Choose one or more of the charge distributions given below: ($\alpha$ and $k$ are constants with appropriate dimensions.)

1. A positively charged (dielectric) spherical shell of inner radius $a$ and outer radius $b$ with a spherically symmetric internal charge density $\rho(\vec r)=\alpha\, r^3$.
2. A positively charged (dielectric) spherical shell of inner radius $a$ and outer radius $b$ with a spherically symmetric internal charge density $\rho(\vec r)=\alpha\, e^{(kr)^3}$.
3. A positively charged (dielectric) spherical shell of inner radius $a$ and outer radius $b$ with a spherically symmetric internal charge density $\rho(\vec r)=\alpha\, {1\over r^2}\, e^{kr}$.
4. An infinite positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density $\rho(\vec r)=\alpha\, r^3$.
5. An infinite positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density $\rho(\vec r)=\alpha\, e^{(kr)^2}$.
6. An infinite positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density $\rho(\vec r)=\alpha\, {1\over r}\, e^{kr}$.

For each chosen distribution, answer each of the following questions:

1. Use Gauss's Law and symmetry arguments to find the electric field at each of the three radii below:
   1. $r_1>b$
   2. $a<r_2<b$
   3. $r_3<a$
2. What dimensions do $\alpha$ and $k$ have?
3. For $\alpha=1$, $k=1$, sketch the magnitude of the electric field as a function of $r$.