Unit 8: Electric Charges, Fields, and Gauss's Law

Three small charged spheres are fixed in the xy-plane: A (+4.0 μC) at the origin, B (+3.0 μC) on the x-axis at x = 0.40 m, and C (−2.0 μC) on the y-axis at y = 0.30 m. What is the magnitude of the net electric force that B and C exert on A?

A charge +q is fixed at x = 0 and a charge +4q is fixed at x = d on the x-axis. At what position on the x-axis would a third small charge feel no net electric force?

Two protons are 1.0 × 10⁻¹⁰ m apart. Which statement about the ratio of the electric force between them to the gravitational force between them is correct?

A neutral metal sphere stands on an insulating base. A negatively charged rod is held near the sphere without touching it. While the rod is held there, the sphere is connected to the ground by a wire; the wire is then removed, and finally the rod is taken away. What is the sphere's final charge?

Three identical small metal spheres sit on insulating stands. A carries +8.0 nC, B carries −4.0 nC and C is neutral. A is touched to B and separated, then B is touched to C and separated. What is the final charge on C?

A charge +Q is fixed at x = 0 and a charge −4Q is fixed at x = d on the x-axis. At what finite position on the x-axis is the net electric field zero?

The diagram shows the electric field lines near two small charged spheres, A and B. Which claim about their charges is best supported by the diagram?

A solid aluminum sphere of radius R carries a net charge +Q and is in electrostatic equilibrium. What is the magnitude of the electric field at a point inside the sphere, a distance R/2 from its center?

An electron is released from rest in a uniform electric field of magnitude 2.0 × 10³ N/C directed in the +x direction, as shown. Ignoring gravity, what is the electron's acceleration?

A thin rod of length L carries charge +Q spread uniformly. Point P is a distance y from the rod's center, on its perpendicular bisector. What is the magnitude of the electric field at P?

A thin rod lies on the x-axis from x = a to x = 2a. Its linear charge density is λ = βx, where β is a positive constant. What is the magnitude of the electric field at the origin?

A thin ring of radius R carries charge +Q spread uniformly. At what distance z from the ring's center, along its axis, is the magnitude of the electric field largest?

A thin plastic rod bent into a semicircle of radius R carries charge +Q spread uniformly along it. What is the magnitude of the electric field at the center of the semicircle?

A very long straight wire along the x-axis carries uniform charge per unit length λ. A student finds the field at point P, a distance r from the wire, by adding the fields of pieces dq = λ dx at positions x. Which integral gives the field at P, and why?

A thin ring of radius R carries charge +Q spread uniformly. At a point on its axis a distance R from its center, the ring's field has magnitude E₁. If the same charge Q were instead a point charge at the ring's center, the field at that point would be E₂. What is E₁/E₂?

A thin ring of radius R carries positive charge with uniform density λ. A short piece of length s (much less than R) is cut out at the top, as shown, and the rest of the ring keeps its charge. What is the electric field at the center of the ring?

The graph shows the magnitude of the electric field along the axis of a thin, uniformly charged ring, against the distance z from the ring's center. Which claim about the ring's radius R is best supported by the graph?

A flat surface of area 0.020 m² sits in a uniform electric field of magnitude 4.0 × 10³ N/C. The normal to the surface makes an angle of 60° with the field, as shown. What is the magnitude of the electric flux through the surface?

A point charge +q is inside two closed surfaces: surface 1 is a sphere of radius r centered on the charge, and surface 2 is a cube of side 4r that also encloses the charge, but not at its center. How does the electric flux Φ₂ through the cube compare with the flux Φ₁ through the sphere?

A point charge +q sits outside a closed spherical surface, as shown. Which statement about the surface is correct?

A uniform electric field E points along the axis of an open hemispherical bowl of radius R, as shown. A flat disk across the bowl's opening would close it into a closed surface. How does the magnitude of the flux through the bowl's curved surface compare with the magnitude of the flux through that flat opening?

In a region of space the electric field is E = bx, pointing in the +x direction, where b is a positive constant. A closed cube of side a has its faces perpendicular to the axes, with one face at x = a and the opposite face at x = 2a. How much charge is inside the cube?

A solid insulating sphere of radius R has charge density ρ = ρ₀r/R, where r is the distance from its center and ρ₀ is a positive constant. What is the magnitude of the electric field a distance r < R from the center?

An insulating sphere of radius R has charge density ρ = A/r, where r is the distance from its center and A is a positive constant. What is the sphere's total charge?

The graph shows the magnitude of the electric field against the distance r from the center of a spherically symmetric charge distribution. Which distribution is best supported by the graph?

A uniformly charged insulating sphere has charge Q and radius R. The field at a distance R/3 from its center is E₁. It is replaced by another uniformly charged sphere with charge 2Q and radius R/2. In terms of E₁, what is the field at the new sphere's surface?

A point charge +q sits at the center of a thick, hollow metal sphere (inner radius a, outer radius b) that carries a net charge of −3q. In electrostatic equilibrium, what charge is on the sphere's outer surface?

A solid insulating sphere of radius 0.10 m has a uniform charge density of 2.0 μC/m³. What is the magnitude of the electric field 0.050 m from its center?

A very large, flat insulating slab of thickness 2d has uniform positive charge density. At a point inside it, a distance d/2 from the slab's central plane, the field magnitude is E₁. What is the field magnitude at a point outside the slab, a distance 3d from the central plane?