Charge is distributed within a solid sphere of radius $r_0$ in such a way that the charge density is a function of the radial position within the sphere of the form: $\rho_{\mathrm{E}}(r)=\rho_0\left(r / r_0\right)$. If the total charge within the sphere is $Q$ (and positive), what is the electric field everywhere within the sphere in terms of $Q, r_0$, and the radial position $r$ ?

A very long solid nonconducting cylinder of radius $R_0$ and length $\ell\left(R_0 \ll \ell\right)$ possesses a uniform volume charge density $\rho_{\mathrm{E}}\left(\mathrm{C} / \mathrm{m}^3\right)$. Determine the electric field at points (a) outside the cylinder $\left(R>R_0\right).$ Do only for points far from the ends and for which $R \ll \ell$.

A very long solid nonconducting cylinder of radius $R_1$ is uniformly charged with a volume charge density $\rho_{\mathrm{E}}$. It is surrounded by a concentric cylindrical tube of inner radius $R_2$ and outer radius $R_3$ and it too carries a uniform charge density $\rho_{\mathrm{E}}$. Determine the electric field as a function of the distance $R$ from the center of the cylinders for; (a) $0<R<R_1$. Assume the cylinders are very long compared to $R_3$.

A solid nonconducting sphere of radius $12.0 \mathrm{~cm}$ is made of two layers. The innermost section has a radius of $6.0 \mathrm{~cm}$ and a uniform charge density of $-4.0 \mathrm{C} / \mathrm{m}^3$. The outer layer has a uniform charge density of $+9.0 \mathrm{C} / \mathrm{m}^3$. Determine the electric field for $(a) 0<r<6.0 \mathrm{~cm}$.

A flat square sheet of thin aluminum foil, $25 \mathrm{~cm}$ on a side, carries a uniformly distributed $245 \mathrm{nC}$ charge. What, approximately, is the electric field $1.0 \mathrm{~cm}$ above the center of the sheet?

Suppose the nonconducting sphere has a spherical cavity of radius $r_1$ centered at the sphere's center. Assuming the charge $Q$ is distributed uniformly in the "shell" (between $r=r_1$ and $r=r_0$) and there is also a charge $-q$ at the center of the cavity. Determine the electric field for $(c) r>r_0$.

Suppose the nonconducting sphere has a spherical cavity of radius $r_1$ centered at the sphere's center. Assuming the charge $Q$ is distributed uniformly in the "shell" (between $r=r_1$ and $r=r_0$) and there is also a charge $-q$ at the center of the cavity. Determine the electric field for (b) $r_1<r<r_0$.

Suppose the nonconducting sphere has a spherical cavity of radius $r_1$ centered at the sphere's center. Assuming the charge $Q$ is distributed uniformly in the "shell" (between $r=r_1$ and $r=r_0$) and there is also a charge $-q$ at the center of the cavity. Determine the electric field for $(a) 0<r<r_1$.

A cube of side $\ell$ is placed in a uniform field $E_0$ with edges parallel to the field lines. What is the net flux through the cube?

In $\text{Fig. 29–20}a$, suppose that the emitter resistance is split into two resistors as shown in $\text{Fig. 29–20}b$. Calculate a. $A_\text{V}$. b. $V_\text{out}$.

A flat disk-like ring (inner radius $R_0$, outer radius $4 R_0$ ) is uniformly charged. In terms of the total charge $Q$, determine the electric field on the axis at points $\text{(b) } 75 R_0$ from the center of the ring. [Hint: The ring can be replaced with two oppositely charged superposed disks.]

A flat disk-like ring (inner radius $R_0$, outer radius $4 R_0$ ) is uniformly charged. In terms of the total charge $Q$, determine the electric field on the axis at points $\text{(a) } 0.25 R_0$ from the center of the ring. [Hint: The ring can be replaced with two oppositely charged superposed disks.]

Suppose the nonconducting sphere has a spherical cavity of radius $r_1$ centered at the sphere's center. Assuming the charge $Q$ is distributed uniformly in the "shell" (between $r=r_1$ and $r=r_0$ ), determine the electric field as a function of $r$ for

(c) $r>r_0$.

Suppose the nonconducting sphere has a spherical cavity of radius $r_1$ centered at the sphere's center. Assuming the charge $Q$ is distributed uniformly in the "shell" (between $r=r_1$ and $r=r_0$ ), determine the electric field as a function of $r$ for

(b) $r_1<r<r_0$.