A spherical probe of radius is immersed in a plasma of electron temperature and Debye…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Plasma Physics · The plasma state and single-particle motion

Plasma Physics, Ch What makes an ionised gas a plasma: shielding, the plasma parameter and the natural frequencies

Problem

A spherical probe of radius

a = 3.0\,\mathrm{mm}

is immersed in a plasma of electron temperature

T_e = 50\,\mathrm{eV}

and Debye length

\lambda_D = 5.0\,\mathrm{mm}

, and is held at a potential

\phi_0 = 2.0\,\mathrm{V}

above the surrounding plasma potential. Because

e\phi_0 \ll T_e

, the linearised Debye-shielding equation

\lambda_D^2\nabla^2\phi = \phi

applies in the plasma outside the probe (where

r > a

). ((a)) Solve

\lambda_D^2\nabla^2\phi=\phi

in spherical symmetry for

r > a

and apply the boundary conditions

\phi(a)=\phi_0

and

\phi\to0

r\to\infty

to show that

\phi(r) = \phi_0\,\frac{a}{r}\,e^{-(r-a)/\lambda_D}.

((b)) Use Gauss's law at the probe surface to find the free charge

Q

that must sit on the probe, and show that it exceeds the vacuum (isolated-sphere) value

Q_{\rm vac}=4\pi\epsilon_0a\phi_0

by the factor

(1+a/\lambda_D)

. ((c)) Evaluate

Q

for the stated numbers.

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