Student handout: Navigating a Hill

AIMS Maxwell AIMS 21
  • group Small Group Activity schedule 30 min. build Tabletop Whiteboard with markers,Computers with Maple (optional),A handout for each student description Student handout (PDF)
What students learn
  • The gradient is perpendicular to the level curves.
  • The gradient is a local quantity, i.e. it only depends on the values of the function at infinitesimally nearby points.
  • Although students learn to chant that "the gradient points uphill," the gradient does not point to the top of the hill.
  • The gradient path is not the shortest path between two points.
  • accessibility_new Acting Out the Gradient

    accessibility_new Kinesthetic

    10 min.

    Acting Out the Gradient
    AIMS Maxwell AIMS 21 Static Fields Winter 2021

    gradient vector fields electrostatics

    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".
  • assignment The Gradient for a Point Charge

    assignment Homework

    The Gradient for a Point Charge
    AIMS Maxwell AIMS 21 Static Fields Winter 2021

    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 Directional Derivative

    assignment Homework

    Directional Derivative
    AIMS Maxwell AIMS 21 Static Fields Winter 2021

    Imagine you're standing on a landscape with a local topography 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 \((1~\mathrm{km}, 2~\mathrm{km})\). At the spot you're standing, what is the slope of the ground in the direction of the cottage? Plot the function \(f(x, y)\) and also its level curves in your favorite plotting software. Does your result makes sense from the graph?

  • assignment Gradient Practice

    assignment Homework

    Gradient Practice
    AIMS Maxwell AIMS 21

    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}

  • assignment Contours

    assignment Homework

    Contours
    AIMS Maxwell AIMS 21 Static Fields Winter 2021

    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 \((x,y)=(3,-2)\).
    4. A contour map for a different function is shown above. On a printout of this contour map, sketch a field vector map of the gradient of this function (sketch vectors for at least 10 different points). The direction and magnitude of your vectors should be qualitatively accurate, but do not calculate the gradient for this function.

  • assignment Electric Field of a Finite Line

    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 Number of Paths

    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 Line Sources Using the Gradient

    assignment Homework

    Line Sources Using the Gradient
    AIMS Maxwell AIMS 21 Static Fields Winter 2021
    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}

  • group Work By An Electric Field (Contour Map)

    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 Vectors

    assignment Homework

    Vectors
    vector geometry AIMS Maxwell AIMS 21

    Let \begin{align} \boldsymbol{\vec a} &= \boldsymbol{\hat x}-3\boldsymbol{\hat y}-\boldsymbol{\hat z}\\ \boldsymbol{\vec b} &= \boldsymbol{\hat x}+\boldsymbol{\hat y}+2\boldsymbol{\hat z}\\ {\boldsymbol{\vec c}} &= -2\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z}\\ \boldsymbol{\vec d} &= -\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z} \end{align}

    Which pairs (if any) of these vectors

    1. Are perpendicular?
    2. Are parallel?
    3. Have an angle less than \(\pi/2\) between them?
    4. Have an angle of more than \(\pi/2\) between them?

Suppose you are standing on a hill. You have a topographic map, which uses rectangular coordinates \((x,y)\) measured in miles. Your global positioning system says your present location is at one of the following points (pick one): \[A:(1,4),\quad B:(4,-9),\quad C:(-4,9),\quad D:(1,-4),\quad E:(2,0),\quad F:(0,3)\] Your guidebook tells you that the height \(h\) of the hill in feet above sea level is given by \[h=a-bx^2-cy^2\] where \(a=5000~\mathrm{ft}\), \(b=30\,\mathrm{\frac{ft}{mi^2}}\), and \(c=10\,\mathrm{\frac{ft}{mi^2}}\).

  1. Starting at your present location, in what map direction (2-dimensional unit vector) do you need to go in order to climb the hill as steeply as possible? Draw this vector on your topographic map.
  2. How steep is the hill if you start at your present location and go in this compass direction? Draw a picture which shows the slope of the hill at your present location.


Learning Outcomes