Supplementary Sheet 2 Solutions

Supplementary Sheet 2 Solutions (Ch 24)

Question 1

A charged particle with charge q is located inside a cubical Gaussian surface with no other charges nearby.
(i) If the particle is at the center of the cube, determine the flux through each face of the cube. The possible answers are:

  • (a) 0
  • (b) ( \frac{q}{2 ext{ϵ}_0} )
  • (c) ( \frac{q}{6 ext{ϵ}_0} )
  • (d) ( \frac{q}{8 ext{ϵ}_0} )
  • (e) depends on the size of the cube

Answer: (i) The correct answer is (c). Equal amounts of flux pass through each of the six faces of the cube.
(ii) If the particle can be moved to any point within the cube, the maximum value that the flux through one face can approach is:
Answer: (b). By moving the charge very close to one face, half the flux can pass through that face while the other half passes through the remaining five faces.

Question 2

A solid insulating sphere of radius 5 cm carries electric charge uniformly distributed throughout its volume. Concentric with the sphere is a conducting spherical shell with no net charge, where the inner radius is 10 cm and the outer radius is 15 cm.

(a) Ranking of the electric field at various points A, B, C, and D located at radii 4 cm, 8 cm, 12 cm, and 16 cm, respectively. The ranking is from largest to smallest.

Answer: A > B > D > C. Let q represent the charge of the insulating sphere:

  • The field at A is ( \frac{(4/5)^3 q}{4 \pi(4 ext{ cm})^2 \text{ϵ}_0} ).
  • The field at B is ( \frac{q}{4 \pi(8 ext{ cm})^2 \text{ϵ}_0} ).
  • The field at C is 0.
  • The field at D is ( \frac{q}{4 \pi(16 ext{ cm})^2 \text{ϵ}_0} ).

    (b) Ranking of the electric flux through concentric spherical surfaces at points A, B, C, and D.

Answer: The ranking is: B = D > A > C. The flux through the 4 cm sphere is ( \frac{(4/5)^3 q}{\text{ϵ}_0} ). The flux through the 8 cm sphere and the 16 cm sphere is ( \frac{q}{\text{ϵ}_0} ) because they enclose the same amount of charge. The flux through the 12 cm sphere is 0 because the electric field is zero inside the conductor.

Question 3

Consider an electric field that is uniform in direction throughout a certain volume. Determine if it can be uniform in magnitude and if it must be uniform in magnitude under two scenarios: (a) filling the volume with an insulating material carrying charge, and (b) assuming the volume is empty space.

Answer:
(a) If the volume charge density is nonzero, then the electric field cannot be uniform in magnitude. This can be demonstrated using a Gaussian surface in the shape of a rectangular box with two faces perpendicular to the direction of the field. The net flux out of the box would be nonzero, indicating that the field must be stronger on one side than the other, hence it cannot be uniform in magnitude.

(b) When the volume contains no charge, the net flux out of the box equals zero. This indicates that the flux entering is equal to the flux exiting, and under these conditions, the electric field must be uniform in magnitude along any line in the direction of the field. However, the field may vary between different points in a plane perpendicular to the field lines.

Question 4

What happens to the total electric flux through a surface surrounding a point charge q if the following changes occur:
(a) the charge is doubled,
(b) the volume of the cube is doubled,
(c) the surface is changed to a sphere,
(d) the charge is moved to another location inside the surface,
(e) the charge is moved outside the surface.

Answer: The flux through a closed surface is proportional to the total charge contained within the surface:
(a) The flux is doubled because the charge is doubled.
(b) The flux remains the same because the total charge is unchanged.
(c) The flux remains the same because it still reflects the charge contained.
(d) The flux remains the same because the charge inside the surface is unchanged.
(e) The flux becomes zero because no charge is now enclosed within the surface.

Question 5

In a region of space containing no charges, a uniform electric field exists. What can be concluded about the net electric flux through a Gaussian surface placed in this region?

Answer: The net flux through any Gaussian surface is zero. This can be reasoned in two ways: (1) Any surface contains zero charge, hence, according to Gauss's law, the total flux is zero. (2) Because the field is uniform, the field lines entering one side of the closed surface will emerge from the other side, resulting in a net flux of zero.

Question 6

If the total charge inside a closed surface is known but the distribution of this charge is unspecified, can Gauss's law be utilized to find the electric field?

Answer: No, Gauss's law cannot be applied to determine the electric field at various points on the surface if the field is not uniform across that surface. If the symmetry of an electric field permits us to state that ( E \, \cos \theta \, dA = E \, \cos \theta \, dA ), where E is a constant across the surface, then Gauss's law can be used. However, if the electric field is generalized as some unknown function ( E(x, y, z) ), simplifications cannot be made effectively.

Question 7

Explain why the electric flux through a closed surface with a given enclosed charge remains independent of the size or shape of the surface.

Answer: The electric flux is independent of the closed surface's size and shape that contains the charge because by Gauss's law, all field lines from the enclosed charge pass through the surface. Hence, the total flux only depends on the total charge enclosed, regardless of the contours of the surface.

Question 8

If more electric field lines leave a Gaussian surface than enter it, what does this imply about the net charge enclosed by that surface?

Answer: This means that the surface encloses a positive total charge. Electric field lines originate from positive charge and terminate at negative charge.

