- Flux Through a Cone
Find the flux through a cone (with no cap!) of height \(H\) and radius \(R\) due to the vector field \(\vec{F} = C\,z\,\hat{z}\). The cone has its vertex at the origin, opens upward symmetrically around the z-axis. Orient your area elements so that upward pointing fields contribute positive flux.
- Gauss's Law on a Cylindrical Shell
Consider an infinite positively charged (dielectric) cylindrical shell of inner radius \(a\)
and outer radius \(b\) with a cylindrically symmetric internal charge density
\(\rho(\vec r)=\alpha\, e^{(ks)^2}\).
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:
- \(s>b\)
- \(a<s<b\)
- \(s<a\)
- What dimensions do \(\alpha\) and \(k\) have?
- For \(\alpha=1\), \(k=1\), sketch the magnitude of the electric field as a
function of \(s\).
- Spherical Shell Step Functions
One way to write volume charge densities without using piecewise functions is to use step \((\Theta)\) or delta (\(\delta\)) functions.
Consider a spherical shell with charge density
\[\rho (\vec{r})=\alpha3e^{(k r)^3} \]
between the inner radius \(a\) and the outer radius \(b\). The charge density is zero everywhere else.
- What are the dimensions of the constants \(\alpha\) and \(k\)?
- By hand, sketch a graph the charge density as a function of \(r\) for \(\alpha > 0\) and \(k>0\) .
- Use step functions to write this charge density as a single function valid everywhere in space.
- Mass of a Slab
Determine the total mass of each of the slabs below.
-
A square slab of side length \(L\) with thickness \(h\), resting on a
table top at \(z=0\), whose mass density is given by
\begin{equation*}
\rho=A\pi\sin\left[\tfrac{\pi z}h\right].
\end{equation*}
-
A square slab of side length \(L\) with thickness \(h\), resting on a
table top at \(z=0\), whose mass density is given by
\begin{equation*}
\rho = 2A \Big[\Theta(z)-\Theta(z-h) \Big]
\end{equation*}
-
An infinitesimally thin square sheet of side length \(L\), resting on
a table top at \(z=0\), whose surface density is given by
\(\sigma=2Ah\).
-
An infinitesimally thin square sheet of side length \(L\), resting on
a table top at \(z=0\), whose mass density is given by
\(\rho=2Ah\,\delta(z)\).
-
What are the dimensions of \(A\)?
-
Write several sentences comparing your answers to the different cases above.