In a plasma globe, a hollow glass sphere is filled with low-pressure gas and a small spherical metal electrode is located at its center. Assume an ac voltage source of peak voltage $V_{0}$ and frequency $f$ is applied between the metal sphere and the ground, and that a person is touching the outer surface of the globe with a fingertip, whose approximate area is $1.0 \mathrm{~cm}^{2}$. The equivalent circuit for this situation is shown in Fig. $30-35,$ where $R_{\mathrm{G}}$ and $R_{\mathrm{p}}$ are the resistances of the gas and the person, respectively, and $C$ is the capacitance formed by the gas, glass, and finger. (a) Determine $C$ assuming it is a parallel-plate capacitor. The conductive gas and the person's fingertip form the opposing plates of area $A=1.0 \mathrm{~cm}^{2}$. The plates are separated by glass (dielectric constant $K=5.0$ ) of thickness $d=2.0 \mathrm{~mm} .$ (b) In a typical plasma globe, $f=12 \mathrm{kHz}$ Determine the reactance $X_{C}$ of $C$ at this frequency in $\mathrm{M} \Omega$. (c) The voltage may be $V_{0}=2500 \mathrm{~V}$. With this high voltage, the dielectric strength of the gas is exceeded and the gas becomes ionized. In this "plasma" state, the gas emits light ("sparks") and is highly conductive so that $R_{\mathrm{G}} \ll X_{C}$. Assuming also that $R_{\mathrm{p}} \ll X_{C},$ estimate the peak current that flows in the given circuit. Is this level of current dangerous? $(d)$ If the plasma globe operated at $f=1.0 \mathrm{MHz},$ estimate the peak current that would flow in the given circuit. Is this level of current dangerous?