to perform a magnetic vector potential calculation using the superposition principle;
to decide which form of the superposition principle to use, depending on the dimensions of the current density;
how to find current from total charge \(Q\), period \(T\), and the geometry of the problem, radius \(R\);
to write the distance formula \(\vec{r}-\vec{r'}\) in both the numerator and denominator of the superposition principle in an appropriate mix of cylindrical coordinates and rectangular basis vectors;
Students work in small groups to use the Biot-Savart law
\[\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\]
to find an integral expression for the magnetic field, \(\vec{B}(\vec{r})\), due to a spinning ring of charge.
In an optional extension, students find a series expansion for \(\vec{B}(\vec{r})\) 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.
Consider a paramagnet, which is a
material with \(n\) spins per unit volume each of which may each be
either “up” or “down”. The spins have energy \(\pm mB\) where
\(m\) is the magnetic dipole moment of a single spin, and there is no
interaction between spins. The magnetization \(M\) is defined as the
total magnetic moment divided by the total volume. Hint: each
individual spin may be treated as a two-state system, which you have
already worked with above.
Plot of magnetization vs. B field
Find the Helmholtz free energy of a paramagnetic system (assume
\(N\) total spins) and show that \(\frac{F}{NkT}\) is a function of
only the ratio \(x\equiv \frac{mB}{kT}\).
Use the canonical ensemble (i.e. partition function and
probabilities) to find an exact expression for the total
magentization \(M\) (which is the total dipole moment per unit
volume) and the susceptibility \begin{align}
\chi\equiv\left(\frac{\partial M}{\partial
B}\right)_T
\end{align} as a function of temperature and magnetic field for the
model system of magnetic moments in a magnetic field. The result for
the magnetization is \begin{align}
M=nm\tanh\left(\frac{mB}{kT}\right)
\end{align} where \(n\) is the number of spins per unit volume. The figure shows what this magnetization looks like.
Show that the susceptibility is \(\chi=\frac{nm^2}{kT}\) in the
limit \(mB\ll kT\).
Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.
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 \(E=0\).
Find the Helmholtz free energy \(F = U-TS\) for a crystal
containing \(N\) nuclei which do not interact with each other.
Find an expression for the entropy as a function of
temperature for this system. (Hint: use results of part a.)
Indicate what your results predict for the entropy at the
extremes of very high temperature and very low temperature.
Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear \(\vec{I}\), surface \(\vec{K}\), and volume \(\vec{J}\) current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.
Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.
electric fieldcone Found in: Computational Physics Lab II course(s)Found in: Computational integrating charge distributions sequence(s)
Students work in small groups 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.
Each group will be given one of the charge distributions given below: (\(\alpha\) and \(k\) are constants with dimensions appropriate for the specific example.)
Spherical Symmetery
A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density
\(\rho (\vec{r}) = \alpha\, r^{3}\)
A positively charged (dielectric) spherical shell of inner radius \(a\)
and outer radius \(b\) with a spherically symmetric internal charge density
\(\rho (\vec{r}) =\alpha\, e^{(kr)^{3}}\)
A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, \frac{1}{r^{2}}\, e^{(kr)}\)
Cylindrical Symmetry
A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, s^{3}\)
A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho (\vec{r}) =\alpha\, e^{(ks)^{2}}\)
A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, \frac{1}{s}\, e^{(ks)}\)
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 \(k\).
Instructor's Guide
Introduction
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.
(4pts)
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*}
Consider a column of atoms each of mass \(M\) at temperature \(T\) in
a uniform gravitational field \(g\). Find the thermal average
potential energy per atom. The thermal average kinetic energy is
independent of height. Find the total heat capacity per atom. The
total heat capacity is the sum of contributions from the kinetic
energy and from the potential energy. Take the zero of the
gravitational energy at the bottom \(h=0\) of the column. Integrate
from \(h=0\) to \(h=\infty\). You may assume the gas is ideal.
(4pts) 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 \((0,0,2L)\).
Found in: AIMS Maxwell, Static Fields, Problem-Solving course(s)
Consider the finite line with a uniform charge density from class.
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.
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!)
The gravitational field due to a spherical shell of matter (or equivalently, the
electric field due to a spherical shell of charge) is given by:
\begin{equation}
\vec g =
\begin{cases}
0&\textrm{for } r<a\\
-G \,\frac{M}{b^3-a^3}\,
\left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\
-G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\
\end{cases}
\end{equation}
This problem explores the consequences of the divergence
theorem for this shell.
Using the given description of the gravitational field, find the divergence of the
gravitational field everywhere in space. You will need to divide this
question up into three parts: \(r<a\), \(a<r<b\), and \(r>b\).
Briefly discuss the physical meaning of the divergence in this particular
example.
For this gravitational field, verify the divergence theorem on a
sphere, concentric with the shell, with radius \(Q\), where \(a<Q<b\).
("Verify" the divergence theorem means calculate the integrals from both sides of the divergence theorem and show that they give the same answer.)
Briefly discuss how this example would change if you were discussing the
electric field of a uniformly charged spherical shell.
Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s)
Students write python programs to compute the potential due to a square of surface charge, and then to visualize the result. This activity can be used to introduce students to the process of integrating numerically.
Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity Electric Potential of Two Charged Plates before this activity.
Students should know that
objects with like charge repel and opposite charge attract,
object tend to move toward lower energy configurations
The potential energy of a charged particle is related to its charge: \(U=qV\)
The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)
Students work in small groups to use the superposition principle
\[V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}\]
to find an integral expression for the electrostatic potential, \(V(\vec{r})\), everywhere in space, due to a ring of charge.
In an optional extension, students find a series expansion for \(V(\vec{r})\) 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.
These lecture notes from the ninth week of https://paradigms.oregonstate.edu/courses/ph441 cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.
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.
Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.
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.
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.
Students explore the effects of putting a point charge at various places inside, outside, and on the surface of a cubical Gaussian surface. The Mathematica worksheet or Sage activity shows the electric field due to the charge, then plots the the flux integrand on the top surface of the box, calculates the flux through the top of the box, and the value of the flux through the whole cube.
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.