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TEMPERATURE VARIATION 77

velocities. The point of immediate interest, however, is the
first term, on the right-hand side of (23), which is always posi-
tive and is independent of the emission velocities. This shows
that owing to the action of the magnetic field due to the heat-
ing current, a definite potential is necessary in order to drag
the electrons across to the electrode. With thin wires, which
require only a small current to heat them, this potential differ-
ence is unimportant. Thus if b\a =200 and j = i ampere,
V_x is only about 0'2 volt. On the other hand, with thick
wires, which require large heating currents, the necessary values
of V-L may be quite large.

Another important factor which has to be taken into ac-
count, especially with thin wires, is the drop of potential along
the wire due to the flow of the heating current. This is usually
comparable with I volt per cm. In order to ensure that no
part of the wire is at a positive potential compared with the
cylinder, it is necessary that the positive end of the wire should
be at a potential at least as low as that of the cylinder. If
the potentials are applied at the negative end of the wire, it
will appear from this cause alone that an additional negative
potential equal to the fall along the hot wire has to be applied
in order to ensure complete saturation. Where the conditions
are such that the current would be varying as the 3/2 power
of the voltage if the cathode were an equipotential surface the
effect of the fall of potential due to the current flow can be
allowed for by the following method. Let V_x be the
potential difference between the positive end of the cathode
filament and the anode, let V₀ be the potential drop in the
filament whose length is /. Let the current per unit length
in general be CV^3/2 then the potential at a point distant x from

the positive end of the filament is V_x + ^V₀ and the current
from a length dx at this point is

% \3/a

i + ₇v₀)

and the total current is

cf (v, ₊ $



	TEMPERATURE VARIATION 77 velocities. The point of immediate interest, however, is the first term, on the right-hand side of (23), which is always posi- tive and is independent of the emission velocities. This shows that owing to the action of the magnetic field due to the heat- ing current, a definite potential is necessary in order to drag the electrons across to the electrode. With thin wires, which require only a small current to heat them, this potential differ- ence is unimportant. Thus if b\a =200 and j = i ampere, V_x is only about 0'2 volt. On the other hand, with thick wires, which require large heating currents, the necessary values of V-L may be quite large. Another important factor which has to be taken into ac- count, especially with thin wires, is the drop of potential along the wire due to the flow of the heating current. This is usually comparable with I volt per cm. In order to ensure that no part of the wire is at a positive potential compared with the cylinder, it is necessary that the positive end of the wire should be at a potential at least as low as that of the cylinder. If the potentials are applied at the negative end of the wire, it will appear from this cause alone that an additional negative potential equal to the fall along the hot wire has to be applied in order to ensure complete saturation. Where the conditions are such that the current would be varying as the 3/2 power of the voltage if the cathode were an equipotential surface the effect of the fall of potential due to the current flow can be allowed for by the following method. Let V_x be the potential difference between the positive end of the cathode filament and the anode, let V₀ be the potential drop in the filament whose length is /. Let the current per unit length in general be CV^3/2 then the potential at a point distant x from the positive end of the filament is V_x + ^V₀ and the current from a length dx at this point is

	% \3/a i + ₇v₀)

	and the total current is cf (v, ₊ $