assignment Homework

Polar vs. Spherical Coordinates
Central Forces 2023 (3 years)

Show that the plane polar coordinates we have chosen are equivalent to spherical coordinates if we make the choices:

  1. The direction of \(z\) in spherical coordinates is the same as the direction of \(\vec L\).
  2. The \(\theta\) of spherical coordinates is chosen to be \(\pi/2\), so that the orbit is in the equatorial plane of spherical coordinates.

assignment Homework

Series Notation 2

Power Series Sequence (E&M)

Static Fields 2022 (5 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 Homework

Memorize \(d\vec{r}\)
Static Fields 2022 (2 years)

Write \(\vec{dr}\) in rectangular, cylindrical, and spherical coordinates.

  1. Rectangular: \begin{equation} \vec{dr}= \end{equation}
  2. Cylindrical: \begin{equation} \vec{dr}= \end{equation}
  3. Spherical: \begin{equation} \vec{dr}= \end{equation}

group Small Group Activity

30 min.

Black space capsule
Contemporary Challenges 2022 (3 years)

stefan-boltzmann blackbody radiation

In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.

assignment Homework

Extensive Internal Energy
Energy and Entropy 2021 (2 years)

Consider a system which has an internal energy \(U\) defined by: \begin{align} U &= \gamma V^\alpha S^\beta \end{align} where \(\alpha\), \(\beta\) and \(\gamma\) are constants. The internal energy is an extensive quantity. What constraint does this place on the values \(\alpha\) and \(\beta\) may have?

assignment Homework

Properties of Logarithms and Exponents
Logarithms Exponents Static Fields 2022
  1. Simplify the following expressions:
    1. \(\ln{x}+\ln{y}\)

    2. \(\ln{a}-\ln{b}\)

    3. \(2\ln{f}+3\ln{f}\)

    4. \(e^{m}e^{k}\)

    5. \(\frac{e^{c}}{e^{d}}\)

  2. Expand the following expressions:
    1. \(e^{(3h-j)}\)

    2. \(e^{2(c+w)}\)

    3. \(\ln{h/g}\)

    4. \(\ln(kT)\)

    5. \(\ln{\sqrt{\frac{q}{r}}}\)

assignment Homework

Spherical Shell Step Functions
step function charge density Static Fields 2022 (5 years)

One way to write volume charge densities without using piecewise functions is to use step \((\Theta)\) or \(\delta\) functions. If you need to review this, see the following link in the math-physics book: https://paradigms.oregonstate.edu

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. Use step functions to write this charge density as a single function valid everywhere in space.

face Lecture

5 min.

Wavelength of peak intensity
Contemporary Challenges 2022 (3 years)

Wein's displacement law blackbody radiation

This very short lecture introduces Wein's displacement law.

assignment_ind Small White Board Question

10 min.

Possible Worldlines
Theoretical Mechanics (4 years)

Special Relativity Spacetime Diagrams Worldlines Postulates of Relativity

Student consider several curves on a spacetime diagram and have to judge which curves could be worldlines for an object.

assignment Homework

Derivative of Fermi-Dirac function
Fermi-Dirac function Thermal and Statistical Physics 2020 Derivative of Fermi-Dirac function Show that the magnitude of the slope of the Fermi-Direc function \(f\) evaluated at the Fermi level \(\varepsilon =\mu\) is inversely proportional to its temperature. This means that at lower temperatures the Fermi-Dirac function becomes dramatically steeper.

assignment Homework

Mass Density
Static Fields 2022 (3 years) Consider a rod of length \(L\) lying on the \(z\)-axis. Find an algebraic expression for the mass density of the rod if the mass density at \(z=0\) is \(\lambda_0\) and at \(z=L\) is \(7\lambda_0\) and you know that the mass density increases
  • linearly;
  • like the square of the distance along the rod;
  • exponentially.

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 \(T\):
  1. Find the volume current density.
  2. Find the total current.

face Lecture

10 min.

Basics of heat engines
Contemporary Challenges 2022 (4 years) This brief lecture covers the basics of heat engines.

assignment Homework

Zapping With d 1
Energy and Entropy 2021 (2 years)

Find the differential of each of the following expressions; zap each of the following with \(d\):

  1. \[f=3x-5z^2+2xy\]

  2. \[g=\frac{c^{1/2}b}{a^2}\]

  3. \[h=\sin^2(\omega t)\]

  4. \[j=a^x\]

  5. \[k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)\]

assignment Homework

Flux through a Plane
Static Fields 2022 (3 years) Find the upward pointing flux of the vector field \(\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}} +\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}\) through the rectangle \(R\) with one edge along the \(y\) axis and the other in the \(xz\)-plane along the line \(z=x\), with \(0\le y\le2\) and \(0\le x\le3\).

assignment Homework

Circle Trig Complex
Complex Numbers Exponential Form Rectangular Form Polar Form Unit Circle Quantum Fundamentals 2022

Find the rectangular coordinates of the point where the angle \(\frac{5\pi}{3}\) meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)

assignment Homework

Central Force Definition
Central Forces 2023 (3 years)

Which of the following forces can be central forces? which cannot? If the force CAN be a central force, explain the circumstances that would allow it to be a central force.

  1. The force on a test mass \(m\) in a gravitational field \(\vec{g~}\), i.e. \(m\vec g\)
  2. The force on a test charge \(q\) in an electric field \(\vec E\), i.e. \(q\vec E\)
  3. The force on a test charge \(q\) moving at velocity \(\vec{v~}\) in a magnetic field \(\vec B\), i.e. \(q\vec v \times \vec B\)

assignment Homework

Normalization of Quantum States
Central Forces 2023 (3 years) Show that if a linear combination of ring energy eigenstates is normalized, then the coefficients must satisfy \begin{equation} \sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1 \end{equation}

assignment Homework

Static Fields 2022 (5 years)

Shown above is a two-dimensional vector field.

Determine whether the divergence at point A and at point C is positive, negative, or zero.

assignment Homework

Bottle in a Bottle
irreversible helium internal energy work first law Energy and Entropy 2021 (2 years)

The internal energy of helium gas at temperature \(T\) is to a very good approximation given by \begin{align} U &= \frac32 Nk_BT \end{align}

Consider a very irreversible process in which a small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated. What is the change in temperature when this process is complete? How much of the helium will remain in the small bottle?