462- Prof. J. J. Thomson on the Discharge of Electricity 



Let ■.:; - ... : 



2afi 1 



m[\ + i) g m (i+0* — e —m(i+(.)h 



:a + «|8. 



Then, taking only the real part of the expression for E 2 , we 

 have 



E 3 = e m &+ x ) { a cos (pt + m (h + x) ) — 13 sin (/?* + m (A + #) ) } , 

 _l_ € -7«(^+x)| a cos (pt-. m (h + #) ) _£ s i n (^ + m (A + #))}. 



The rate of production of heat per unit volume of the plate 

 is E 2 2 /cr; the mean value of this at a point whose coordinate 

 is x is 



1 



2(7 



(» 2 + /3 2 ) { € 2m(A+aO + e -2w»(*+*) + 2 C OS 171 (A + A') }. 



To get the rate of heat-production per unit area of the 

 plate, we must integrate this with respect to x between the 

 limits x = 0, x=—h. Performing this integration we find 

 that the rate of production of heat per unit area of the plate is 



-i-(a 2 + /3 2 ){6 2 ^-6- 2 ^ + 2sin2mA}. 



Substituting for a 2 + /3 2 its value this becomes 

 2a V (€ 2mh — e - 2mh + 2 sin 2mh) 



4o-m J (e 2mh + e 



-2«iA. 



2 cos 2/72 A) 



To compare this with the expression we found for the solid 

 cylinder, we notice that if H e 1 ^ is the magnetic force in the 

 incident wave at the surface of the plate, 



H y = a 2 . 



So that the rate of heat-production per unit area may be 

 written 



1 H W (e 2mh 



2 am 



■ 2mh 



+ 2 sin 2mA) 



2mh _i_ c —2mh. 



(e 2mh + e 



2 cos 2mh) ' 



± Hp 2 'fjuipV f(e 



2mA . 



-2jhA 



-f 2 sin 2mA) 1 



8^/2 ( 7T 3 J 1 "(e 2mA + e" 2wiA — 2 cos 2mA) J * 



The part of this expression involving A is equal to unity 

 when mh is very great, that is for a thick plate ; it is, however, 

 greater than unity for some values, the smaller values of A 

 for which it is a maximum satisfying the equation 



sin 2m/i(< 



2mh 



■2mh 



)=0. 



