6.4 Conductors in electrostatic equilibrium  (Page 7/9)

A vector field $\overrightarrow{\text{E}}$ (not necessarily an electric field; note units) is given by $\overrightarrow{\text{E}}=3x^2\widehat{\text{k}}.$ Calculate ${\displaystyle {\int }_S\overrightarrow{\text{E}}·\widehat{\text{n}}da,}$ where S is the area shown below. Assume that $\widehat{\text{n}}=\widehat{\text{k}}.$

Repeat the preceding problem, with $\overrightarrow{\text{E}}=2x\widehat{i}+3x^2\widehat{\text{k}}.$

A circular area S is concentric with the origin, has radius a , and lies in the yz -plane. Calculate ${\displaystyle {\int }_S\overrightarrow{\text{E}}·\widehat{\text{n}}dA}$ for $\overrightarrow{\text{E}}=3z^2\widehat{i}.$

(a) Calculate the electric flux through the open hemispherical surface due to the electric field $\overrightarrow{\text{E}}=E_0\widehat{\text{k}}$ (see below). (b) If the hemisphere is rotated by $90\text{°}$ around the x -axis, what is the flux through it?

Suppose that the electric field of an isolated point charge were proportional to $1\text{/}{r}^{2+\sigma }$ rather than $1\text{/}{r}^2.$ Determine the flux that passes through the surface of a sphere of radius R centered at the charge. Would Gauss’s law remain valid?

The electric field in a region is given by $\overrightarrow{\text{E}}=a\text{/}(b+cx)\widehat{i},$ where $a=200\text{N}·\text{m/C,}$ $b=2.0\text{m,}$ and $c=2.0.$ What is the net charge enclosed by the shaded volume shown below?

Two equal and opposite charges of magnitude Q are located on the x -axis at the points + a and – a , as shown below. What is the net flux due to these charges through a square surface of side 2 a that lies in the yz -plane and is centered at the origin? ( Hint: Determine the flux due to each charge separately, then use the principle of superposition. You may be able to make a symmetry argument.)

A fellow student calculated the flux through the square for the system in the preceding problem and got 0. What went wrong?

A $10\text{cm}\times 10\text{cm}$ piece of aluminum foil of 0.1 mm thickness has a charge of $20\mu \text{C}$ that spreads on both wide side surfaces evenly. You may ignore the charges on the thin sides of the edges. (a) Find the charge density. (b) Find the electric field 1 cm from the center, assuming approximate planar symmetry.

Two $10\text{cm}\times 10\text{cm}$ pieces of aluminum foil of thickness 0.1 mm face each other with a separation of 5 mm. One of the foils has a charge of $\text{+30}\mu \text{C}$ and the other has $\text{−}30\mu \text{C}$ . (a) Find the charge density at all surfaces, i.e., on those facing each other and those facing away. (b) Find the electric field between the plates near the center assuming planar symmetry.

Two large copper plates facing each other have charge densities $\pm 4.0{\text{C/m}}^2$ on the surface facing the other plate, and zero in between the plates. Find the electric flux through a $3\text{cm}\times 4\text{cm}$ rectangular area between the plates, as shown below, for the following orientations of the area. (a) If the area is parallel to the plates, and (b) if the area is tilted $\theta =30\text{°}$ from the parallel direction. Note, this angle can also be $\theta =180\text{°}+30\text{°}.$