## Series Notation 1

• This problem is used in the following sequences
• assignment Series Notation 2

assignment Homework

##### Series Notation 2

Power Series Sequence (E&M)

Static Fields 2022 (4 years)

Write (a good guess for) the following series using sigma $\left(\sum\right)$ notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)

1. $1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots$

2. $\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots$

• assignment Mass of a Slab

assignment Homework

##### Mass of a Slab
Static Fields 2022 (4 years)

Determine the total mass of each of the slabs below.

1. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by $$\rho=A\pi\sin(\pi z/h).$$
2. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by $$\rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big)$$
3. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose surface density is given by $\sigma=2Ah$.
4. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose mass density is given by $\rho=2Ah\,\delta(z)$.
5. What are the dimensions of $A$?
6. Write several sentences comparing your answers to the different cases above.

• assignment Total Charge

assignment Homework

##### Total Charge
charge density curvilinear coordinates

Integration Sequence

Static Fields 2022 (4 years)

For each case below, find the total charge.

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

• group Magnetic Field Due to a Spinning Ring of Charge

group Small Group Activity

30 min.

##### Magnetic Field Due to a Spinning Ring of Charge
Static Fields 2022 (5 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use the Biot-Savart law $\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}$ to find an integral expression for the magnetic field, $\vec{B}(\vec{r})$, due to a spinning ring of charge.

In an optional extension, students find a series expansion for $\vec{B}(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• group Magnetic Vector Potential Due to a Spinning Charged Ring

group Small Group Activity

30 min.

##### Magnetic Vector Potential Due to a Spinning Charged Ring
Static Fields 2022 (4 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use the superposition principle $\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}$ to find an integral expression for the magnetic vector potential, $\vec{A}(\vec{r})$, due to a spinning ring of charge.

In an optional extension, students find a series expansion for $\vec{A}(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• assignment Linear Quadrupole (w/ series)

assignment Homework

Power Series Sequence (E&M)

Static Fields 2022 (4 years)

Consider a collection of three charges arranged in a line along the $z$-axis: charges $+Q$ at $z=\pm D$ and charge $-2Q$ at $z=0$.

1. Find the electrostatic potential at a point $\vec{r}$ in the $xy$-plane at a distance $s$ from the center of the quadrupole. The formula for the electrostatic potential $V$ at a point $\vec{r}$ due to a charge $Q$ at the point $\vec{r'}$ is given by: $V(\vec{r})=\frac{1}{4\pi\epsilon_0} \frac{Q}{\vert \vec{r}-\vec{r'}\vert}$ Electrostatic potentials satisfy the superposition principle.
2. Assume $s\gg D$. Find the first two non-zero terms of a power series expansion to the electrostatic potential you found in the first part of this problem.

• assignment Sum Shift

assignment Homework

##### Sum Shift
Central Forces 2022 (2 years)

In each of the following sums, shift the index $n\rightarrow n+2$. Don't forget to shift the limits of the sum as well. Then write out all of the terms in the sum (if the sum has a finite number of terms) or the first five terms in the sum (if the sum has an infinite number of terms) and convince yourself that the two different expressions for each sum are the same:

1. $$\sum_{n=0}^3 n$$
2. $$\sum_{n=1}^5 e^{in\phi}$$
3. $$\sum_{n=0}^{\infty} a_n n(n-1)z^{n-2}$$

• keyboard Sinusoidal basis set

keyboard Computational Activity

120 min.

##### Sinusoidal basis set
Computational Physics Lab II 2022 (2 years)

Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.
• assignment Gibbs sum for a two level system

assignment Homework

##### Gibbs sum for a two level system
Gibbs sum Microstate Thermal average energy Thermal and Statistical Physics 2020
1. Consider a system that may be unoccupied with energy zero, or occupied by one particle in either of two states, one of energy zero and one of energy $\varepsilon$. Find the Gibbs sum for this system is in terms of the activity $\lambda\equiv e^{\beta\mu}$. Note that the system can hold a maximum of one particle.

2. Solve for the thermal average occupancy of the system in terms of $\lambda$.

3. Show that the thermal average occupancy of the state at energy $\varepsilon$ is \begin{align} \langle N(\varepsilon)\rangle = \frac{\lambda e^{-\frac{\varepsilon}{kT}}}{\mathcal{Z}} \end{align}

4. Find an expression for the thermal average energy of the system.

5. Allow the possibility that the orbitals at $0$ and at $\varepsilon$ may each be occupied each by one particle at the same time; Show that \begin{align} \mathcal{Z} &= 1 + \lambda + \lambda e^{-\frac{\varepsilon}{kT}} + \lambda^2 e^{-\frac{\varepsilon}{kT}} \\ &= (1+\lambda)\left(1+e^{-\frac{\varepsilon}{kT}}\right) \end{align} Because $\mathcal{Z}$ can be factored as shown, we have in effect two independent systems.

• face Entropy and Temperature

face Lecture

120 min.

##### Entropy and Temperature
Thermal and Statistical Physics 2020

These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.
• Static Fields 2022 (4 years)

Write out the first four nonzero terms in the series:

1. $\sum\limits_{n=0}^\infty \frac{1}{n!}$

2. $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}$
3. $$\sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}}$$