Students consider the dimensions of spin-state kets and position-basis kets.

1. Find the chemical potential of an ideal monatomic gas in two dimensions, with \(N\) atoms confined to a square of area \(A=L^2\). The spin is zero.

2. Find an expression for the energy \(U\) of the gas.

3. Find an expression for the entropy \(\sigma\). The temperature is \(kT\).

• 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 solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance \(d\) down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.