format_list_numbered Sequence

Gradient Sequence
This sequence starts with an introduction to partial derivatives and continues through gradient. While some of the activities/problems are pure math, a number of other activities/problems are situated in the context of electrostatics. This sequence is intended to be used intermittently across multiple days or even weeks of a course or even multiple courses.

computer Mathematica Activity

30 min.

Visualising the Gradient
Static Fields 2022 (6 years)

Gradient Sequence

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.

accessibility_new Kinesthetic

10 min.

Acting Out the Gradient
Static Fields 2022 (6 years)

gradient vector fields electrostatics

Gradient Sequence

Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".

group Small Group Activity

30 min.

DELETE Navigating a Hill
Static Fields 2022 (5 years) In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.

group Small Group Activity

30 min.

Directional Derivatives
Vector Calculus I 2022

Directional derivatives

Gradient Sequence

This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.

assignment Homework

Gradient Practice

Gradient Sequence

Static Fields 2022 (4 years)

Find the gradient of each of the following functions:

  1. \begin{equation} f(x,y,z)=e^{(x+y)}+x^2 y^3 \ln \frac{x}{z} \end{equation}
  2. \begin{equation} \sigma(\theta,\phi)=\cos\theta \sin^2\phi \end{equation}
  3. \begin{equation} \rho(s,\phi,z)=(s+3z)^2\cos\phi \end{equation}

group Small Group Activity

30 min.

The Hillside
Vector Calculus I 2022

Gradient Sequence

Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).

group Small Group Activity

30 min.

The Hill
Vector Calculus II 2022 (4 years)

Gradient

Gradient Sequence

In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.

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.

assignment Homework

Contours

Gradient Sequence

Static Fields 2022 (6 years)

Shown below is a contour plot of a scalar field, \(\mu(x,y)\). Assume that \(x\) and \(y\) are measured in meters and that \(\mu\) is measured in kilograms. Four points are indicated on the plot.

  1. Determine \(\frac{\partial\mu}{\partial x}\) and \(\frac{\partial\mu}{\partial y}\) at each of the four points.
  2. On a printout of the figure, draw a qualitatively accurate vector at each point corresponding to the gradient of \(\mu(x,y)\) using your answers to part a above. How did you choose a scale for your vectors? Describe how the direction of the gradient vector is related to the contours on the plot and what property of the contour map is related to the magnitude of the gradient vector.
  3. Evaluate the gradient of \(h(x,y)=(x+1)^2\left(\frac{x}{2}-\frac{y}{3}\right)^3\) at the point \((x,y)=(3,-2)\).

group Small Group Activity

30 min.

Number of Paths

E&M Conservative Fields Surfaces

Student discuss how many paths can be found on a map of the vector fields \(\vec{F}\) for which the integral \(\int \vec{F}\cdot d\vec{r}\) is positive, negative, or zero. \(\vec{F}\) is conservative. They do a similar activity for the vector field \(\vec{G}\) which is not conservative.

assignment Homework

The Gradient for a Point Charge

Gradient Sequence

Static Fields 2022 (6 years)

The electrostatic potential due to a point charge at the origin is given by: \begin{equation} V=\frac{1}{4\pi\epsilon_0} \frac{q}{r} \end{equation}

  1. Find the electric field due to a point charge at the origin as a gradient in rectangular coordinates.
  2. Find the electric field due to a point charge at the origin as a gradient in spherical coordinates.
  3. Find the electric field due to a point charge at the origin as a gradient in cylindrical coordinates.

assignment Homework

Directional Derivative

Gradient Sequence

Static Fields 2022 (6 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 Homework

Electric Field of a Finite Line

Consider the finite line with a uniform charge density from class.

  1. Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
  2. Perform the integral to find the \(z\)-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the \(s\)-component as well!)

group Small Group Activity

30 min.

Work By An Electric Field (Contour Map)

E&M Path integrals

Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.

assignment Homework

Line Sources Using the Gradient

Gradient Sequence

Static Fields 2022 (6 years)
  1. Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: \begin{equation} V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right) \end{equation}

assignment Homework

Power from the Ocean
heat engine efficiency Energy and Entropy 2021 (2 years)

It has been proposed to use the thermal gradient of the ocean to drive a heat engine. Suppose that at a certain location the water temperature is \(22^\circ\)C at the ocean surface and \(4^{o}\)C at the ocean floor.

  1. What is the maximum possible efficiency of an engine operating between these two temperatures?

  2. If the engine is to produce 1 GW of electrical power, what minimum volume of water must be processed every second? Note that the specific heat capacity of water \(c_p = 4.2\) Jg\(^{-1}\)K\(^{-1}\) and the density of water is 1 g cm\(^{-3}\), and both are roughly constant over this temperature range.

assignment Homework

Nucleus in a Magnetic Field
Energy and Entropy 2021 (2 years)

Nuclei of a particular isotope species contained in a crystal have spin \(I=1\), and thus, \(m = \{+1,0,-1\}\). The interaction between the nuclear quadrupole moment and the gradient of the crystalline electric field produces a situation where the nucleus has the same energy, \(E=\varepsilon\), in the state \(m=+1\) and the state \(m=-1\), compared with an energy \(E=0\) in the state \(m=0\), i.e. each nucleus can be in one of 3 states, two of which have energy \(E=\varepsilon\) and one has energy \(E=0\).

  1. Find the Helmholtz free energy \(F = U-TS\) for a crystal containing \(N\) nuclei which do not interact with each other.

  2. Find an expression for the entropy as a function of temperature for this system. (Hint: use results of part a.)

  3. Indicate what your results predict for the entropy at the extremes of very high temperature and very low temperature.

assignment Homework

Effective Potentials: Graphical Version
Central Forces 2023 (2 years)

Consider a mass \(\mu\) in the potential shown in the graph below. You give the mass a push so that its initial angular momentum is \(\ell\ne 0\) for a given fixed value of \(\ell\).

  1. Give the definition of a central force system and briefly explain why this situation qualifies.
  2. Make a sketch of the graph of the effective potential for this situation.
  3. How should you push the puck to establish a circular orbit? (i.e. Characterize the initial position, direction of push, and strength of the push. You do NOT need to solve any equations.)
  4. BRIEFLY discuss the possible orbit shapes that can arise from this effective potential. Include a discussion of whether the orbits are open or closed, bound or unbound, etc. Make sure that you refer to your sketch of the effective potential in your discussions, mark any points of physical significance on the sketch, and describe the range of parameters relevant to each type of orbit. Include a discussion of the role of the total energy of the orbit.

assignment Homework

Linear Quadrupole (w/ series)

Power Series Sequence (E&M)

Static Fields 2022 (6 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.