Instructions for 2022: You will need to complete this assignment in a 15 minute appointment on Zoom or in person with one of the members of the teaching team between 1/21 and 10 pm on 1/26. Here is a link to a sign-up page.
You are required to watch a sample video for how to make symmetry arguments here. As demonstrated in the video you should bring with you to the meeting a cylinder, an observer, and a vector.
Use good symmetry arguments to find the possible direction for
the electric field due to a charged wire. Also, use good symmetry
arguments to find the possible functional dependence of the electric field
due to a charged wire. Rather than writing this up to turn in, you
should find a member of the teaching team
and make the arguments to them verbally.
Found in: Static Fields, AIMS Maxwell, Problem-Solving, None course(s)
Show that
\begin{align}
f(\mu+\delta) &= 1 - f(\mu-\delta)
\end{align}
This means that the probability that an orbital above the
Fermi level is occupied is equal to the probability an orbital
the same distance below the Fermi level being empty. These unoccupied orbitals are called holes.
SymmetryOrbitals Found in: Thermal and Statistical Physics course(s)
Solve your assigned system of equations using any algebraic method. Show you work and be ready to explain how you solved it.
Also graph the system of equations and show how the solution appears on your graph. You may use graphing technology such as Desmos.
Group Roles
Facilitator: Read the directions out loud and check whether everyone understands each other.
“How should we start?” “How do you know?”
Team Captain: Help your team members step up and step back.
“How do you know?” “What do you think?”
Resource Manager: Help your group get unstuck.
“Is this working?” “What else could we try?” “Should we ask a team question?”
Recorder/Reporter: Be prepared to share out in the whole class discussion.
“How should I explain...?”
Problems
\[y=-3x\\4x+y=2\]
\[y=7x-5\\2x+y=13\]
\[x=-5y-4\\x-4y=23\]
\[x+y=10\\y=x-4\]
\[y=5-x\\4x+2y=10\]
\[3x+5y=23\\y=x+3\]
\[y=-x-2\\2x+3y=-9\]
\[y=2x-3\\-2x+y=1\]
\[x=\frac{1}{2}y+\frac{1}{2}\\2x+y=-1\]
\[a=2b+4\\b-2a=16\]
\[y=3-2x\\4x+2y=6\]
\[y=x+1\\x-y=1\]
(Adapted from CPM Core Connections)
Whole Class Directions
Each group will share out how you solved your system of equations.
Listen to each group and think about similarities and differences.
Ask questions about anything you do not understand or you disagree with.
You do not need to write anything during the whole class discussion, but you will have an exit ticket to see what you learned from the discussion.
Exit Ticket: Systems of Equations Compare and Contrast
Sheila missed class today. She tried to solve Problem 8 on her own, but she thinks she made a mistake because -3 does not equal 1.
\begin{align}
&y=2x-3\\
&-2x+y=1
\end{align}
\begin{align}
&-2x+(2x-3)=1\\
&-2x+2x-3=1\\
&0-3=1\\
&-3=1
\end{align}
Explain to Sheila what happened, using as much detail as possible to help her understand this type of problem.
Introduction
This Compare and Contrast activity is based on the College Preparatory Mathematics (CPM) Core Connections Algebra Parent Guide with Extra Practice, freely available here. CPM is a problem based curriculum with many conceptual problems for students to work on in small groups in class. The parent guide provides examples, exercises, and solutions for students to work alone and/or with parent support if they miss class or need extra practice. As such, the parent guide is one aspect of the CPM curriculum most focused on practice of procedures. The attached problem set is copied exactly from the CPM Parent Guide; the surrounding student instructions were written by Alyssa Sayavedra.
Special Cases of note
Problems 8 and 12 have no solution while Problem 11 has infinite solutions. It is important to include these problems, but be prepared for small groups to get tripped up by them. Many students, when solving equations, expect the “answer” to be a value. They may struggle to interpret an equation that is always or never true.
All other problems have one solution with integer coordinates.
Some problems in this set are easier than others. If any group finishes early, they can be encouraged to complete a second problem. Problem 1 is the most straightforward since y is equal to only one term. The next easiest problems are 2, 3, 4, and 8, because they do not require distribution after substitution.
