## Find Force Law: Spiral Orbit

• assignment Differential Form of Gauss's Law

assignment Homework

##### Differential Form of Gauss's Law
Static Fields 2023 (6 years)

For an infinitesimally thin cylindrical shell of radius $b$ with uniform surface charge density $\sigma$, the electric field is zero for $s<b$ and $\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s$ for $s > b$. Use the differential form of Gauss' Law to find the charge density everywhere in space.

• assignment Find Force Law: Logarithmic Spiral Orbit

assignment Homework

##### Find Force Law: Logarithmic Spiral Orbit
Central Forces 2023 (3 years)

In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a $1/r^2$ force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.

Find the force law for a mass $\mu$, under the influence of a central-force field, that moves in a logarithmic spiral orbit given by $r = ke^{\alpha \phi}$, where $k$ and $\alpha$ are constants.

• accessibility_new Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

accessibility_new Kinesthetic

30 min.

##### Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
Quantum Fundamentals 2021

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.
• assignment Isolength and Isoforce Stretchability

assignment Homework

##### Isolength and Isoforce Stretchability
Energy and Entropy 2021 (2 years)

In class, you measured the isolength stretchability and the isoforce stretchability of your systems in the PDM. We found that for some systems these were very different, while for others they were identical.

Show with algebra (NOT experiment) that the ratio of isolength stretchability to isoforce stretchability is the same for both the left-hand side of the system and the right-hand side of the system. i.e.: \begin{align} \frac{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{x_R}}{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{F_R}} &= \frac{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{x_L}}{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{F_L}} \label{eq:ratios} \end{align}

##### Hint
You will need to make use of the cyclic chain rule: \begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{C} = -\left(\frac{\partial {A}}{\partial {C}}\right)_{B}\left(\frac{\partial {C}}{\partial {B}}\right)_{A} \end{align}
##### Hint
You will also need the ordinary chain rule: \begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{D} = \left(\frac{\partial {A}}{\partial {C}}\right)_{D}\left(\frac{\partial {C}}{\partial {B}}\right)_{D} \end{align}

• keyboard Kinetic energy

keyboard Computational Activity

120 min.

##### Kinetic energy
Computational Physics Lab II 2022

Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.
• assignment Charge on a Spiral

assignment Homework

##### Charge on a Spiral
Static Fields 2023 (2 years) A charged spiral in the $x,y$-plane has 6 turns from the origin out to a maximum radius $R$ , with $\phi$ increasing proportionally to the distance from the center of the spiral. Charge is distributed on the spiral so that the charge density increases linearly as the radial distance from the center increases. At the center of the spiral the linear charge density is $0~\frac{\textrm{C}}{\textrm{m}}$. At the end of the spiral, the linear charge density is $13~\frac{\textrm{C}}{\textrm{m}}$. What is the total charge on the spiral?
• assignment Energy, Entropy, and Probabilities

assignment Homework

##### Energy, Entropy, and Probabilities
Energy Entropy Probabilities Thermodynamic identity

The goal of this problem is to show that once we have maximized the entropy and found the microstate probabilities in terms of a Lagrange multiplier $\beta$, we can prove that $\beta=\frac1{kT}$ based on the statistical definitions of energy and entropy and the thermodynamic definition of temperature embodied in the thermodynamic identity.

The internal energy and entropy are each defined as a weighted average over microstates: \begin{align} U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i \end{align}: We saw in clase that the probability of each microstate can be given in terms of a Lagrange multiplier $\beta$ as \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} & Z &= \sum_i e^{-\beta E_i} \end{align} Put these probabilities into the above weighted averages in order to relate $U$ and $S$ to $\beta$. Then make use of the thermodynamic identity \begin{align} dU = TdS - pdV \end{align} to show that $\beta = \frac1{kT}$.

• assignment Energy, Entropy, and Probabilities

assignment Homework

##### Energy, Entropy, and Probabilities
Thermal and Statistical Physics 2020

The goal of this problem is to show that once we have maximized the entropy and found the microstate probabilities in terms of a Lagrange multiplier $\beta$, we can prove that $\beta=\frac1{kT}$ based on the statistical definitions of energy and entropy and the thermodynamic definition of temperature embodied in the thermodynamic identity.

The internal energy and entropy are each defined as a weighted average over microstates: \begin{align} U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i \end{align} We saw in clase that the probability of each microstate can be given in terms of a Lagrange multiplier $\beta$ as \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} & Z &= \sum_i e^{-\beta E_i} \end{align} Put these probabilities into the above weighted averages in order to relate $U$ and $S$ to $\beta$. Then make use of the thermodynamic identity \begin{align} dU = TdS - pdV \end{align} to show that $\beta = \frac1{kT}$.

• assignment Ring Function

assignment Homework

##### Ring Function
Central Forces 2023 (3 years) Consider the normalized wavefunction $\Phi\left(\phi\right)$ for a quantum mechanical particle of mass $\mu$ constrained to move on a circle of radius $r_0$, given by: $$\Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)}$$ where $N$ is the normalization constant.
1. Find $N$.

2. Plot this wave function.
3. Plot the probability density.
4. Find the probability that if you measured $L_z$ you would get $3\hbar$.
5. What is the expectation value of $L_z$ in this state?
• face Systems of Particles Lecture Notes

face Lecture

10 min.

##### Systems of Particles Lecture Notes
Central Forces 2023 (3 years)
• Central Forces 2023 (3 years)

In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a $1/r^2$ force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.

Find the force law for a central-force field that allows a particle to move in a spiral orbit given by $r=k\phi^2$, where $k$ is a constant.