Circle Vector, Version 2

    • assignment Tetrahedron

      assignment Homework

      Tetrahedron
      Static Fields 2022 (5 years)

      Using a dot product, find the angle between any two line segments that join the center of a regular tetrahedron to its vertices. Hint: Think of the vertices of the tetrahedron as sitting at the vertices of a cube (at coordinates (0,0,0), (1,1,0), (1,0,1) and (0,1,1)---you may need to build a model and play with it to see how this works!)

    • assignment Divergence through a Prism

      assignment Homework

      Divergence through a Prism
      Static Fields 2022 (5 years)

      Consider the vector field \(\vec F=(x+2)\hat{x} +(z+2)\hat{z}\).

      1. Calculate the divergence of \(\vec F\).
      2. In which direction does the vector field \(\vec F\) point on the plane \(z=x\)? What is the value of \(\vec F\cdot \hat n\) on this plane where \(\hat n\) is the unit normal to the plane?
      3. Verify the divergence theorem for this vector field where the volume involved is drawn below. (“Verify” means calculate both sides of the divergence theorem, separately, for this example and show that they are the same.)

    • assignment Scattering

      assignment Homework

      Scattering
      Central Forces 2023 (3 years)

      Consider a very light particle of mass \(\mu\) scattering from a very heavy, stationary particle of mass \(M\). The force between the two particles is a repulsive Coulomb force \(\frac{k}{r^2}\). The impact parameter \(b\) in a scattering problem is defined to be the distance which would be the closest approach if there were no interaction (See Figure). The initial velocity (far from the scattering event) of the mass \(\mu\) is \(\vec v_0\). Answer the following questions about this situation in terms of \(k\), \(M\), \(\mu\), \(\vec v_0\), and \(b\). ()It is not necessarily wise to answer these questions in order.)

      1. What is the initial angular momentum of the system?
      2. What is the initial total energy of the system?
      3. What is the distance of closest approach \(r_{\rm{min}}\) with the interaction?
      4. Sketch the effective potential.
      5. What is the angular momentum at \(r_{\rm{min}}\)?
      6. What is the total energy of the system at \(r_{\rm{min}}\)?
      7. What is the radial component of the velocity at \(r_{\rm{min}}\)?
      8. What is the tangential component of the velocity at \(r_{\rm{min}}\)?
      9. What is the value of the effective potential at \(r_{\rm{min}}\)?
      10. For what values of the initial total energy are there bound orbits?
      11. Using your results above, write a short essay describing this type of scattering problem, at a level appropriate to share with another Paradigm student.

    • assignment Center of Mass for Two Uncoupled Particles

      assignment Homework

      Center of Mass for Two Uncoupled Particles
      Central Forces 2023 (3 years)

      Consider two particles of equal mass \(m\). The forces on the particles are \(\vec F_1=0\) and \(\vec F_2=F_0\hat{x}\). If the particles are initially at rest at the origin, find the position, velocity, and acceleration of the center of mass as functions of time. Solve this problem in two ways, with or without theorems about the center of mass motion. Write a short description comparing the two solutions.

    • assignment Directional Derivative

      assignment Homework

      Directional Derivative

      Gradient Sequence

      Static Fields 2022 (5 years)

      You are on a hike. The altitude nearby is described by the function \(f(x, y)= k x^{2}y\), where \(k=20 \mathrm{\frac{m}{km^3}}\) is a constant, \(x\) and \(y\) are east and north coordinates, respectively, with units of kilometers. You're standing at the spot \((3~\mathrm{km},2~\mathrm{km})\) and there is a cottage located at \((1~\mathrm{km}, 2~\mathrm{km})\). You drop your water bottle and the water spills out.

      1. Plot the function \(f(x, y)\) and also its level curves in your favorite plotting software. Include images of these graphs. Special note: If you use a computer program written by someone else, you must reference that appropriately.
      2. In which direction in space does the water flow?
      3. At the spot you're standing, what is the slope of the ground in the direction of the cottage?
      4. Does your result to part (c) make sense from the graph?

    • assignment Surface temperature of the Earth

      assignment Homework

      Surface temperature of the Earth
      Temperature Radiation Thermal and Statistical Physics 2020 Calculate the temperature of the surface of the Earth on the assumption that as a black body in thermal equilibrium it reradiates as much thermal radiation as it receives from the Sun. Assume also that the surface of the Earth is a constant temperature over the day-night cycle. Use the sun's surface temperature \(T_{\odot}=5800\text{K}\); and the sun's radius \(R_{\odot}=7\times 10^{10}\text{cm}\); and the Earth-Sun distance of \(1.5\times 10^{13}\text{cm}\).
    • assignment Cone Surface

      assignment Homework

      Cone Surface
      Static Fields 2022 (5 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.

    • assignment Gradient Point Charge

      assignment Homework

      Gradient Point Charge

      Gradient Sequence

      Consider the fields at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\).

      1. Write down an expression for the electrostatic potential \(V(\vec{r})\) at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\). (There is nothing to calculate here.)
      2. Write down an expression for the electric field \(\vec{E}(\vec{r})\) at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\). (There is nothing to calculate here.)
      3. Working in rectangular coordinates, compute the gradient of \(V\).
      4. Write several sentences comparing your answers to the last two questions.

    • group Energy and Angular Momentum for a Quantum Particle on a Ring

      group Small Group Activity

      30 min.

      Energy and Angular Momentum for a Quantum Particle on a Ring

      central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum energy hermitian operators probability superposition representations notations degeneracy

      Quantum Ring Sequence

      Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.
    • assignment Sum Shift

      assignment Homework

      Sum Shift
      Central Forces 2023 (3 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. \begin{equation} \sum_{n=0}^3 n \end{equation}
      2. \begin{equation} \sum_{n=1}^5 e^{in\phi} \end{equation}
      3. \begin{equation} \sum_{n=0}^{\infty} a_n n(n-1)z^{n-2} \end{equation}

  • Static Fields 2022 (5 years)

    Learn more about the geometry of \(\vert \vec{r}-\vec{r'}\vert\) in two dimensions.

    1. Make sketches of the following functions, by hand, on the same axes: \begin{align} y &= \sin x\\ y &= \sin(2+x) \end{align} Briefly describe the role that the number 2 plays in the shape of the second graph
    2. Make a sketch of the graph \begin{equation} \vert \vec{r} - \vec{a} \vert = 2 \end{equation}

      for each of the following values of \(\vec a\): \begin{align} \vec a &= \vec 0\\ \vec a &= 2 \hat x- 3 \hat y\\ \vec a &= \text{points due east and is 2 units long} \end{align}

    3. Derive a more familiar equation equivalent to \begin{equation} \vert \vec r - \vec a \vert = 2 \end{equation} for arbitrary \(\vec a\), by expanding \(\vec r\) and \(\vec a\) in rectangular coordinates. Simplify as much as possible. (Ok, ok, I know this is a terribly worded question. What do I mean by “more familiar"? What do I mean by “simplify as much as possible"? Why am I making you read my mind? Try it anyway. Real life is not full of carefully worded problems. Bonus points to anyone who can figure out a better way of wording the question that doesn't give the point away.)
    4. Write a brief description of the geometric meaning of the equation \begin{equation} \vert \vec r - \vec a \vert = 2 \end{equation}

  • Media & Figures
    • figures/sinx.png
    • figures/circ3_sAyVCEw.png