Problems 1, 2, 3, 4, 5, 8, 9 and 11 can be solved using the Equal Values Method without introducing new fractions. The Equal Values Method is a variant of substitution in which students solve both equations for the same variable, then set the equations equal to each other, resulting in a single equation in one variable. This method is easier for many students because it results in a simpler one variable equation and is less prone to distribution errors. But it is usually not worth introducing fractions into the problem in order to use this method.
Problems 5, 9 and 11 can be simplified by either multiplying or dividing an entire equation by 2. It is unusual for students to think of this strategy at this stage, but it can be a helpful preview of the elimination method. This method also removes the fractions in Problem 9.
Small variations in notation can easily trip students up. Problem 10 uses a and b instead of x and y. Problems 3 and 9 have one equation solved for x instead of y. Problems 4 and 6 have the second equation solved for y instead of the first. Do not be surprised if some students still solve the first equation for y and plug it into the second.
Suggestions for Facilitating Small Group Work
Remind students of class norms for productive and respectful group work. Assign one problem to each group, including at least problems 2, 4, 6, 8, 11 and 12. Walk once or twice around the class within the first five minutes to make sure all small groups understand how to get started and are making progress. Make sure students understand the directions and have started to dig into the mathematics, but avoid giving strategic suggestions at this stage. The purpose of the small group time is for students to wrestle with the tricky bits of one problem. If a group chooses an inefficient strategy or makes an error, monitor their frustration level, but try to allow them to pursue it in some detail before suggesting there may be an easier method. The first 3 questions (from Schoenfeld) assist students with metacognitive monitoring of their own problem solving process. Whenever possible, allow students to check their own work using graphing technology and/or substitution of their answers rather than checking it for them.
Some good questions to ask groups during this time are:
“What are you doing?”
“Why are you doing that?”
“Is it working?”
“Are you done?”
“Have you found values for all the unknowns?”
“How could you check your work?”
“Can you graph the problem to check your work?”
“Can you substitute these numbers back in to check your work?”
“What would you expect to see on the graph?”
Suggestions for Facilitating Whole Class Discussion
Remind students of their norms for active listening during presentations, respect for presenters and treating mistakes as learning opportunities. Ask the reporters from at least 4-6 groups to share out their work (the reporter role should rotate regularly, even every class period). If not all groups will present, give priority to students or groups who present less often but who have done excellent work, to groups that have tried innovative strategies or made important revisions, and to the most important special cases. When sequencing the presentations, start with easier and/or typical examples. Often, it should work well to simply present the examples you choose in numerical order. Close with an exit ticket like “Explain one way you revised your work or thinking today” or “Use Jorge's method to solve this new problem.” You can also create an exit ticket in advance, such as the one attached.
Consider a system of \(n\) different masses \(m_i\), interacting with each other and being acted on by external forces. We can write Newton's second law for the positions \(\vec{r}_i\) of each of these masses with respect to a fixed origin \(\cal{O}\), thereby obtaining a system of equations governing the motion of the masses.
\begin{align}
m_1 \frac{d^2\, \vec{r}_1}{dt^2}
&=\vec{F}_1+\;\; 0\;\, +\vec{f}_{12}+\vec{f}_{13}+\;\,\dots\;\, +\vec{f}_{1n}\nonumber\\
m_2 \frac{d^2\, \vec{r}_2}{dt^2}
&=\vec{F}_2+\vec{f}_{21}+\;\; 0\;\, +\vec{f}_{23}+\;\,\dots\;\, +\vec{f}_{2n}
\label{NewtonSystem}\\
\vdots\nonumber\\
m_n \frac{d^2\, \vec{r}_n}{dt^2}
&=\vec{F}_n+\vec{f}_{n1}+\vec{f}_{n2}+\dots+\vec{f}_{n(n-1)}+0\quad\nonumber
\end{align}
Here, we have chosen the notation \(\vec{F}_i\) for the net external forces acting on mass \(m_i\) and \(\vec{f}_{ij}\) for the internal force of mass \(m_j\) acting on \(m_i\).
In general, each internal force \(\vec{f}_{ij}\) will depend on the positions of the particles \(\vec{r}_i\) and \(\vec{r}_j\) in some complicated way, making
\((\ref{NewtonSystem})\),
a set of coupled differential equations. To solve
\((\ref{NewtonSystem})\),
we first need to decouple the differential equations, i.e. find an equivalent set of differential equations in which each equation contains only one variable.
