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<, until we get close to the cathode
the fall of potential in this part of the discharge will be very
approximately equal to i/k2e.[root]([alpha]l/q), where l is the
distance between the electrodes. Close to the cathode, the electric
force when i0 is not nearly equal to i is approximately given by the
equation
i /[alpha]\^1/2
X = --------- (---------) [epsilon]^{-4[pi]e^2k2qx/[alpha]i},
e(k1k2)^1/2 \ q / ,
and the fall of potential at the cathode is equal approximately to
_[oo]
/
| X dx,
_/0
that is to
i /[alpha]\^1/2 [alpha]i
--------- (---------) -----------.
e(k1k2)^1/2 \ q / 4[pi]e^2k2q
The potential difference between the plates is the sum of the fall of
potential in the uniform part of the discharge plus the fall at the
cathode, hence
/[alpha]\^1/2 i / i[alpha]^2 1 \
V = (---------) --- ( il + ---------- ------------ ).
\ q / ek2 \ 4[pi]e^2q [root](k1k2)/
The fall of potential at the cathode is proportional to the square of
the current, while the fall in the rest of the circuit is directly
proportional to the current. In the case of flames or hot gases, the
fall of potential at the cathode is much greater than that in the rest
of the circuit, so that in such cases the current through the gas
varies nearly as the square root of the potential difference. The
equation we have just obtained is of the form
V = Ai + Bi^2,
and H. A. Wilson has shown that a relation of this form represents the
results of his experiments on the conduction of electricity through
flames.
The expression for the fall of potential at the cathode is inversely
proportional to q^(3/2), q being the number of ions produced per cubic
centimetre per second close to the cathode; thus any increase in the
ionization at the cathode will diminish the potential fall at the
cathode, and as practically the whole potential difference between the
electrodes occurs at the cathode, a diminution in the potential fall
there will be much more important than a diminution i]]>
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