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

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                                (93) If we com<s>pare<\s>bine the equations of Magnetic Force (B) with those 
of Electric currents (C) and put for brevity 

dF/dx + dG/dy + dH/dz = J and d<sup>2<\sup>/dx<sup>2<\sup> + d<sup>2<\sup>/dy<sup>2<\sup> + d<sup>2<\sup>dz<sup>2<\sup> = [del operator] <sup>2<\sup> (63) 
4[pi][mu]p' = dJ/dx - <s>d<sup>2<\sup>[mu]<sup>2<\sup>/dx<sup>2<\sup><\s> [del operator]<sup>2<\sup>F 
4[pi][mu]q' = dJ/dy - [del operator]<sup>2<\sup>G 
4[pi][mu]r' = dJ/dz - [del operator]<sup>2<\sup>H } (64) 

If the medium in the field is a perfect dielectric there is no 
true conduction and the currents p' q' r' are only variations 
in the dielectric displacement or, by the equations of Total Currents (A) 
p'=df/dt q'=dg/dt r'=dh/dt (65) 
But these electric displacements are caused by electromotive forces 
and by the equations of Eectric Elasticity (E) 

P=kf Q=kg R=kh (66) 

These electromotive forces are due to the variations either of the electromagnetic 
or the electrostatic functions as there is no motion of conductors in 
the field, so that the equations of electromotive force (D) are 

P= -dF/dt - d[psi]/dx 
Q= -dG/dt - d[psi]dy 
R = -dH/dt - d[psi]/dz } (67) 

(94) Combining these equations we obtain the following 

k(dJ/dx) - [del operator]<sup>2<\sup>F) + 4[pi][mu](d<sup>2<\sup>F/dt<sup>2<\sup> + d<sup>2<\sup>[psi]/dxdt) = 0 
k(dJ/dy) - [del operator]<sup>2<\sup>G) + 4[pi][mu](d<sup>2<\sup>G/dt<sup>2<\sup> + d<sup>2<\sup>[psi]/dydt) = 0
k(dJ/dz) - [del operator]<sup>2<\sup>H) + 4[pi][mu](d<sup>2<\sup>H/dt<sup>2<\sup> + d<sup>2<\sup>[psi]/dzdt) = 0 } (68) 

If we differentiate the third of these equations with respect to y and the 
second with respect to y[sic, z?] and subtract J & 4[H?] disappear and 
be remembering the equations (B) of magnetic force the results 
may be written 

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James Clerk Maxwell
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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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