The weak form of Newton's third law states that the force \(\vec{f}_{12}\) of \(m_2\) on \(m_1\) is equal and opposite to the force
\(\vec{f}_{21}\) of \(m_1\) on \(m_2\). We see that each internal force appears twice in the system of equations \((\ref{NewtonSystem})\),
once with a positive sign and once with a negative sign. Therefore, if we add all of the equations
together, the internal forces will all cancel, leaving:
\begin{equation}
\sum_{i=1}^n m_i \frac{d^2 \vec{r}_i}{dt^2}
=\sum_{i=1}^n\vec{F}_i\label{NewtonCOM}
\end{equation}
Notice what a surprising equation
\((\ref{NewtonCOM})\)
is. The right-hand side directs us to add up all of the external forces, each of which acts on a different mass; something you were taught never to do in introductory physics.
The left-hand side of
\((\ref{NewtonCOM})\)
directs us to add up (the second derivatives of) \(n\) “weighted" position vectors pointing from the origin to different masses. We can simplify the left-hand side of
\((\ref{NewtonCOM})\)
if we multiply and divide by the total mass \(M=m_1+m_2+\dots+m_n\) and use the linearity of differentiation to “factor out” the derivative operator:
\begin{align}
\sum_{i=1}^n m_i \frac{d^2 \vec{r}_i}{dt^2}
&=M\frac{d^2}{dt^2}
\left(\sum_{i=1}^n \frac{m_i}{M}\, \vec{r}_i\right)\label{CenterOfMass1}\\
&=M\frac{d^2 \vec{R}_{cm}}{dt^2}\label{CenterOfMass2}
\end{align}
We recognize (or define) the quantity in the parentheses on the right-hand side of
\((\ref{CenterOfMass1})\)
as the position vector \(\vec{R}_{cm}\) from the origin to the “center of mass” of the system of particles, i.e.
\begin{equation}
\vec{R}_{cm}=\sum_{i=1}^n\frac{m_i}{M}\, \vec{r}_i\label{CenterOfMass3}
\end{equation}
With these simplifications, equation (\ref{NewtonCOM}) becomes:
\begin{equation}
M \frac{d^2 \vec{R}_{cm}}{dt^2}
=\sum_{i=1}^n\vec{F}_i\label{NewtonCOM2}
\end{equation}
which has the form of Newton's 2nd Law for a fictitious particle with mass \(M\) sitting at the center of mass of the system of particles and acted on by all of the external forces from the original system.
We can define the momentum of the center of mass as the total mass times the time derivative of the position of the center of mass:
\begin{equation}
\vec{P}_{cm}=M\frac{d\vec{R}_{cm}}{dt}
\end{equation}
If there are no external forces acting, then the acceleration of the center of mass is zero and the momentum of the center of mass is constant in time (conserved).
\begin{equation}
M\frac{d^2 \vec{R}_{cm}}{dt^2}=\frac{d\vec{P}_{cm}}{dt}=0
\label{MomentumConservation}
\end{equation}
Notice that the entire discussion above applies even if all of the internal forces are zero \(\vec{f}_{ij}=0\), i.e. none of the particles have any way of knowing that the others are even present. Such particles are called non-interacting. The position of the center of mass of the system will still move according to equation \((\ref{NewtonCOM2})\).
Find the angle between the diagonal of a cube (connecting opposite corners) and the diagonal of one of its faces (connecting opposite corners of one square face).
Found in: Vector Calculus I, Surfaces/Bridge Workshop, Problem-Solving course(s)
This small group activity is designed to help students visual the process of chopping, adding, and multiplying in single integrals.
Students work in small groups to determine the volume of a cylinder in as many ways as possible.
The whole class wrap-up discussion emphasizes the equivalence of different ways of chopping the cylinder.
A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula.
\[\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}\]
Consider the rectangle in the first quadrant of the \(xy\)-plane as in the figure with thick black lines.
Label the bottom horizontal edge of the rectangle \(y=c\).
Label the sides of the rectangle \(\Delta x\) and \(\Delta y\).
