. D Electricity & Magnetism Lecture 6, Slide 35

36 CheckPoint: Electric Field Lines 2 The field-line representation of the E-field in a certain region in space is shown below. The dashed lines represent equipotential lines. Compare the work done moving a negative charge from A to B and from C to D. Which one requires more work? A. More work is required to move a negative charge from A to B than from C to D B. More work is required to move a negative charge from C to D than from A to B C. The same amount of work is required to move a negative charge from A to B as to move it from C to D D. Cannot determine without performing the calculation Electricity & Magnetism Lecture 6, Slide 36

37 What we thought.(2) The field-line representation of the E-field in a certain region in space is shown below. The dashed lines represent equipotential lines. Compare the work done moving a negative charge from A to B and from C to D. Which one requires more work? A. More work is required to move a negative charge from A to B than from C to D B. More work is required to move a negative charge from C to D than from A to B C. The same amount of work is required to move a negative charge from A to B as to move it from C to D D. Cannot determine without performing the calculation A. There is more work required moving from A to B because of the density of field lines in that area. The thicker density indicates a stronger E-field which would require more work to move the charge.. B. The charge is moving to the same potential but the charge is traveling over a greater distance from C to D than with A to B. C. A and C are on the same equipotential line, and B and D are on the same equipotential line. The work to move a negative charge will be the same whether its from A to B or C to D. Unit 6, Slide 37

38 CheckPoint: Electric Field Lines 3 The field-line representation of the E-field in a certain region in space is shown below. The dashed lines represent equipotential lines. Compare the work done moving a negative charge from A to B and from A to D. Which one requires more work? A. More work is required to move a negative charge from A to B than from A to D B. More work is required to move a negative charge from A to D than from A to B C. The same amount of work is required to move a negative charge from A to B as to move it from A to D D. Cannot determine without performing the calculation Electricity & Magnetism Lecture 6, Slide 38

39 What we thought..(3) The field-line representation of the E-field in a certain region in space is shown below. The dashed lines represent equipotential lines. Compare the work done moving a negative charge from A to B and from A to D. Which one requires more work? A. More work is required to move a negative charge from A to B than from A to D B. More work is required to move a negative charge from A to D than from A to B C. The same amount of work is required to move a negative charge from A to B as to move it from A to D D. Cannot determine without performing the calculation A. Moving the charge from A to D would be more "distance" and therefore more work. B. B and D both lie on the same equipotential line. Therefore, whether moving from A to B or A to D, each has the same change in potential. Therefore, the amount of work for each will be the same. Electricity & Magnetism Lecture 6, Slide 39

40 Question The diagram to the right shows equally space equipotential lines 1V apart. Where on the above diagram is the electric field the strongest? 10V 12V 14V A. On the left hand side B. On the right hand side C. In the middle D. It s constant throughout E. We can t tell about the electric field from the equipotential lines.

41 Question An electric field is represented by the equipotential surface lines shown to the right. If a positive charge is place in this field, how will it move? -100V 0V A. Constant velocity to the left B. Constant velocity to the right C. Constant acceleration to the left D. Constant acceleration to the right E. None of the above (Each line represent a change of 20V.)

42 Question Which of the below drawings of possible equipotential surfaces could represent the electrical field lines shown to the right? E A. (a) B. (b) C. (c) (b) (a) D. (a) or (c) depending on the sign of the test charge (c)

43 Question Which of the below drawings of possible equipotential surfaces could represent the electrical field lines shown to the right? E 50V 10V -10V 10V 50V -50V A. (a) B. (b) C. (c) D. (a) or (c) E. (a) or (b) (a) (b) (c)