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

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                                (98) Let us now go back upon equations in (94) in which the quantities 
J and [psi] occur to see whether any other kind of disturbance can be 
propagated through the medium depending on these quantities which 
disappeared from the final equations. 

If we determine [chi] from the equation 

[del operator]<sup>2<\sup>[chi] = d<sup>2<\sup>x/dx<sup>2<\sup> + d<sup>2<\sup>[chi]/dy<sup>2<\sup> + d<sup>2<\sup>[chi]/dz<sup>2<\sup> = J (73) 

and F' G' H' from the equations 

F' = F - d[chi]/dx G' = G — d[chi]/dy H' = H — d[chi]/dz (74) 
then dF'/dx + dG'/dy + dH'/dz = 0 (75) 
and the equations in (94) become of the form 
k[del operator]<sup>2<\sup>F' = 4[pi][mu](d<sup>2<\sup>F'/dt<sup>2<\sup> + d/dxdt ([psi] + d[chi]/dt)) (76) 

Differentiating the three equations with respect to x y and z and 
adding we find that 
[psi] = -d[chi]/dt + [phi](x y z) (77) 
and that k[del operator]<sup>2<\sup>F' = 4[pi][mu] d<sup>2<\sup>F'/dt<sup>2<\sup> 
k[del operator]<sup>2<\sup>G' = 4[pi][mu] d<sup>2<\sup>G'/dt<sup>2<\sup>
k[del operator]<sup>2<\sup>H' = 4[pi][mu] d<sup>2<\sup>H'/dt<sup>2<\sup> }(78) 
Hence the disturbances indicated by F' G' H' are propagated with 
the velocity V=[square root[k/4[pi][mu] through the field and since 
dF'/dx + dG'/dy + dH'/dz = 0 
the resultant of these disturbances is in the plane of the wave 

(99) The remaining part of the total disturbances F,G,H being the 
part depending on [chi] is subject to no condition except that expressed 
in the equation d[psi]/dt + d<sup>2<\sup>[chi]/dt<sup>2<\sup> = 0 <s>and the<\s> If we perform the operation [del operator]<sup>2<\sup> on this 
equation it becomes ke = dJ/dt - k[del operator]<sup>2<\sup>[phi](x y z) (79) 
Hence J is either zero or it continually increases or diminishes 
with the time, if e remains constant, which no physical quantity 
can do. Hence J is zero and the only disturbance propagated is that indicated 
by F', G', H' which is wholly transversal 

                            
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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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