What is the area of the rectangle?
There are also 2 rectangles whose base is the \(x\)-axis, the larger of which
contains both the smaller and the original rectangle. Express the area of the
original rectangle as the difference between the areas of these 2 rectangles.
On the grid below, draw any simple, closed, piecewise smooth curve \(C\), all of
whose segments \(C_i\) are parallel either to the \(x\)-axis or to the \(y\)-axis.
Your curve should not be a rectangle. Pick an origin and label it,
and assume that each square is a unit square.
Compute the area of the region \(D\) inside \(C\) by counting the number of
squares inside \(C\).
Evaluate the line integral
\(\displaystyle \oint_C y\,\boldsymbol{\hat{x}}\cdot d\boldsymbol{\vec{r}}\)
by noticing that along each segment either \(x\) or \(y\) is constant, so that the
integral is equal to
\(\sum_{C_i} y\,\Delta x\).
Can you relate this to Problem 1?
Are your answers to the preceding two calculations the same?
Would any of your answers change if you replaced \(y\,\boldsymbol{\hat{x}}\) by \(x\,\boldsymbol{\hat{y}}\) in part
(b)?
Main ideas
Understanding different ways of expressing area using integration.
Concrete example of Area Corollary to Green's/Stokes' Theorem.
We originally used this activity after covering Green's Theorem; we now skip
Green's Theorem and do this activity shortly before Stokes' Theorem.
Prerequisites
Familiarity with line integrals.
Green's Theorem is not a prerequisite!
Warmup
The first problem is a good warmup.
Props
whiteboards and pens
a planimeter if available
Wrapup
Emphasize the magic -- finding area by walking around the boundary!
Point out that this works for any closed curve, not just the rectangular
regions considered here.
Demonstrate or describe a planimeter, used for instance to measure the area of
a region on a map by tracing the boundary.
Details
In the Classroom
Make sure students use a consistent orientation on their path.
Make sure students explicitly include all segments of their path, including
those which obviously yield zero.
Students in a given group should all use the same curve.
Students should be discouraged from drawing a curve whose longest side is
along a coordinate axis.
Students may need to be reminded that \(\oint\) implies the counterclockwise
orientation. But it doesn't matter what orientation students use so long as
they are consistent!
A geometric argument that the orientation should be reversed when
interchanging \(x\) and \(y\) is to rotate the \(xy\)-plane about the line \(y=x\).
(This explains the minus sign in Green's Theorem.)
Students may not have seen line integrals of this form (see below).
Students do very well on this lab, particularly after working in groups for several weeks. Resist the urge to intervene.
Make sure everyone sees the reason \(y\,\boldsymbol{\hat{x}}\cdot d\boldsymbol{\vec{r}}\) is zero on vertical pieces.
The issue of the negative will come up. Suggest students make a quick sketch of the vector field.
It is well worthwhile to do an example with a circle together as a class. The line integral should pose no trouble for them and the area of a circle is something they believe.
Emphasize the connection between the boundary and the interior. This is a concrete display of this relationship.
Subsidiary ideas
Orientation of closed paths.
Line integrals of the form \(\int P\,dx+Q\,dy\).
We do not discuss such integrals in class! Integrals of this form
almost always arise in applications as \(\int\boldsymbol{\vec{F}}\cdot d\boldsymbol{\vec{r}}\).
Homework
Determine the area of a triangle or an ellipse by integrating along the boundary.
Essay questions
Describe times in your life when you needed to know area (or imagine such a time). Maybe buying carpet or painting a room. What is the first step in computing area? How does this lab truly differ, if at all?
Enrichment
Write down Green's Theorem.
Go to 3 dimensions --- bend the curve out of the plane and stretch the region
like a butterfly net or rubber sheet. This is the setting for Stokes'
Theorem!
Students work in groups to measure the steepest slope and direction at a given point on a plastic surface and to
compare their result with the gradient vector, obtained by measuring its
components (the slopes in the coordinate directions).
Found in: Vector Calculus I course(s)Found in: Gradient Sequence, Workshop Presentations 2023 sequence(s)
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).
This small group activity using surfaces introduces a geometric interpretation of partial derivatives in terms of measured ratios of small changes.
Students work in small groups to identify locations on their surface with particular properties.
