24. Electric Force & Field; Gauss' Law

A spherically symmetric charge distribution produces the electric field $\overrightarrow{E}=\left(5000r^2\right)\hat{r}$ N/C, where r is in m. How much charge is inside this 40-cm-diameter spherical surface?

A flat sheet of paper of area $0.250$ m² is oriented so that the normal to the sheet is at an angle of $60$° to a uniform electric field of magnitude $14$ N/C. For what angle $\varphi$ between the normal to the sheet and the electric field is the magnitude of the flux through the sheet (i) largest and (ii) smallest? Explain your answers.

The cube in FIGURE EX24.7 contains negative charge. The electric field is constant over each face of the cube. Does the missing electric field vector on the front face point in or out? What strength must this field exceed?

Given a uniform electric field perpendicular to the plane of an annular ring (surface $3$) with inner radius $a$ and outer radius $b$, what is the electric flux $\Phi 3_{}$ through the ring if the electric field has magnitude $E$ and points outward from the surface?

Which of the following correctly describes the electric flux $\Phi$ through a flat surface of area $A$ placed perpendicular to a uniform electric field $E$?

What is the total flux through the two surfaces depicted in the following figure? Note that surface 1 has an area of 50 cm² and surface 2 has an area of 100 cm² , and E = 500 N/C.

A point charge of $q$ is placed at the center of a closed spherical surface. What is the electric flux through the surface?

All examples of Gauss’s law have used highly symmetric surfaces where the flux integral is either zero or EA. Yet we’ve claimed that the net Φₑ = Qᵢₙ / ϵ₀ is independent of the surface. This is worth checking. FIGURE CP24.57 shows a cube of edge length L centered on a long thin wire with linear charge density λ. The flux through one face of the cube is not simply EA because, in this case, the electric field varies in both strength and direction. But you can calculate the flux by actually doing the flux integral. Now integrate dΦ to find the total flux through this face.

A uniform electric field $E$ points vertically upward. Cylinder (a) in Figure 1 is oriented so its axis is horizontal and its ends are perpendicular to the field lines. What is the net electric flux through cylinder (a)?

A 10 nC charge is at the center of a 2.0 m x 2.0 m x 2.0 m cube. What is the electric flux through the top surface of the cube?

Find the electric fluxes Φ_A to Φ_E through surfaces A to E in FIGURE P24.29.

An infinite line of charge with linear charge density $\lambda$ passes along the axis of a cylindrical surface of radius $r$ and length $L$. What is the electric flux through the curved surface of the cylinder due to the line of charge?

A uniform electric field of magnitude $E$ points perpendicular to the plane of a circular loop of radius $r$. What is the electric flux $\Phi$ through the loop?