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Champlain College - St-Lambert!

Physics 203-NYB

Problem Session Field, Flux and Gauss's Law

1. What is the electric flux through the surface shown in the following figures? For each surface the area vector is considered pointing upward, the area of the surface is 10 cm x 10 cm, and the electric field is 200 N/C, oriented at 30° from the surface.

2. A 2.0 cm x 3.0 cm rectangle lies in the xy-plane. What is the electric flux through the rectangle if ^ ^ E a) = (50i + 100k) N/C ^ ^ E b) = (50i + 100 j) N/C 3. A 4.0-cm-diameter circle lies in the xy-plane in a region where the electric field is ^ = (1000i + 1000 j + 1000k) N/C. What is the electric flux through the circle. ^ ^ E 4. A 1.0 cm x 1.0 cm x 1.0 cm box is between two parallel conducting plates charged with opposite signs charges. Two faces of the box are perpendicular to the electric field. The electric field strength is 1000 N/C. What is the net electric flux through the box? 5. What is the net electric flux through the two following cylinders shown in the figures below? Each gauss cylinder has a radius R, a length L, and the field has a magnitude E.

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6. A thin, metallic, spherical shell of radius a carries a charge qa. Concentric with it is another thin, metallic, spherical shell of radius b (b > a) carrying a charge qb. Use gauss' law to find the electric field at radial points where: a) r < a b) a < r < b c) r > b d) Discuss the criterion one would use to determine how the charges are distributed on the inner and outer surfaces of each shell. 7. A sphere of radius R has a constant volume charge density . Use Gauss' law to derive expressions for the electric field at a distance r from the center of the sphere for: a) r < R b) r > R 8. A thin-walled cylindrical shell of radius R and infinite length carries a uniform charge density . a) Show that the electric field is zero for r < R. R b) Show that for r > R the electric field has the magnitude E = . 0 r 9. An infinitely long cylinder of radius R has a uniform volume charge density . R2 a) Show that E = for r R . 20 r r b) Show that E = for r R . 20

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c) Sketch E versus r. 10. The figure below shows a spherical non-conducting shell of charge of uniform density . Find the electric field everywhere.

11. The following figure shows a section through two infinitely long concentric cylinders of radii a and b (b > a). The cylinders carry equal and opposite charges per unit length . Use Gauss's law to prove:

a) that E = 0 for r > b and r < a. b) that between the cylinders E is given by E =

directed inward. 20 r

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Answers

1. a) 1.0 N m2/C, b) -1.0 N m2/C 2. a) 6.0 × 10-2 N m2/C, b) 0 N m2/C 3. 1.26 N m2/C 4. 0 N m2/C 5. a) 0 N m2/C, b) 2R2 E

qa + qb qa , c) 40 r2 40 r2

6. b)

10.

r > b: a < r < b: r < a:

kQ k 4 = 2 (b3 - a3 ) r2 r 3 (r3 - a3 ) E= 3r2 0 E=0 E=

(b3 - a3 ) a3 11. for a < r < b: E = . r - 2 , for r > b: E = 30 r2 30 r

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