5: Electric Flux and Gauss'

5: Electric Flux and Gauss’ Law

Worksheet 6 −→ 1. A surface has the area vector A = (2î + 4ˆj) m2. −→ A. What is the flux of a uniform electric field through it if the field is E = 4î N/C? −→ B. What is the flux of a uniform electric field through it if the field is E = 4kˆ N/C? 2. The figure shows, in cross section, a central metal ball, two spherical metal shells, and three spherical Gaussian surfaces of radii R, 2R, and 3R, all with the same center. The uniform charges on the three objects are: ball, Q; smaller shell, 3Q; larger shell, 5Q. Rank the Gaussian surfaces according to the magnitude of the electric field

at any point on the surface, greatest first. −→ 3. A uniform vector field E = (3.00E5î − 4.00E5ˆj + 5.00E5kˆ) N/C fills all of space. A. Imagine a virtual square centered in the xy plane of edge length 2.00 m. What is the electric flux through this square, ΦE? B Now replace the square with a cube of edge length 2.00 m centered on the origin. What is the electric flux ΦE through: i. the face that cuts across the +x axis? ii. the face that cuts across the -x axis? iii. how about the entire cube? 4. Use Gauss’ law and your visulatization of the electric field lines* to prove our mathematical equation for the electric field due to a point charge. Note: Your visualization of field lines is important. It turns out that Gauss’ law alone is not a complete statement of electrostatics. In electrtostatics, electric fields have zero curl (loop integrals always evaluate to zero); Gauss’ law says nothing about this. The bottom line is that the fields you visualize knowing some basics about electric field lines (that they start on + charges and terminate on - charges) gaurantees zero curl. Your visualization is bringing in some essential physics not included in Gauss’ law alone.

5. Imagine a virtual bubble. It can be of any shape, size, and location. Imagine that the inside of the bubble are two charges -3Q and +Q. Outsides the bubble is a single charge +5Q. What is the electric flux ΦE through the bubble? −→ 6. An electric field given by E = 1.9î − 1.7(y2 + 2.1)ˆj pierces the Gaussian cube of edge length 0.180 m and positioned as shown in the figure. (The magnitude E is in newtons per coulomb and the position x is in meters.) What is the electric flux through the (a) top face, (b) bottom face, (c) left face, and (d) back face? (e) What is the net electric flux through the cube?

1 7. There is a Gaussian surface in the shape of a cube with edge length 1.70 m. What are (a) the net flux through the surface and (b) the net charge qenc enclosed by the surface if the electric field in the region is in the positive y-direction and has a magnitude that is given by E = 3.71y N/C? What are (c) the net flux and (d) net enclosed charge if the electric field is in the xy plane and has components Ex = -3.76 N/C and Ey = (6.81 + 3.71y) N/C? 8. There is a Gaussian surface in the shape of a cube with edge length 1.93 m and is oriented as shown in a region of uniform electric field. Find the electric flux through the right face if the electric field (a) is in the positive x direction and has magnitude 7.56 N/C, (b) is in the negative y-direction and has magnitude 2.86 N/C, and (c) is in the xz plane, has x-component -3.04 N/C, and has z-component 4.30 N/C. (d) What is the total flux through the cube’s surface for each field? 9. The electric field in a certain region of Earth’s atmosphere is directed vertically down. At an altitude of 409 m the field has magnitude 74.5 N/C; at an altitude of 318 m, the magnitude is 110 N/C. Find the net amount of charge contained in a cube 91 m on edge, with horizontal faces at altitudes of 318 and 409 m. 10. A charged particle is suspended at the center of two concentric spherical shells that are very thin and made of nonconducting material. Figure (a) shows a cross section. Figure (b) gives the net flux (Φ) through a Gaussian sphere centered on the particle as a function of the radius r of the sphere. The scale of the vertical axis is set 2 by Φs = 3.5E5 N· m /C. (a) What is the charge of the central particle? What are the net charges of (b) shell A and (c) shell B?

1 Answers

1.(A) 8 N·m (B) 0 N·m 2. E(R) = E(2R) = E(3R) 3.(A.) 20E5 N·m2/C (Bi) 12E5 N·m2/C (Bii) -12E5 N·m2/C (Biii) 0 N·m2/C 2 4. E = kqinc/r 5.ΦE = -2Q/0 6. (a)-0.117 N·m2/C (b) 0.116 N·m2/C (c) -0.062 N·m2/C (d) 0 N·m2/C (e) -0.00178 N·m2/C 7. (a) 18.23 N·m2/C (b) 1.6131E-10 C (c) 18.23 N·m2/C (d) 1.6131E-10 C 8. (a) 0 N·m2/C (b) -10.654 N·m2/C (c) 0 N·m2/C (d) 0 N·m2/C 9. 2.6E6 C 10. (a) 1.239E-6 C (b) -3.717E-6 C (c) 6.195E-6 C

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