The whole class wrap-up discussion emphasizes the equivalence of multiple representations of partial derivatives.
You are climbing a hill along the steepest path, whose slope at your current location is \(1\over5\). There is another path branching off at an angle of \(30^\circ\) (\(\pi\over6\)). How steep is it?
Found in: Vector Calculus I, Problem-Solving course(s)Found in: Gradient Sequence sequence(s)
A pretzel is to be dipped in chocolate. The pretzel is in the shape of a quarter circle,
consisting of a straight segment from the origin to the point (2,0), a circular arc from there
to (0,2), followed by a straight segment back to the origin; all distances are in centimeters.
The (linear) density of chocolate on the pretzel is given by \(\lambda = 3(x^ 2 + y^2 )\) in grams per
centimeter. Find the total amount of chocolate on the pretzel.
Main ideas
Calculating (scalar) line integrals.
Use what you know!
Prerequisites
Familiarity with \(d\boldsymbol{\vec{r}}\).
Familiarity with “Use what you know” strategy.
Warmup
It is not necessary to explicitly introduce scalar line integrals,
before this lab; figuring out that the (scalar) line element must be
\(|d\boldsymbol{\vec{r}}|\) can be made part of the activity (if time permits).
Props
whiteboards and pens
“linear” chocolate covered candy (e.g. Pocky)
Wrapup
Emphasize that students must express each integrand in terms of a single
variable prior to integration.
Emphasize that each integral must be positive!
Discuss several different ways of doing this problem (see below).
Details
In the Classroom
Make sure the shape of the pretzel is clear! It might be worth drawing it on
the board.
Some students will work geometrically, determining \(ds\) on each piece by
inspection. This is fine, but encourage such students to try using \(d\vec{r}\)
afterwards.
Polar coordinates are natural for all three parts of this problem, not just
the circular arc.
Many students will think that the integral “down” the \(y\)-axis should be
negative. They will argue that \(ds=dy\), but the limits are from \(2\) to \(0\).
The resolution is that \(ds = |dy\,\boldsymbol{\hat x}|=|dy|=-dy\) when integrating in this
direction.
Unlike work or circulation, the amount of chocolate does not depend on which
way one integrates, so there is in fact no need to integrate “down” the
\(y\)-axis at all.
Some students may argue that \(d\boldsymbol{\vec{r}}=\boldsymbol{\hat T}\,ds\Longrightarrow ds=d\boldsymbol{\vec{r}}\cdot\boldsymbol{\hat T}\),
and use this to get the signs right. This is fine if it comes up, but the
unit tangent vector \(\boldsymbol{\hat T}\) is not a fundamental part of our approach.
There is of course a symmetry argument which says that the two “legs” along
the axes must have the same amount of chocolate --- although some students
will put a minus sign into this argument!
The depth of a puddle in millimeters is given by
\[h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)\]
Your path through the puddle is given by
\[x=3t \qquad y=4t\]
and your current position is \(x=3\), \(y=4\), with \(x\) and \(y\) also in millimeters, and \(t\) in seconds.
At your current position, how fast is the depth of water through which you are walking changing per unit time?
At your current position, how fast is the depth of water through which you are walking changing per unit distance?
FOOD FOR THOUGHT (optional)
There is a walkway over the puddle at \(x=10\). At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.
differentials Found in: AIMS Maxwell, Static Fields, Vector Calculus I, Surfaces/Bridge Workshop, Problem-Solving course(s)
This small group activity is designed to provide practice with the multivariable chain rule.
Students determine a particular rate of change using given information involving other rates of change.
The discussion emphasizes the equivalence of a variety of approaches, including the use of differentials.
Good “review” problem; can also be used as a homework problem.
Found in: Vector Calculus I, Surfaces/Bridge Workshop course(s)
Write down as many integrals as you can think of for the area of the triangular region \(R\) shown, with vertices at \((0,0)\), \((1,0)\), and \((0,2)\).
This small group activity has students reasoning about how the Planck distribution shifts when the temperature is doubled. This leads to a qualitative argument for the Stefan-Boltzmann law.
