CONDUCTIVITY. 93 



Definition of Conductivity. We may now give a precise signifi- 

 cation to the term as follows : The conductivity of a substance is the 

 quantity of heat conducted per second through a square centimetre in 

 the substance, when the temperature changes in a direction perpendicular 

 to the area at the rate of 1 C. per centimetre. If then AB (Fig. 64) is 

 a plane over which the temperature is ; CD, EF two parallel planes 

 at cm. distance, one on each side, over which the temperatures are 6 + | 

 and Q -\ respectively, the quantity of heat passing in one second through 

 1 square centimetre of the plane AB is the conductivity of the sub- 

 stance at the temperature 0. We shall denote this quantity by k. If 

 the area be A square centimetres and the time t seconds, the quantity 

 passing through must be If At, for, the circumstances are the same for 

 each square centimetre and for each successive second. 



It is not easy to make direct exact experiments on the flow of heat 

 with different slopes of temperature, but general experience might lead 

 us to expect it to be proportional to the steepness of slope, or the fall of 

 temperature per centimetre. We may roughly verify this by immersing 

 a thermometer in warm water, and noting the rate of 

 rise. If we keep the thermometer moving in the liquid 

 all the time, the outside layer of the glass has probably 

 nearly the same temperature as the liquid, and the rise 

 per second is nearly proportional to the distance of the 

 top of the column from the final point reached that is, 

 the quantity of heat received per second is nearly pro- 

 portional to the difference of temperature between the 

 mercury and the water outside. 



This is evidently complicated by the expansion of 



3 



the glass, a thermometer giving true indications only FIG. 64. 

 when both glass and mercury are at the same tempera- 

 ture. A still better verification consists in immersing a small calorimeter 

 containing water in an outer vessel also containing water, but at a different 

 temperature, keeping the contents of both vessels well stirred, and it will 

 be found that the quantity of heat passing from one vessel to the other is 

 roughly proportional to the difference of temperature. We shall, there- 

 fore, assume that the quantity of heat conducted is proportional to the 

 slope of temperature. Such experiments as these bear to the verification 

 of the law the same relation that experiments with Atwood's machine 

 bear to the verification of the laws of motion. We must regard them as 

 suggesting, rather than verifying. The more exact verification is in the 

 agreement of experiment with calculations based on the assumption of the 

 truth of the law. 



If the temperature-difference between two neighbouring points, d apart 

 at equal distances, one on each side of the area A through which heat is 



flowing, is equal to r, then -. is the rate of variation per centimetre or 



61 



the slope of temperature, and assuming that the flow of heat is propor- 

 tional to this, the total quantity flowing through A in time t is given by 



a 

 where Jc is the conductivity at 6, the temperature of A. 



