E From Hollow and Solid Spheres

Transcription

1 E From Hollow and Solid Spheres PHYS David Blasing Wednesday June 19th 1 / 32

2 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions Note: What we are doing today is only an approximation when you get very close to the surface of an object. 2 / 32

3 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions Note: What we are doing today is only an approximation when you get very close to the surface of an object. Effects from precisely how the individual charges have arranged themselves (each electron s current position) become important when you are less than about 1000 atomic diameters (< 10 7 m) away from the surface. 2 / 32

4 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions Note: What we are doing today is only an approximation when you get very close to the surface of an object. Effects from precisely how the individual charges have arranged themselves (each electron s current position) become important when you are less than about 1000 atomic diameters (< 10 7 m) away from the surface. Outside this range, the charges can be treated as uniform sheets of charge rather than individual charges. This is how we are treating them from here on out. 2 / 32

5 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions Note on Gauss s Law 1 Gauss s law can get E net from certain highly symmetric charge distributions very easily. 3 / 32

6 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions Note on Gauss s Law 1 Gauss s law can get E net from certain highly symmetric charge distributions very easily. 2 Gauss s law is a topic covered later, and we will be able to prove the following then. 3 / 32

7 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions Note on Gauss s Law 1 Gauss s law can get E net from certain highly symmetric charge distributions very easily. 2 Gauss s law is a topic covered later, and we will be able to prove the following then. 3 It can also be done through direct integration, but its messy. For the sake of time I won t present the derivation, but you should go through it on your own. 3 / 32

8 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions E From Hollow Sphere Q is the total charge on the sphere s surface, R is the radius of the sphere For r > R, E sphere = 1 4πɛ 0 Q r 2 ˆr For r < R, E sphere = 0 4 / 32

9 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions A spherical shell of charge Note: a uniformly charged spherical shell (as long as its stays uniformly charged) interacts with external fields like a point charge The shell must be spherically symmetric and thin enough as to neglect any effects arising from this thickness 5 / 32

10 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions A spherical shell of charge Note: a uniformly charged spherical shell (as long as its stays uniformly charged) interacts with external fields like a point charge The shell must be spherically symmetric and thin enough as to neglect any effects arising from this thickness It does not matter what the spherical shell is made of It only matters that the charge is distributed in a thin, spherically symmetric shell 5 / 32

11 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions A spherical shell of charge A qualitative drawing showing that E net sums to that of a point charge outside the spherical shell of charge: 6 / 32

12 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions A spherical shell of charge A qualitative drawing showing that E net sums to 0 inside a uniform spherical shell of charge: 7 / 32

13 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions A spherical shell of charge Group question: you have a spherical shell that is uniformly charged on only its surface and filled with a plastic. What is the polarization of the molecules in the plastic inside of the charged shell? Why? Is P different very close to the charged surface of the sphere as compared to the center? Why? 8 / 32

14 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions A spherical shell of charge Answer: Recall that P = α E. So no E, no P. It doesn t matter at all if there are charges around, it only matters what the local E net is at the location of the plastic/dielectric material. Since the E net = 0 inside a uniformly charged spherical shell, P = 0 every in the plastic Even if the molecules are very close to the charged surface, E net is still zero. There will be no polarization of those molecules. 9 / 32

15 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions Summary: E from a thin uniformly charged shell: for r < R, E = 0 for r > R, E = 1 Q 4πɛ 0 r 2 ˆr Outside, E is exactly that of a point charge with the total charge (Q) located at the center 10 / 32

16 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions A spherical shell of charge 1 E of Hollow Shell vs Distance Q R ) 2 E 1 (Units of 4π ǫ Radial Distance (Units of R) 11 / 32

17 Clicker Question 1 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions 12 / 32

18 Clicker Question 2 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions 13 / 32

19 Clicker Question 3 Validity Note on Gauss s Law E From Hollow Sphere Clicker Questions 14 / 32

20 A Vague Group Question Discuss in small group the following question, why can t E = 1 Q 4πɛ 0 ˆr be the whole story for the E field from a point charge? r 2 15 / 32

21 A Vague Group Question Discuss in small group the following question, why can t E = 1 Q 4πɛ 0 ˆr be the whole story for the E field from a point charge? r 2 It is undefined at r=0. E can be made arbitrarily big by making r arbitrarily small. Fields take energy to set up, so that formula says the field from point charges carry an unlimited amount of energy. 15 / 32

