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A hollow conducting spherical shell has radii of 0.80 m and 1.20 m, as shown in the figure. The sphere carries an excess charge of -500 nC. A point charge of +300 nC is present at the center. The surface charge density on the inner spherical surface is closest to A hollow conducting spherical shell has radii of 0.80 m and 1.20 m, as shown in the figure. The sphere carries an excess charge of -500 nC. A point charge of +300 nC is present at the center. The surface charge density on the inner spherical surface is closest to A) zero. B) +4.0 × 10<sup>-8</sup> C/m<sup>2</sup>. C) +6.0 × 10<sup>-8</sup> C/ m<sup>2</sup>. D) -4.0 × 10<sup>-8</sup> C/ m<sup>2</sup>. E) -6.0 × 10<sup>-8</sup> C/ m<sup>2</sup>.


A) zero.
B) +4.0 × 10-8 C/m2.
C) +6.0 × 10-8 C/ m2.
D) -4.0 × 10-8 C/ m2.
E) -6.0 × 10-8 C/ m2.

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A solid nonconducting sphere of radius R carries a uniform charge density throughout its volume. At a radial distance r1 = R/4 from the center, the electric field has a magnitude E0. What is the magnitude of the electric field at a radial distance r2 = 2R?


A) E0/4
B) zero
C) E0/2
D) E0
E) 2E0

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A hollow conducting spherical shell has radii of 0.80 m and 1.20 m, as shown in the figure. The sphere carries a net excess charge of -500 nC. A point charge of +300 nC is present at the center. (k = 1/4πε0 = 8.99 × 109 N ∙ m2/C) The radial component of the electric field at a point that is 0.60 m from the center is closest to A hollow conducting spherical shell has radii of 0.80 m and 1.20 m, as shown in the figure. The sphere carries a net excess charge of -500 nC. A point charge of +300 nC is present at the center. (k = 1/4πε<sub>0</sub> = 8.99 × 10<sup>9</sup> N ∙ m<sup>2</sup>/C) The radial component of the electric field at a point that is 0.60 m from the center is closest to A) zero. B) +5000 N/C. C) +7500 N/C. D) -5000 N/C. E) -7500 N/C.


A) zero.
B) +5000 N/C.
C) +7500 N/C.
D) -5000 N/C.
E) -7500 N/C.

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Two concentric spheres are shown in the figure. The inner sphere is a solid nonconductor and carries a charge of +5.00 µC uniformly distributed over its outer surface. The outer sphere is a conducting shell that carries a net charge of -8.00 µC. No other charges are present. The radii shown in the figure have the values R1 = 10.0 cm, R2 = 20.0 cm, and R3 = 30.0 cm. (k = 1/4πε0 = 8.99 × 109 N ∙ m2/C2) (a) Find the total excess charge on the inner and outer surfaces of the conducting sphere. (b) Find the magnitude and direction of the electric field at the following distances r from the center of the inner sphere: (i) r = 9.5 cm, (ii) r = 15.0 cm, (iii) r = 27.0 cm, (iv) r = 35.0 cm. Two concentric spheres are shown in the figure. The inner sphere is a solid nonconductor and carries a charge of +5.00 µC uniformly distributed over its outer surface. The outer sphere is a conducting shell that carries a net charge of -8.00 µC. No other charges are present. The radii shown in the figure have the values R<sub>1</sub> = 10.0 cm, R<sub>2</sub> = 20.0 cm, and R<sub>3</sub> = 30.0 cm. (k = 1/4πε<sub>0</sub> = 8.99 × 10<sup>9</sup> N ∙ m<sup>2</sup>/C<sup>2</sup>) (a) Find the total excess charge on the inner and outer surfaces of the conducting sphere. (b) Find the magnitude and direction of the electric field at the following distances r from the center of the inner sphere: (i) r = 9.5 cm, (ii) r = 15.0 cm, (iii) r = 27.0 cm, (iv) r = 35.0 cm.

