assignment Homework

Cross Triangle
Static Fields 2022 (5 years)

Use the cross product to find the components of the unit vector \(\mathbf{\boldsymbol{\hat n}}\) perpendicular to the plane shown in the figure below, i.e.  the plane joining the points \(\{(1,0,0),(0,1,0),(0,0,1)\}\).

group Small Group Activity

5 min.

Acting Out Flux
Static Fields 2022 (3 years)

flux electrostatics vector fields

Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.

group Small Group Activity

30 min.

Visualization of Divergence
Vector Calculus II 2022 (8 years) Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the definition of the divergence of a vector field at a point: The divergence of a vector field at a point is flux per unit volume through an infinitesimal box surrounding that point. Optionally, students can use a Mathematica notebook to verify their predictions.

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

Curl
Static Fields 2022 (5 years)

Shown above is a two-dimensional cross-section of a vector field. All the parallel cross-sections of this field look exactly the same. Determine the direction of the curl at points A, B, and C.

assignment Homework

Vector Sketch (Rectangular Coordinates)
vector fields Static Fields 2022 (3 years) Sketch each of the vector fields below.
  1. \(\boldsymbol{\vec F} =-y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}\)
  2. \(\boldsymbol{\vec G} = x\,\boldsymbol{\hat x} + y\,\boldsymbol{\hat y}\)
  3. \(\boldsymbol{\vec H} = y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}\)

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

Electric Field from a Rod
Static Fields 2022 (4 years) Consider a thin charged rod of length \(L\) standing along the \(z\)-axis with the bottom end on the \(xy\)-plane. The charge density \(\lambda\) is constant. Find the electric field at the point \((0,0,2L)\).

group Small Group Activity

30 min.

Vector Integrals (Contour Map)

E&M Path integrals

assignment_ind Small White Board Question

10 min.

Vector Differential--Rectangular
Static Fields 2022 (7 years)

vector differential rectangular coordinates math

Integration Sequence

In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

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 Homework

Vector Sketch (Curvilinear Coordinates)
Static Fields 2022 Sketch each of the vector fields below.
  1. \(\boldsymbol{\vec K}=s\,\boldsymbol{\hat s}\)
  2. \(\boldsymbol{\vec L}=\frac1s\boldsymbol{\hat\phi}\)
  3. \(\boldsymbol{\vec M}=\sin\phi\,\boldsymbol{\hat s}\)
  4. \(\boldsymbol{\vec N}=\sin(2\pi s)\,\boldsymbol{\hat\phi}\)

assignment Homework

Divergence
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.

computer Mathematica Activity

30 min.

Visualising the Gradient
Static Fields 2022 (5 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 (5 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".

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!)

group Small Group Activity

30 min.

Electric Field Due to a Ring of Charge
Static Fields 2022 (7 years)

coulomb's law electric field charge ring symmetry integral power series superposition

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

assignment Homework

Contours

Gradient Sequence

Static Fields 2022 (5 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 \((x,y)=(3,-2)\).

assignment Homework

Line Sources Using Coulomb's Law
Static Fields 2022 (5 years)
  1. Find the electric field around a finite, uniformly charged, straight rod, at a point a distance \(s\) straight out from the midpoint, starting from Coulomb's Law.
  2. Find the electric field around an infinite, uniformly charged, straight rod, starting from the result for a finite rod.

accessibility_new Kinesthetic

10 min.

Curvilinear Basis Vectors
Static Fields 2022 (8 years)

symmetry curvilinear coordinate systems basis vectors

Curvilinear Coordinate Sequence

Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).