A1 Find and express them in terms of and . · OlympiadBench #1115

Physics · electromagnetism

Problem

A1 Find

_{0}, a, b

and express them in terms of

Z_{\text {ext }}

and

r

. Given constants: Problem3. Simplest model of gas discharge An electric current flowing through a gas is called a gas discharge. There are many types of gas discharges including glow discharge in lighting lamps, arc discharge inwelding and the well known spark discharge that occurs between the clouds and the earth in the form of lightning. Part A. Non-self-sustained gas discharge In this part of the problem the so-called non-self-sustained gas discharge is studied. To maintain it a permanent operationan external ionizer is needed, which creates

Z_{\text {ext }}

pairs of singly ionized ions and free electrons per unit volume and per unit time uniformly in the volume. When an external ionizer is switched on, the number of electrons and ions starts to grow. Unlimited increase in the number densities of electrons and ions in the gas is prevented by the recombination process in which a free electron recombineswith an ion to form a neutral atom. The number of recombining events

Z_{\text {rec }}

that occurs in the gas per unit volume and per unit time is given by

Z_{\mathrm{rec}}=r n_{e} n_{i},

where

r

is a constant called the recombination coefficient, and

n_{e}, n_{i}

denotethe electron and ion number densities, respectively. Suppose that at time

t=0

the external ionizer is switched on and the initial number densities of electrons and ions in the gas are both equal to zero. Then, the electron number density

n_{e}(t)

depends on timetas follows:

n_{e}(t)=n_{0}+a \tanh b t

where

n_{0}, a

and

b

are some constants, and

\tanh x

stands for the hyperbolic tangent.

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