Activity: Gauss’s Law on Cylinders and Spheres

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Section 13.5 Activity: Gauss’s Law on Cylinders and Spheres

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

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

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

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