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(a) -5.00 µC (inner surface), ...

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Under electrostatic conditions, the electric field just outside the surface of any charged conductor


A) is always parallel to the surface.
B) is always zero because the electric field is zero inside conductors.
C) is always perpendicular to the surface of the conductor.
D) is perpendicular to the surface of the conductor only if it is a sphere, a cylinder, or a flat sheet.
E) can have nonzero components perpendicular to and parallel to the surface of the conductor.

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The cross section of a long coaxial cable is shown in the figure, with radii as given. The linear charge density on the inner conductor is The cross section of a long coaxial cable is shown in the figure, with radii as given. The linear charge density on the inner conductor is and the linear charge density on the outer conductor is The inner and outer cylindrical surfaces are respectively denoted by A, B, C, and D, as shown. The radial component of the electric field at a point that 34 mm from the axis is closest to A) -16,000 N/C. B) +16,000 N/C. C) -37,000 N/C. D) +37,000 N/C. E) zero. and the linear charge density on the outer conductor is The cross section of a long coaxial cable is shown in the figure, with radii as given. The linear charge density on the inner conductor is and the linear charge density on the outer conductor is The inner and outer cylindrical surfaces are respectively denoted by A, B, C, and D, as shown. The radial component of the electric field at a point that 34 mm from the axis is closest to A) -16,000 N/C. B) +16,000 N/C. C) -37,000 N/C. D) +37,000 N/C. E) zero. The inner and outer cylindrical surfaces are respectively denoted by A, B, C, and D, as shown. The cross section of a long coaxial cable is shown in the figure, with radii as given. The linear charge density on the inner conductor is and the linear charge density on the outer conductor is The inner and outer cylindrical surfaces are respectively denoted by A, B, C, and D, as shown. The radial component of the electric field at a point that 34 mm from the axis is closest to A) -16,000 N/C. B) +16,000 N/C. C) -37,000 N/C. D) +37,000 N/C. E) zero. The radial component of the electric field at a point that 34 mm from the axis is closest to The cross section of a long coaxial cable is shown in the figure, with radii as given. The linear charge density on the inner conductor is and the linear charge density on the outer conductor is The inner and outer cylindrical surfaces are respectively denoted by A, B, C, and D, as shown. The radial component of the electric field at a point that 34 mm from the axis is closest to A) -16,000 N/C. B) +16,000 N/C. C) -37,000 N/C. D) +37,000 N/C. E) zero.


A) -16,000 N/C.
B) +16,000 N/C.
C) -37,000 N/C.
D) +37,000 N/C.
E) zero.

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As shown in the figure, a square insulating slab 5.0 mm thick measuring 2.0 m × 2.0 m has a charge of 8.0 × 10-11 C distributed uniformly throughout its volume. Use Gauss's law to determine the electric field at point P, which is located within the slab beneath its center, 1.0 mm from one of the faces. As shown in the figure, a square insulating slab 5.0 mm thick measuring 2.0 m × 2.0 m has a charge of 8.0 × 10<sup>-11</sup> C distributed uniformly throughout its volume. Use Gauss's law to determine the electric field at point P, which is located within the slab beneath its center, 1.0 mm from one of the faces. A) 0.68 N/C B) 14 N/C C) 23 N/C D) 34 N/C E) 57 N/C As shown in the figure, a square insulating slab 5.0 mm thick measuring 2.0 m × 2.0 m has a charge of 8.0 × 10<sup>-11</sup> C distributed uniformly throughout its volume. Use Gauss's law to determine the electric field at point P, which is located within the slab beneath its center, 1.0 mm from one of the faces. A) 0.68 N/C B) 14 N/C C) 23 N/C D) 34 N/C E) 57 N/C