The function \(\theta(x)\) (the Heaviside or unit
step function) is a defined as:
\begin{equation*}
\theta(x) =\begin{cases}
1 & \textrm{for}\; x>0 \\
0 & \textrm{for}\; x<0
\end{cases}
\end{equation*}
This function is discontinuous at \(x=0\) and is generally taken to
have a value of \(\theta(0)=1/2\).
Make sketches of the following functions, by hand, on axes with the
same scale and domain. Briefly describe, using good scientific
writing that includes both words and equations, the role that the
number two plays in the shape of each graph:
\(y = \theta (x)\)
\(y = 2+\theta (x)\)
\(y = \theta(2+x)\)
\(y = 2\theta (x)\)
\(y = \theta (2x)\)
Found in: Static Fields, AIMS Maxwell, Problem-Solving, None course(s)
In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.
A positively charged (dielectric) spherical shell of inner radius
\(a\) and outer radius \(b\) with a spherically symmetric internal
charge density
\begin{equation*}
\rho(\vec{r})=3\alpha\, e^{(kr)^3}
\end{equation*}
A positively charged (dielectric) cylindrical shell of inner radius
\(a\) and outer radius \(b\) with a cylindrically symmetric internal
charge density
\begin{equation*}
\rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks}
\end{equation*}
Each group will be given one of the charge distributions given below: (\(\alpha\) and \(k\) are constants with dimensions appropriate for the specific example.)
For your group's case, answer the following questions:
Find the total charge. (If the total charge is infinite, decide what you should calculate instead to provide
a meaningful answer.)
Find the dimensions of the constants \(\alpha\) and \(k\).
Spherical Symmetry - A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density:
Cylindrical Symmetry - A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density:
We usually start with a mini-lecture reminder that total charge is calculated by integrating over the charge density by chopping up the charge density, multiplying by the appropriate geometric differential (length, area, or volume element), and adding up the contribution from each of the pieces. Chop, Multiply, Add is a mantra that we want students to use whenever they are doing integration in a physical context.
The students should already know formulas for the volume elements in cylindrical and spherical coordinates. We recommend Scalar Surface and Volume Elements as a prerequisite.
We start the activity with the formulas \(Q=\int\rho(\vec{r}')d\tau'\), \(Q=\int\sigma(\vec{r}')dA'\), and \(Q=\int\lambda(\vec{r}')ds'\) written on the board. We emphasize that choosing the appropriate formula by looking at the geometry of the problem they are doing, is part of the task.
This activity helps students practice the mechanics of making total charge calculations.
Order of Integration When doing multiple integrals, students rarely think about the geometric interpretation of the order of integration. If they do the \(r\) integral first, then they are integrating along a radial line. What about \(\theta\) and \(\phi\). If this topic does not come up in the small groups, it makes a rich discussion in the wrap-up.
Limits of Integration some students need some practice determining the limits of the integrals. This issue becomes especially important for the groups working with a cylinder - the handout does not give the students a height of the cylinder. There are two acceptable resolutions to this situation. Students can “name the thing they don't know” and leave the height as a parameter of the problem. Students can also give the answer as the total charge per unit length. We usually talk the groups through both of these options.
Dimensions Students have some trouble determining the dimensions of constants. Making students talk through their reasoning is an excellent exercise. In particular, they should know that the argument of the exponential function (indeed, the argument of any special fuction other than the logarithm) must be dimensionless.
Integration Some students need a refresher in integrating exponentials and making \(u\)-substitutions.
Wrap-up
You might ask two groups to present their solutions, one spherical and one cylindrical so that everyone can see an example of both. Examples (b) and (f) are nice illustrative examples.
A current \(I\) flows down a cylindrical wire of radius \(R\).
If it is uniformly distributed over the surface, give a formula for
the surface current density \(\vec K\).
If it is distributed in such a way that the volume current density,
\(|\vec J|\), is inversely proportional to the distance from the axis,
give a formula for \(\vec J\).
Current \(I\) flows down a wire with square cross-section. The length of the square side is \(L\). If the current is uniformly distributed over the entire area, find the
current density
.
If the current is uniformly distributed over the outer surface only, find the
current density
.
Consider the following diagram of \(T\) vs \(V\) at different \(p\). The diagram illustrates the relationship between pressure, volume and temperature for an unknown substance (do not assume this is an ideal gas).
