Spectra and Planck's Law. 435 



not at some stages alter appreciably, while at other stages 

 an increase in the energy of the cathode rays produced con- 

 siderable increase in the hardness of the radiation. This is 

 what would happen if this radiation were a mixture of definite 



types (a, b, c, d ) of characteristic radiations from the 



target struck by the cathode rays. The type a not being- 

 excited unless the energy of the cathode rays exceeded e^ 

 b not being excited unless the energy exceeded e 2 , and so on. 

 Thus, when the energy of the cathode rays was between e^ 

 and e 2 , the radiation would be confined to the a type, it 

 would be a mixture of the a and b types as soon as the 

 energy of the cathode rays exceeded e 2 , when it exceeded e B 

 the c type of radiation would be added, and so on. 



Number of Waves in a Train of Waves, 



When an electron falls into a position of equilibrium and 

 rotates round the lines of magnetic force, the energy it 

 acquires by the fall is gradually converted into radiant 

 energy, and the electron gradually comes to rest. 



If/ is the acceleration of an electron, the rate at which it 

 emits energy is 



2 e 2 f 2 



3 V ' 



where e is the charge on the electron and V the velocity of 

 light. 



If H is the magnetic force, v the velocity of the electron, 

 then when the orbit is at right angles to the magnetic force 



Wev 



J m 



where n is the frequency of the vibration of the electrons, 

 when this is governed by the magnetic force. Substituting 

 this value for f] we find that the rate at which energy is 

 emitted from the electron is equal to 



77* 



2. ,2 



e*irv 



E, 



8V 3Vo »» 



where E is the kinetic energy of the electron ; hence when 

 the loss of energy by the electron is entirely due to the 

 Tadiation, we have 



^ E _ 1 W ! 2^ e= _ 



dt 3V m ■ ' J 



Thus E = ce- kt . 



2H2 



