24. Electric Force & Field; Gauss' Law

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

Where does the normal vector point for a spherical shell?

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 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.

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.

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.

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?

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?

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?

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 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?

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