The Plasma Globe's Dancing Arc · AI Generator #GEN-BDA26757

Physics · electromagnetism

Problem

A plasma globe contains a central electrode surrounded by low-pressure gas. When powered by a high-frequency AC source, beautiful arcs of plasma dance between the central electrode and the glass surface. The central spherical electrode has radius

r = 2.0\;\text{cm}

and the glass sphere has inner radius

R = 8.0\;\text{cm}

. The space between them is filled with gas that has been ionized into a plasma with conductivity

\sigma = 1.2 \times 10^{-4}\;\text{S/m}

. However, a thin layer of normal gas with thickness

d = 0.5\;\text{mm}

remains unionized near the glass surface, acting as a dielectric with relative permittivity

\epsilon_r = 1.8

. When you touch the glass surface, you complete a circuit and observe that the plasma arcs preferentially form toward your finger. If the AC voltage source provides an RMS voltage of

V_{\text{rms}} = 3000\;\text{V}

at frequency

f = 35\;\text{kHz}

, find the impedance magnitude

|Z|

of the path from the central electrode to the glass surface through the plasma and dielectric layer. *Use

\epsilon_0 = 8.85 \times 10^{-12}\;\text{F/m}

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