Translate the information on this diagram from the T-V plane to the p-V plane (i.e. draw contours of constant \(T\) on a graph of \(p\) vs \(V\)). Include point \(A\) on your p-V graph. Complete your graph by hand using discrete data points that you read from the T-V diagram. Make a fairly accurate sketch of the contours using the attached grid or in some other way making nice square axes with appropriate tick marks. Don't make up data for pressures above 1000 Pa or below 400 Pa.
Are the lines that you drew straight or curved? What feature of the \(TV\) graph would have to change to change this result?
Sketch the line of constant temperature that passes through the point A.
What are the values of all the thermodynamic variables associated with the point A?
For systems of particles, we used the formulas
\begin{align}
\vec{R}_{cm}&=\frac{1}{M}\left(m_1\vec{r}_1+m_2\vec{r}_2\right) \nonumber\\
\vec{r}&=\vec{r}_2-\vec{r}_1
\label{cm}
\end{align}
to describe the system of two objects in terms of the center of mass and relative position instead of the positions of each object. After solving for the equations of motion in the center-of-mass coordinates, you may want to transform back to the original coordinate system to examine the motion of each object.
Find the positions of the two objects in terms of the position of the center of mass and the relative position, i.e. solve for:
\begin{align}
\vec{r}_1&=\\
\vec{r}_2&=
\end{align}
Hint: The system of equations (\ref{cm}) is linear, i.e. each variable is to the first power, even though the variables are vectors. In this case, you can use all of the methods you learned for solving systems of equations while keeping the variables vector valued, i.e. you can safely ignore the fact that the \(\vec{r}\)s are vectors while you are doing the algebra as long as you don't divide by a vector.
(Sketch limiting cases) Purpose: For two central force systems that share the same reduced mass system, discover how the motions of the original systems are the same and different.
The figure below shows the position vector \(\vec r\) and the orbit of
a “fictitious” reduced mass \(\mu\).
Suppose \(m_1=m_2\), Sketch the position vectors and orbits for \(m_1\) and \(m_2\) corresponding to \(\vec{r}\). Describe a common physics example of central force motion for which \(m_1=m_2\).
List the properties that define a central force system.
Calculate a reduced mass for a two-body system and describe why it is important.
Use the solution (algebraic or geometric) to a reduced mass system to describe the motion of the original system.
Describe the role that conservation of energy and angular momentum play in a central force system. In particular, where do these properties appear in the solutions of the equations of motion?
Use an effective potential diagram to predict the possible orbits in a central force system: which orbits are bound or unbound? which are closed or open? where will the turning points be?
In this unit, you will explore the most common partial differential equations that arise in physics contexts. You will learn the separation of variables procedure to solve these equations.
Motivating Questions
How are partial differential equations (PDEs) different from ordinary differential equations (ODEs)?
What new kinds of physics can we learn from solving partial differential equations?
What can we learn about physics and geometry from the separation of variables procedure?
In this unit, you will explore the electrostatic potential \(V(\vec{r})\) due to one or more discrete charges and the gravitational potential \(\Phi(\vec{r})\) due to one or more discrete masses. How does the potential vary in space? How do equipotential surfaces and the superposition principle help you answer these questions graphically? How does the value of the potential fall-off as you move away from the charges? How do power series approximations help you answer these questions algebraically?
Describe the important similarities and differences between the electrostatic potential and the gravitational potential.
Sketch the potential due to a small number of discrete charges or masses, showing important regions of interest and qualitatively depict the correct spacing between equipotential surfaces (or curves).
Compute power and Laurent series expansions from a real-world problem using simple, memorized power series.
Truncate a series properly at a given order by keeping all the terms up to that order and none of the terms of higher order.
Discuss in detail the relationship between the graphical, algebraic, and power series representations of the potentials.
Describe the energy eigenstates for the ring system algebraically and graphically.
List the physical measurables for the system and give expressions for the corresponding operators in bra/ket, matrix, and position representations.
Give the possible quantum numbers for the quantum ring system and describe any degeneracies.
For a given state, use the inner product in bra/ket, matrix, and position representations, to find the probability of making any physically relevant measurement, including states with degeneracy.
Use an expansion in energy eigenstates to find the time dependence of a given state.
Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).
