Figure shows a section of a long, thin-walled metal tube of radius $R=2.50 \mathrm{cm},$ with a charge per unit length of $\lambda =2.00\times {10}^{-8} \mathrm{C} {\mathrm{m}}^{-1}$. What is the magnitude $E$ of the electric field at radial distance (a) $r=\frac{R}{2.00}$ and (b) $r=2.00 R?$ (c) Graph $E$ versus $r$ for the range $r=0$ to $2.00 R$.

 

The cube in figure has edge length $1.40 \mathrm{m}$ and is oriented as shown in a region of uniform electric field. Find the electric flux through the right face if the electric field, in newtons per coulomb, is given by (a) $19.0\hat{\mathrm{i}},$ (b)$-2.00\hat{\mathrm{j}},$  and (c) $-20.0\hat{\mathrm{i}}+4.00\hat{\mathrm{k}}$  (d) What is the total flux through the cube for each field? (e) What is the total flux if the edge length is doubled?

At each point on the surface of the cube shown in the figure, the electric field is parallel to the $z$ axis. The length of each edge of the cube is $4.0 \mathrm{m}$. On the top face of the cube the field of $\overrightarrow{E}=-34 \hat{\mathrm{k}}\mathrm{N} {\mathrm{C}}^{-1}$ and on the bottom face it is $\overrightarrow{E}=+20 \hat{\mathrm{k}}\mathrm{N} {\mathrm{C}}^{-1}$ . Determine the net charge contained within the cube.

Figure shows a Gaussian surface in the shape of a cube with edge length $5.60 \mathrm{m}$. What are (a) the net flux $\Phi$ through the surface and (b) the net charge $q_{\mathrm{enc}}$ enclosed by the surface if $\overrightarrow{E}=\left(3.00\mathrm{y}\hat{\mathrm{j}}\right) \mathrm{N} {\mathrm{C}}^{-1}$ with $y$ in meters? What are (c) $\varphi$ and (d) $q_{\mathrm{enc}}$ if $\overrightarrow{E}=[-17.0\hat{\mathrm{i}}+\left(6.00+3.00y\right)\hat{\mathrm{j}} ] \mathrm{N} {\mathrm{C}}^{-1}?$

In the figure below, a proton is at a distance $d/2$ directly above the center $r$ of a square of side $d.$ What is the magnitude of the electric flux through the square? (Think of the square as one of the faces of a cube with edge $d.$)

The figure shows cross sections through two large, parallel, nonconducting sheets with identical distributions of positive charge with surface charge density $\mathit{\text{σ}}=2.31\times {10}^{-22} \mathrm{C} {\mathrm{m}}^{-2}$. In unit-vector notation, what is $\overrightarrow{E}$ at points (a) above the sheets, (b) between them, and (c) below them?