24. Electric Force & Field; Gauss' Law

A uniform electric field of magnitude $E$ passes perpendicularly through a flat surface of area $A$. What is the electric flux $\Phi$ through the surface?

If a uniform electric field of magnitude $E$ passes perpendicularly through a flat surface of area $A$, the electric flux through the shaded surface is

Given three point charges enclosed by a closed surface $a_1$, what is the electric flux through the surface $a_1$ according to Gauss's law?

A uniform electric field points in the positive x direction and has a magnitude of 40N/C. What is the net flux through a cube with sides 4.0cm? The cube has one corner on the origin and the positive x, y, and z axes form three of its edges.

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 ${\Phi }_e=Q_{in}/{ϵ}_0$ 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. Consider the face parallel to the yz-plane. Define area $d\overrightarrow{A}$ as a strip of width dy and height L with the vector pointing in the 𝓍-direction. One such strip is located at position y. Use the known electric field of a wire to calculate the electric flux dΦ through this little area. Your expression should be written in terms of y, which is a variable, and various constants. It should not explicitly contain any angles.

The electric flux through the surface shown in FIGURE EX24.10 is 25 N m²/C. What is the electric field strength?

Given a uniform electric field $E$ perpendicular to the plane of an annular ring with inner radius $r_1$ and outer radius $r_2$, what is the electric flux $\Phi$ through the ring?

A 2.0 cm × 3.0 cm rectangle lies in the $xz$-plane with unit vector $\hat{n}$ pointing in the +y-direction. What is the electric flux through the rectangle if the electric field is $\overrightarrow{E}=(4000\hat{i}-2000\hat{k})$ N/C?

Where does the normal vector point for a spherical shell?

A 12 cm × 12 cm rectangle lies in the first quadrant of the xy-plane with one corner at the origin. Unit vector $\hat{n}$ points in the +𝒵-direction. What is the electric flux through the rectangle if the electric field is $E=\left(2000m^{-1}\right)x\hat{k}$ N/C? Hint: Divide the rectangle into narrow strips of width.

A hemispherical surface with radius $r$ in a region of uniform electric field $\overrightarrow{E}$ has its axis aligned parallel to the direction of the field. Calculate the flux through the surface.

The electric flux through each surface of a cube is given below. Which surfaces of the cube does the electric field run parallel to? 

Φ₁ = 100 Nm² /C      Φ₄ = 0 Nm² /C

Φ₂ = 20 Nm² /C       Φ₅ = −40 Nm² /C

Φ₃ = 0 Nm² /C        Φ₆ = −80 Nm² /C

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.

(a) Find the magnitude of the electric flux through the sheet.

(b) Does the answer to part (a) depend on the shape of the sheet? Why or why not?