22 A Vague Group Question Discuss in small group the following question, why can t E = 1 Q 4πɛ 0 ˆr be the whole story for the E field from a point charge? r 2 It is undefined at r=0. E can be made arbitrarily big by making r arbitrarily small. Fields take energy to set up, so that formula says the field from point charges carry an unlimited amount of energy. When specifically do you think this formula breaks down? 15 / 32

23 A Vague Group Question Discuss in small group the following question, why can t E = 1 Q 4πɛ 0 ˆr be the whole story for the E field from a point charge? r 2 It is undefined at r=0. E can be made arbitrarily big by making r arbitrarily small. Fields take energy to set up, so that formula says the field from point charges carry an unlimited amount of energy. When specifically do you think this formula breaks down? This formula is not accurate when the distance becomes smaller than the radius of the point charge. Next, we are going to write down a better approximation for the electric field inside a point charge. 15 / 32

24 E from of a solid sphere (radius R, total charge Q) charged throughout its volume: Model the sphere as a series of concentric spherical shells At a location outside the sphere, we are also outside of all spherical shells Each spherical shell creates an electric field like a point charge would at the center of the shell Question: what is the electric field outside a charged solid sphere? 16 / 32

25 Electric field of a solid sphere charged throughout its volume, for r > R: E = 1 4πɛ 0 Q r 2 17 / 32

26 Electric field of a solid sphere charged throughout its volume: Now r < R, same model as above At a location inside the sphere we are inside some of the spherical shells - they contribute nothing to E net At a location inside the sphere we are outside some of the spherical shells - they contribute to E net 18 / 32

27 For r < R 19 / 32

28 For r < R 20 / 32

29 For r < R 21 / 32

30 For r < R 22 / 32

31 For r < R 23 / 32

32 For r < R 24 / 32

33 For r < R 25 / 32

34 For r < R 26 / 32

35 Summary: E from a solid sphere charged throughout its volume: for r < R, E = 1 Q 4πɛ 0 R 3 r ˆr for r > R, E = 1 Q 4πɛ 0 r 2 ˆr Inside E grows linearly; outside it falls 1 r 2 E of a point charge. and is exactly like the 27 / 32

36 1 E of Solid Sphere vs Distance Q R ) 2 E 1 (Units of 4π ǫ Radial Distance (Units of R) 28 / 32

37 E of Solid Sphere vs Distance Sanity Check: E = 0 at r=0 Q R 2) 1 E (Units of 4π ǫ Radial Distance (Units of R) 29 / 32

38 E of Solid Sphere vs Distance Sanity Check: E = 0 at r=0 Q R 2) 1 E (Units of 4π ǫ Radial Distance (Units of R) E grows as we enclose more charge for r < R 29 / 32

39 E of Solid Sphere vs Distance Sanity Check: E = 0 at r=0 Q R 2) 1 E (Units of 4π ǫ Radial Distance (Units of R) E grows as we enclose more charge for r < R E = E Point Charge for r > R 29 / 32

40 1 E of Solid Sphere vs Distance 1 E of Hollow Shell vs Distance Q R 2) 0.7 Q R 2) 0.7 E 1 (Units of 4π ǫ E 1 (Units of 4π ǫ Radial Distance (Units of R) Radial Distance (Units of R) Inside they are different, but outside both are exactly the same as a point charge 30 / 32

41 1 E of Solid Sphere vs Distance 1 E of Hollow Shell vs Distance Q R 2) 0.7 Q R 2) 0.7 E 1 (Units of 4π ǫ E 1 (Units of 4π ǫ Radial Distance (Units of R) Radial Distance (Units of R) Inside they are different, but outside both are exactly the same as a point charge So the sanity check of reducing to a point charge far away doesen t help in this case 30 / 32

42 Note: E always jumps by σ ɛ 0 across a uniformly charged thin surface (can be proved with Gauss s law). 31 / 32

43 E jumps by σ ɛ 0 across a uniformly thing charged surface 1 E of Hollow Shell vs Distance Q R 2) E 1 (Units of 4π ǫ Radial Distance (Units of R) Group discussion question: use this to verify the jump in the above graph. 32 / 32

44 E jumps by σ ɛ 0 across a uniformly thing charged surface 1 E of Hollow Shell vs Distance Q R 2) E 1 (Units of 4π ǫ Radial Distance (Units of R) Group discussion question: use this to verify the jump in the above graph. For the hollow shell, σ, the charge per area ( Q Q A ), is. So E 4πR 2 jumps from 0 to σ ɛ 0 = Q 4πɛ 0 R 2 32 / 32