J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’

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                                Also, since the order of magnitude of the coefficients is the 
same as that of the indices the value of [psi]' can never change 
sign, but must start from zero, become positive, and finally 
disappear 

(90) Let us next consider the total amount of electricity which 
would pass from the first surface to the second, if the condenser 
after being thoroughly saturated by the current and then discharged 
has its extreme surfaces connected by a conductor of resistance R 
Let p be the current in this conductor then during the 
discharge 
[psi]' = p<sub>1<\sub>r<sub>1<\sub> + p<sub>2<\sub>r<sub>2 + &c = pR (59) 

Integrating with respect to the time, and calling q<sub>1<\sub> q<sub>2<\sub> q 
the quantities of electricity which traverse the different conductors 
q<sub>1<\sub>r<sub>1<\sub> + q<sub>2<\sub>r<sub>2 + &c = qR  (60) 
The quantities of electricity on the several surfaces will be 

e'<sub>1<\sub> - q - g<sub>1<\sub> 
e<sub>2<\sub> + q<sub>1<\sub> - q<sub>2<\sub> 
&c 

and since at last all these quantities vanish, we find 

q<sub>1<\sub> = e'<sub>1<\sub> - q 
q<sub>2<\sub> = e'<sub>1<\sub> + e<sub>2<\sub> - q

whence qR = [psi]/r (r<sub>1<\sub><sup>2<\sup>/a<sub>1<\sub>k<sub>1<\sub>  r<sub>2<\sub><sup>2<\sup>/a<sub>2<\sub>k<sub>2<\sub> + &c) - [psi]r/ak 

or q = [psi]/akrR {a<sub>1<\sub>k<sub>1<\sub>a<sub>2<\sub>k<sub>2<\sub> (r<sub>1<\sub>/a<sub>1<\sub>k<sub>1<\sub> - r<sub>2<\sub>/a<sub>2<\sub>k<sub>2<\sub>)<sup>2<\sup> + a<sub>2<\sub>k<sub>2<\sub>a<sub>3<\sub>k<sub>3<\sub> (r<sub>2<\sub>/a<sub>2<\sub>k<sub>2<\sub> - r<sub>3<\sub>/a<sub>3<\sub>k<sub>3<\sub>)<sup>2<\sup> + &c} (61) 

a quantity essentially positive, so that when the primary electrification 
is in one direction the secondary discharge is always in the same direction as 
the primary discharge. 
                            
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Manuscript details

Author
James Clerk Maxwell
Reference
PT/72/7
Series
PT
Date
1864
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Cite as

J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’, 1864. From The Royal Society, PT/72/7

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