*group*Flux through a Cone*group*Small Group Activity30 min.

##### Flux through a Cone

Static Fields 2021 (4 years) Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .*assignment*Cone Surface*assignment*Homework##### Cone Surface

Static Fields 2023 (6 years)- Find \(dA\) on the surface of an (open) cone in both cylindrical and spherical coordinates. Hint: Be smart about how you coordinatize the cone.
- Using integration, find the surface area of an (open) cone with height \(H\) and radius \(R\). Do this problem in both cylindrical and spherical coordinates.

*keyboard*Electric field for a waffle cone of charge*keyboard*Computational Activity120 min.

##### Electric field for a waffle cone of charge

Computational Physics Lab II 2022 Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.*assignment*Find Area/Volume from $d\vec{r}$*assignment*Homework##### Find Area/Volume from \(d\vec{r}\)

Static Fields 2023 (5 years)Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.

- Rectangular: \begin{align} dA&=\\ d\tau&= \end{align}
- Cylindrical: \begin{align} dA&=\\ d\tau&= \end{align}
- Spherical: \begin{align} dA&=\\ d\tau&= \end{align}

*assignment*Cube Charge*assignment*Homework##### Cube Charge

charge density Static Fields 2023 (6 years)- Charge is distributed throughout the volume of a dielectric cube with charge density \(\rho=\beta z^2\), where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
- On a different cube: Charge is distributed on the surface of a cube with charge density \(\sigma=\alpha z\) where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge on the cube? Don't forget about the top and bottom of the cube.

*assignment*Current in a Wire*assignment*Homework##### Current in a Wire

Static Fields 2023 (4 years) The current density in a cylindrical wire of radius \(R\) is given by \(\vec{J}(\vec{r})=\alpha s^3\cos^2\phi\,\hat{z}\). Find the total current in the wire.*assignment*Current from a Spinning Cylinder*assignment*Homework##### Current from a Spinning Cylinder

A solid cylinder with radius \(R\) and height \(H\) has its base on the \(x,y\)-plane and is symmetric around the \(z\)-axis. There is a fixed volume charge density on the cylinder \(\rho=\alpha z\). If the cylinder is spinning with period \(T\):- Find the volume current density.
- Find the total current.

*assignment*Einstein condensation temperature*assignment*Homework##### Einstein condensation temperature

Einstein condensation Density Thermal and Statistical Physics 2020**Einstein condensation temperature**Starting from the density of free particle orbitals per unit energy range \begin{align} \mathcal{D}(\varepsilon) = \frac{V}{4\pi^2}\left(\frac{2M}{\hbar^2}\right)^{\frac32}\varepsilon^{\frac12} \end{align} show that the lowest temperature at which the total number of atoms in excited states is equal to the total number of atoms is \begin{align} T_E &= \frac1{k_B} \frac{\hbar^2}{2M} \left( \frac{N}{V} \frac{4\pi^2}{\int_0^\infty\frac{\sqrt{\xi}}{e^\xi-1}d\xi} \right)^{\frac23} T_E &= \end{align} The infinite sum may be numerically evaluated to be 2.612. Note that the number derived by integrating over the density of states, since the density of states includes all the states*except*the ground state.**Note:**This problem is solved in the text itself. I intend to discuss Bose-Einstein condensation in class, but will not derive this result.*assignment*Energy of a relativistic Fermi gas*assignment*Homework##### Energy of a relativistic Fermi gas

Fermi gas Relativity Thermal and Statistical Physics 2020For electrons with an energy \(\varepsilon\gg mc^2\), where \(m\) is the mass of the electron, the energy is given by \(\varepsilon\approx pc\) where \(p\) is the momentum. For electrons in a cube of volume \(V=L^3\) the momentum takes the same values as for a non-relativistic particle in a box.

Show that in this extreme relativistic limit the Fermi energy of a gas of \(N\) electrons is given by \begin{align} \varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13} \end{align} where \(n\equiv \frac{N}{V}\) is the number density.

Show that the total energy of the ground state of the gas is \begin{align} U_0 &= \frac34 N\varepsilon_F \end{align}

*assignment*Potential energy of gas in gravitational field*assignment*Homework##### Potential energy of gas in gravitational field

Potential energy Heat capacity Thermal and Statistical Physics 2020 Consider a column of atoms each of mass \(M\) at temperature \(T\) in a uniform gravitational field \(g\). Find the thermal average potential energy per atom. The thermal average kinetic energy is independent of height. Find the total heat capacity per atom. The total heat capacity is the sum of contributions from the kinetic energy and from the potential energy. Take the zero of the gravitational energy at the bottom \(h=0\) of the column. Integrate from \(h=0\) to \(h=\infty\).*You may assume the gas is ideal.*-
Static Fields 2023 (6 years)
Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).