An oscillator circuit is important to many applications. A simple oscillator circuit can be built by adding a neon gas tube to an $R C$ circuit, as shown in FIGURE CP32.79 Gas is normally a good insulator, and the resistance of the gas tube is essentially infinite when the light is off. This allows the capacitor to charge. When the capacitor voltage reaches a value $V_{\infty}$, the electric field inside the tube becomes strong enough to ionize the neon gas. Visually, the tube lights with an orange glow. Electrically, the ionization of the gas provides a very-low-resistance path through the tube. The capacitor very rapidly (we can think of it as instantaneously) discharges through the tube and the capacitor voltage drops. When the capacitor voltage has dropped to a value $V_{\text {oft }}$, the electric field inside the tube becomes too weak to sustain the ionization and the neon light turns off. The capacitor then starts to charge again. The capacitor voltage oscillates between $V_{\mathrm{cR}},$ when it starts charging, and $V_{\infty},$ when the light comes on to discharge it.
a. Show that the oscillation period is
$$T=R C \ln \left(\frac{\mathcal{E}-V_{\mathrm{e} \alpha}}{\mathcal{E}-V_{\mathrm{col}}}\right)$$
b. A neon gas tube has $V_{\infty}=80 \mathrm{V}$ and $V_{\mathrm{eff}}=20 \mathrm{V} .$ What resistor value should you choose to go with a $10 \mu F$ capacitor and a 90 V battery to make a 10 Hz oscillator?
CAN'T COPY THE FIGURE