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

View transcription
                                (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 

Please login to transcribe

Manuscript details

James Clerk Maxwell
Open IIIF manifest
(What's this?)
This is a link to the IIIF web URL for this item. You can drag and drop the IIIF image link into other compatible viewers

Cite as

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



Please login to comment