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

```                                (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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## 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