Question 9

Consider a person in a large hollow metallic sphere insulated from the ground.
(a) If a large charge is placed on the sphere, will the person be harmed upon touching the inside of the sphere?
(b) What will happen if the person also has an initial charge opposite to that on the sphere?

Answer: (a) No, the person will not be harmed. The electric field inside the hollow sphere remains zero, meaning no charge is present on the interior wall of the shell.
(b) If the person possesses an initial charge @q@, the electric field inside the sphere would no longer be zero, inducing a charge of -q on the inner wall. Touching the inner surface will result in a minor shock as the charge on the person's body discharges to the metal.

Question 10

Explain how two identical conducting spheres can attract each other even if one has a large positive charge and the other has a smaller positive charge.

Answer: The sphere with the large charge produces a strong electric field that polarizes the smaller charged sphere. The migration of excess charge pushes the like charge to the far side, resulting in a patch of opposite charge adjacent to the large charged sphere, creating a disparity in force due to the varying field strengths. The attraction of the oppositely charged near side exceeds the repulsion due to the similarly charged far side, leading to an attractive force overall.

Question 11

Discuss the impact of the Sun's position relative to the Earth on sunlight flux and weather conditions.

Answer: (a) When the Sun is lower in the sky during winter, the luminous flux to a given area on Earth's surface decreases. This is due to the increased angle between sun rays and the area vector, thus reducing the cosine of the angle.
(b) The decrease in sunlight flux typically results in cooler temperatures and subsequently can influence weather patterns leading to colder conditions.

Question P1

A circular loop of diameter 40.0 cm rotates in a uniform electric field until maximum electric flux is reached, measured ( F = 5.20 \times 10^5 \text{ N} \cdot \text{m}^2/\text{C} ). What is the magnitude of the electric field?

Question P2

Inside a submarine, the following charges exist: 5.00 µC, −9.00 µC, 27.0 µC, and −84.0 µC.
(a) Calculate the net electric flux through the hull of the submarine.
(b) Discuss the imbalance of electric field lines leaving versus entering the submarine.

Question P3

Consider four closed surfaces, S1 to S4, with charges -2Q, Q, and -Q. Calculate the electric flux through each surface as sketched in the scenario.

Question P4

A charge of 170 µC is placed at the center of a cube with an edge of 80.0 cm. Determine:
(a) the flux through each face of the cube,
(b) the total flux through the surface,
(c) discuss changes if the charge were not centered, justifying your reasoning.

Question P5

A charge with 12.0 µC is centered in a spherical shell of radius 22.0 cm. Determine:
(a) the total electric flux through the shell’s surface,
(b) through any hemispherical surface of the shell,
(c) discuss radius independence in results.

Question P6

An infinitely long uniform line charge with charge per unit length ( 𝜆 ) lies a distance d from point O. Calculate the total electric flux through a sphere of radius R centered at point O, analyzing cases where (a) ( R < d ) and (b) ( R > d ).

Question P7

Analyze a charge density of −90.0 µC/m along a long straight filament at distances of (a) 10.0 cm, (b) 20.0 cm, and (c) 100 cm measured perpendicularly from the filament.

Question P8

A uniformly charged straight filament of length 7.00 m has a total positive charge of 2.00 µC. An uncharged cylindrical cardboard of length 2.00 cm and radius 10.0 cm surrounds this filament at its center. Calculate:
(a) the electric field at the cylinder's surface,
(b) the total electric flux through this cylinder.

Question P9

Find the electric field intensity inside a uniform cylindrical charge distribution of radius R with a uniform charge density ρ at distances ( r < R ).

Question P10

A cylindrical shell of radius 7.00 cm and length 2.40 m has charge uniformly distributed on its surface. The electric field magnitude at a point 19.0 cm radially outward is 36.0 kN/C. Find:
(a) net charge on the shell,
(b) electric field at 4.00 cm from the axis, measured outward from the shell's midpoint.

Question P11

For a solid sphere of radius 40.0 cm with a total charge of 26.0 µC uniformly distributed, calculate the electric field magnitudes at:
(a) 0 cm, (b) 10.0 cm, (c) 40.0 cm, and (d) 60.0 cm from the center of the sphere.

Question P12

For a solid metallic sphere of radius a with total charge Q, determine the electric field just outside its surface:
( E = \frac{k_e Q}{a^2} ) occurring radially outward. Examine whether locally, it appears similar to an infinite sheet of charge at the surface.

Question P13

Compare the electric fields produced at the center of the upper surface of a large aluminum plate of area A with that from a glass plate with the same charge Q uniformly distributed over both surfaces.

Question P14

Two identical conducting spheres of radius 0.500 cm are connected via a light conducting wire of 2.00 m. When subjected to a charge of 60.0 µC, determine the tension in the wire.

Question P15

In a hollow metallic cylinder, encircled by wire of charge per unit length( λ ). With a net charge of 2λ, apply Gauss's law to:
(a) determine charge densities on the inner cylinder's surface,
(b) the outer cylinder's surface, and
(c) the electric field existing outside the cylinder at distance r from the axis.

Question P16

For a solid insulating sphere of radius a with uniform charge density ρ and charge Q, analyze with an uncharged, concentric conducting hollow sphere:
(a) charge with ( r < a ), (b) magnitude of the electric field for ( r < a ), (c) charge for ( a < r < b ), (d) electric field for ( a < r < b ), and so forth until (h) determining which spherical surface possesses greater charge density.