
Sensitive Hot- Wire Voltmeter. 31e 
In order to get an idea of the relation of the sag of the wire 
to its increment of length, suppose that it is stretched quite 
straight to begin with ; that its length is 2/, and that we require 
to find » the contraction of the spiral spring for a given 
increment of length ain the wire. This is seen to be given 
by 2?=/a, when a is small. 
To form an idea of the multiplication obtainable in practice, 
suppose that the distance between supports =5 cms., and 
that the wire is heated through 1° C., the initial interior 
angle being 175°, and the coefficient of expansion of the wire 
per degree ‘00002. Calculation shows that in this case the 
magnification is about 22; and of course it increases rapidly 
the more nearly the initial angle approaches 180°. The con- 
dition as to the length of wire which it is best to employ, is 
easily obtained from the following considerations. 
The resistance of the snstrument-cireuit is practically the 
same as the resistance of the fine wire. The diameter of this 
being fixed at the minimum available, the resistance is simply 
proportional to the length. The rate of heat evolution is 
therefore inversely as the length at constant P.D. If the 
length of wire be increased n- ‘fold, the rate at which heat 
. . . f . . . 
is supplied is — of its former value, and the cooling surface is 
7 
increased n-fold; therefore, for small rises of temperature the 
* TNE Ks Sa 
rise of temperature of the wire is 2 of its former value. 
The total increase of length of the wire which is the subject 
of measurement is propor tional both to the length of the wire 
and to the temperature difference ; so that it finally becomes 
1 dei. 
— of the original value. 
VL : 
The sensitiveness of the instrument, in so far as the limit is 
set by the difficulty of measuring small changes of length, 
is therefore inversely as the length of the wire. If the sen- 
sitiveness be regarded as limited by the least amount of sag 
that can be perceived due to heating, we have to consider 
what further condition is imposed. The practical condition 
is that enough sag must be allowed initially to prevent the 
wire getting broken if the current is accidentally t taken off 
while the measuring apparatus is in the position necessary to 
make the sag constant in spite of the heating. In this case 
the magnification is constant, and « depends only on the 
length—giving the same condition for sensitiveness as obtains 
when we consider the measuring apparatus as imposing the 
limit. 
