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.
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.
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 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".
\(\boldsymbol{\vec{K}} = yz \,\boldsymbol{\hat{x}} + xz \,\boldsymbol{\hat{y}}\)
Main ideas
Finding potential functions.
Students love this activity. Some groups will finish in 10 minutes or less;
few will require as much as 30 minutes.
*
Prerequisites
Fundamental Theorem for line integrals
The Murder Mystery Method
Warmup
none
Props
whiteboards and pens
Wrapup
Revisit integrating conservative vector fields along various paths, including
reversing the orientation and integrating around closed paths.
Details
In the Classroom
We recommend having the students work in groups of 2 on this activity, and not
having them turn anything in.
Most students will treat the last example as 2-dimensional, giving the answer
\(xyz\). Ask these students to check their work by taking the gradient; most
will include a \(\boldsymbol{\hat{z}}\) term. Let them think this through. The correct answer
of course depends on whether one assumes that \(z\) is constant; we have
deliberately left this ambiguous.
It is good and proper that students want to add together multivariable terms. Keep returning to the gradient, something they know well. It is better to discover the guidelines themselves.
Subsidiary ideas
3-d vector fields do not necessarily have a \(\boldsymbol{\hat{z}}\)-component!
Homework
A challenging question to ponder is why a surface fails to exist for nonconservative fields. Using an example such as \(y\,\boldsymbol{\hat{x}}+\boldsymbol{\hat{y}}\), prompt students to plot the field and examine its magnitude at various locations. Suggest piecing together level sets. There is serious geometry lurking that entails smoothness. Wrestling with this is healthy.
Essay questions
Write 3-5 sentences describing the connection between derivatives and integrals in the single-variable case. In other words, what is the one-dimensional version of MMM? Emphasize that much of vector calculus is generalizing concepts from single-variable theory.
Enrichment
The derivative check for conservative vector fields can be described using the
same type of diagrams as used in the Murder Mystery Method; this is just
moving down the diagram (via differentiation) from the row containing the
components of the vector field, rather than moving up (via integration). We
believe this should not be mentioned until after this lab.
When done in 3-d, this makes a nice introduction to curl --- which
however is not needed until one is ready to do Stokes' Theorem. We would
therefore recommend delaying this entire discussion, including the 2-d case,
until then.
Work out the Murder Mystery Method using polar basis vectors, by reversing the
process of taking the gradient in this basis.
Revisit the example in the Ampère's Law lab, using the Fundamental Theorem
to explain the results. This can be done without reference to a basis, but
it is worth computing \(\boldsymbol{\vec\nabla}\phi\) in a polar basis.
Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.
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.
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.