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

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                                Let the resistances of the layrs be respectively r<sub>1<\sub> r<sub>2<\sub> &c and 
let r<sub>1<\sub> + r<sub>2<\sub> + &c = r be the resistance of the whole condenser 
to a steady current through it per unit of surface. 

Let the electric displacement in each layer be f<sub>1<\sub> f<sub>2<\sub> &c 

Let the electric current in each layer be p<sub>1<\sub> p<sub>2<\sub> &c 

Let the potential on the first surface be [psi]<sub>1<\sub> and the electricity per unit 
of surface e<sub>1<\sub>. Let the corresponding quantities at the boundary of 
the first and second surfaces be [psi]<sub>2<\sub> & e<sub>2<\sub> and so on - Then 
by equations (G) and (H) 
e<sub>1<\sub> = - f<sub>1<\sub>  de<sub>1<\sub>/dt = - p<sub>1<\sub> 
e<sub>2<\sub> = f<sub>1<\sub> - f<sub>2<\sub>    de<sub>2<\sub>/dt = - p<sub>1<\sub> - p<sub>2<\sub> &c }(51) 

But by equations (E) and (F) 

[psi]<sub>1<\sub> - [psi]<sub>2<\sub> <s> E<sub>1<\sub> - E<sub>2<\sub><\s> =  a<sub>1<\sub>k<sub>1<\sub>f<sub>1<\sub> = - r<sub>1<\sub>p<sub>1<\sub> 
[psi]<sub>2<\sub> - [psi]<sub>3<\sub> <s> E<sub>2<\sub> - E<sub>3<\sub><\s> =  a<sub>2<\sub>k<sub>2<\sub>f<sub>2<\sub> = - r<sub>2<\sub>p<sub>2<\sub>
&c } (52) 

After the electromotive force has ben kept up for a sufficient time 
the current becomes the same in each layer and 

p<sub>1<\sub> = p<sub>2<\sub> = &c = p = [psi]/r 
where [psi] is the total difference of potentials between the extreme layers 
We have then f<sub>1<\sub> = - [psi]/r  r<sub>1<\sub>/a<sub>1<\sub>k<sub>1<\sub> f<sub>2<\sub> = - [psi]/r  r<sub>2<\sub>/a<sub>2<\sub>k<sub>2<\sub> &c 
e<sub>1<\sub> = [psi]/r  r<sub>1<\sub>/a<sub>1<\sub>k<sub>1<\sub> e<sub>2<\sub> = [psi]/r  (r<sub>2<\sub>/a<sub>2<\sub>k<sub>2<\sub> - r<sub>1<\sub>/a<sub>1<\sub>k<sub>1<\sub>) &c }(53) 
these are the quantities of electricity on the different surfaces. 

(87) Now let the condenser be discharged by connecting the 
extreme surfaces through a perfect conductor so that their 
potentials are instantly rendered equal, then the electricity on 
the extreme surfaces will be altered but that on the internal 
surfaces will not have time to escape. The total difference of 
potentials is now 

[psi]' = a<sub>1<\sub>k<sub>1<\sub>e'<sub>1<\sub> + a<sub>2<\sub>k<sub>2<\sub>(e'<sub>1<\sub> + e<sub>2<\sub>) + a<sub>3<\sub>k<sub>3<\sub>(e'<sub>1<\sub> + e<sub>2<\sub> + e<sub>3<\sub>) &c = 0 (54) 
whence e'<sub>1<\sub> = [psi]/r r<sub>1<\sub>/a<sub>1<\sub>k<sub>1<\sub> - [psi]/ak (55) 


                            
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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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