A) 0.68 N/C
B) 14 N/C
C) 23 N/C
D) 34 N/C
E) 57 N/C

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An infinitely long nonconducting cylinder of radius R = 2.00 cm carries a uniform volume charge density of An infinitely long nonconducting cylinder of radius R = 2.00 cm carries a uniform volume charge density of Calculate the electric field at distance r = 1.00 cm from the axis of the cylinder. (ε<sub>0</sub> = 8.85 × 10<sup>-12</sup> C<sup>2</sup>/N ∙ m<sup>2</sup>) A) 2.50 × 10<sup>3</sup> N/C B) 5.10 × 10<sup>3</sup> N/C C) zero D) 2.00 × 10<sup>3</sup> N/C E) 10.2 × 10<sup>3</sup> N/C Calculate the electric field at distance r = 1.00 cm from the axis of the cylinder. (ε0 = 8.85 × 10-12 C2/N ∙ m2)


A) 2.50 × 103 N/C
B) 5.10 × 103 N/C
C) zero
D) 2.00 × 103 N/C
E) 10.2 × 103 N/C

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A solid nonconducting sphere of radius R carries a charge Q distributed uniformly throughout its volume. At a certain distance r1 (r1 < R) from the center of the sphere, the electric field has magnitude E. If the same charge Q were distributed uniformly throughout a sphere of radius 2R, the magnitude of the electric field at the same distance r1 from the center would be equal to


A) E/8.
B) E/2.
C) 2E.
D) 8E.
E) E.

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A

The graph in the figure shows the electric field strength (not the field lines) as a function of distance from the center for a pair of concentric uniformly charged spheres. Which of the following situations could the graph plausibly represent? (There may be more than one correct choice.) The graph in the figure shows the electric field strength (not the field lines) as a function of distance from the center for a pair of concentric uniformly charged spheres. Which of the following situations could the graph plausibly represent? (There may be more than one correct choice.) A) a positively charged conducting sphere within another positively charged conducting sphere B) a positively charged conducting sphere within an uncharged conducting sphere C) a solid nonconducting sphere, uniformly charged throughout its volume, inside of a positively charged conducting sphere D) a positively charged nonconducting thin-walled spherical shell inside of a positively charged conducting sphere E) a positively charged nonconducting thin-walled spherical shell inside of another positively charged nonconducting thin-walled spherical shell


A) a positively charged conducting sphere within another positively charged conducting sphere
B) a positively charged conducting sphere within an uncharged conducting sphere
C) a solid nonconducting sphere, uniformly charged throughout its volume, inside of a positively charged conducting sphere
D) a positively charged nonconducting thin-walled spherical shell inside of a positively charged conducting sphere
E) a positively charged nonconducting thin-walled spherical shell inside of another positively charged nonconducting thin-walled spherical shell

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Charge is distributed uniformly throughout a large insulating cylinder of radius R. The charge per unit length in the cylindrical volume is λ. (a) Use Gauss's law to find the magnitude of the electric field at a distance r from the central axis of the cylinder for r < R. Your answer should be in terms of r, R, λ, ε0 , and π. (b) Check the reasonableness of your answer by evaluating it at the surface of the cylinder.

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(a) E = blured image (b) At r = ...

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A nonconducting spherical shell of inner radius R1 and outer radius R2 contains a uniform volume charge density ρ throughout the shell. Use Gauss's law to derive an equation for the magnitude of the electric field at the following radial distances r from the center of the sphere. Your answers should be in terms of ρ, R1, R2, r, ε0 , and π. (a) r < R1 (b) R1 < r < R2 (c) r > R2

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(a) E = 0 ...

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If a rectangular area is rotated in a uniform electric field from the position where the maximum electric flux goes through it to an orientation where only half the flux goes through it, what has been the angle of rotation?


