2. Electrostatics of an ionized atmosphere. Consider an atmosphere consisting of

ID: 2269820 • Letter: 2

Question

2. Electrostatics of an ionized atmosphere.

Consider an atmosphere consisting of an ionized gas (the outer envelope of a white dwarf, where densities are low

enough to use the ideal gas law without quantum corrections, is a good example). For simplicity, suppose that only

hydrogen is present, so the atmosphere contains protons and electrons. Assign these masses mp and me respectively

(and charges +q and ?q), and number densities np and ne. Let the acceleration due to gravity be g and the vertical

coordinate be denoted by z. Real atmospheres are not isothermal (typically the temperature goes down as you go

up), but for simplicity in this problem, assume temperature T to be a constant.

(a) Write down the pressures Pp of the protons and Pe of the electrons in terms of the temperature T , number

densities, and fundamental constants.

(b) Consider a small rectangular parcel of material of dimensions ?x

a) using ideal gas equation :: PV = nRT

P = (n/V)*RT

where n/V = density

np = moles per unit volume ,

hence for proton , Pp =np*RT

pn = ne*RT

N = avogadro number =6.23*10^23

number of proton per unit volume =N*np

psitive charge density = q* number of proton per unit vlume = qN*np

similarly ::

negative charge density = -qN*ne

total charge density = (qN*np-qN*ne)

differnetial form of gauss law tells :: divergance of Electric filed is equal to the volume charge density multiplied by 1/(epsilon).

hence using gauss law :::

dE/dz = total charge density /(epsilon)

dE/dz = (qN*np-qN*ne)/(epsilon)

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