  We describe a simple AC susceptometer built in-house that can be used to make
high-resolution measurements of the magnetic susceptibility of high-temperature
superconductors in an undergraduate physics lab. Our system, cooled using
liquid nitrogen, can reach a base temperature of 77 K. Our apparatus does not
require gas handling systems or PID temperature controllers. Instead, it makes
use of a thermal circuit that is designed to allow the sample to cool on a time
scale that is suitable for an undergraduate lab. Furthermore, the temperature
drift rate at the superconducting transition temperature $T_\mathrm{c}$ is low
enough to allow for precise measurements of the complex magnetic susceptibility
through $T_\mathrm{c}$, even for single-crystal samples with exceedingly sharp
superconducting transitions. Using an electromagnet, we were able to apply
static magnetic fields up to 63 mT at the sample site. By measuring the change
in susceptibility as a function of the strength of an applied of static
magnetic field, we were able to estimate the lower critical field
$H_\mathrm{c1}$ of a single-crystal sample of optimally-doped
YBa$_2$Cu$_3$O$_{6.95}$ at 77 K. We also investigated the mixed state of a
sintered polycrystalline sample of YBa$_2$Cu$_3$O$_{6+y}$.

Physics Education;Superconductivity