  We calculate the charge sensitivity of a recently demonstrated device where
the Josephson inductance of a single Cooper-pair transistor is measured. We
find that the intrinsic limit to detector performance is set by oscillator
quantum noise. Sensitivity better than $10^{-6}$e$/\sqrt{\mathrm{Hz}}$ is
possible with a high $Q$-value $\sim 10^3$, or using a SQUID amplifier. The
model is compared to experiment, where charge sensitivity $3 \times
10^{-5}$e$/\sqrt{\mathrm{Hz}}$ and bandwidth 100 MHz are achieved.