A) 45°
B) 26.6°
C) 90°
D) 30°
E) 60°

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A hollow conducting spherical shell has radii of 0.80 m and 1.20 m, as shown in the figure. The sphere carries a net excess charge of -500 nC. A point charge of +300 nC is present at the center. (k = 1/4πε0 = 8.99 × 109 N ∙ m2/C) The radial component of the electric field at a point that is 1.50 m from the center is closest to A hollow conducting spherical shell has radii of 0.80 m and 1.20 m, as shown in the figure. The sphere carries a net excess charge of -500 nC. A point charge of +300 nC is present at the center. (k = 1/4πε<sub>0</sub> = 8.99 × 10<sup>9</sup> N ∙ m<sup>2</sup>/C) The radial component of the electric field at a point that is 1.50 m from the center is closest to A) +1200 N/C. B) +2000 N/C. C) -800 N/C. D) -1600 N/C. E) -2000 N/C.


A) +1200 N/C.
B) +2000 N/C.
C) -800 N/C.
D) -1600 N/C.
E) -2000 N/C.

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Two concentric conducting spherical shells produce a radially outward electric field of magnitude 49,000 N/C at a point 4.10 m from the center of the shells. The outer surface of the larger shell has a radius of 3.75 m. If the inner shell contains an excess charge of -5.30 μC, find the amount of charge on the outer surface of the larger shell. (k = 1/4πε0 = 8.99 × 109 N ∙ m2/C2)

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91.6 µC

A huge (essentially infinite) horizontal nonconducting sheet 10.0 cm thick has charge uniformly spread over both faces. The upper face carries +95.0 nC/m2 while the lower face carries -25.0 nC/m2. What is the magnitude of the electric field at a point within the sheet 2.00 cm below the upper face? A huge (essentially infinite) horizontal nonconducting sheet 10.0 cm thick has charge uniformly spread over both faces. The upper face carries +95.0 nC/m<sup>2</sup> while the lower face carries -25.0 nC/m<sup>2</sup>. What is the magnitude of the electric field at a point within the sheet 2.00 cm below the upper face? A) 0.00 N/C B) 3.95 × 10<sup>3</sup> N/C C) 6.78 × 10<sup>3</sup> N/C D) 7.91 × 10<sup>3</sup> N/C E) 1.36 × 10<sup>4</sup> N/C


A) 0.00 N/C
B) 3.95 × 103 N/C
C) 6.78 × 103 N/C
D) 7.91 × 103 N/C
E) 1.36 × 104 N/C

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A non-conducting sphere of radius R = 7.0 cm carries a charge Q = 4.0 mC distributed uniformly throughout its volume. At what distance, measured from the center of the sphere, does the electric field reach a value equal to half its maximum value?


A) 3.5 cm only
B) 4.9 cm only
C) 3.5 cm and 9.9 cm
D) 3.5 cm and 4.9 cm
E) 9.9 cm only

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Two long straight parallel lines, #1 and #2, carry uniform positive linear charge densities. The charge density on line #2 is twice as great as the charge density on line #1. The locus of points where the electric field due to these lines is zero is


A) along a line between the lines closer to line #2 than line #1.
B) at a point midway between the lines.
C) along a line perpendicular to lines #1 and #2.
D) along a line between the lines closer to line #1 than line #2.

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D

Consider a spherical Gaussian surface of radius R centered at the origin. A charge Q is placed inside the sphere. To maximize the magnitude of the flux of the electric field through the Gaussian surface, the charge should be located


A) at x = 0, y = 0, z = R/2.
B) at the origin.
C) at x = R/2, y = 0, z = 0.
D) at x = 0, y = R/2, z = 0.
E) The charge can be located anywhere, since flux does not depend on the position of the charge as long as it is inside the sphere.

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Which of the following statements about Gauss's law are correct? (There may be more than one correct choice.)


A) Gauss's law is valid only for symmetric charge distributions, such as spheres and cylinders.
B) If there is no charge inside of a Gaussian surface, the electric field must be zero at points of that surface.
C) Only charge enclosed within a Gaussian surface can produce an electric field at points on that surface.
D) If a Gaussian surface is completely inside an electrostatic conductor, the electric field must always be zero at all points on that surface.
E) The electric flux passing through a Gaussian surface depends only on the amount of charge inside that surface, not on its size or shape.

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