2
Mr. J. J. Thompson on the Magnetic Effects
placement -currents are due entirely to variations in the electric displacement
(
f
,
g
,
h
)
{\displaystyle (f,g,h)}
caused by the motion of the sphere and the medium, the components
ξ
1
,
η
1
,
ζ
1
{\displaystyle \xi _{1},\eta _{1},\zeta _{1}}
of these currents would be given by
ξ
1
=
u
d
f
d
x
+
v
d
f
d
y
+
(
w
−
w
0
)
d
f
d
z
,
η
1
=
u
d
g
d
x
+
v
d
g
d
y
+
(
w
−
w
0
)
d
g
d
z
,
ζ
1
=
u
d
h
d
x
+
v
d
h
d
y
+
(
w
−
w
0
)
d
h
d
z
.
{\displaystyle {\begin{array}{c}\xi _{1}=u{\dfrac {df}{dx}}+v{\dfrac {df}{dy}}+\left(w-w_{0}\right){\dfrac {df}{dz}},\\\\\eta _{1}=u{\dfrac {dg}{dx}}+v{\dfrac {dg}{dy}}+\left(w-w_{0}\right){\dfrac {dg}{dz}},\\\\\zeta _{1}=u{\dfrac {dh}{dx}}+v{\dfrac {dh}{dy}}+\left(w-w_{0}\right){\dfrac {dh}{dz}}.\end{array}}}
These values, however, do not satisfy the equation
d
ξ
1
d
x
+
d
η
1
d
y
+
d
ζ
1
d
z
=
0
,
{\displaystyle {\frac {d\xi _{1}}{dx}}+{\frac {d\eta _{1}}{dy}}+{\frac {d\zeta _{1}}{dz}}=0,}
unless the dielectric is moving uniformly; so that, if the circuits are to be closed, the motion of the medium must produce some other effect analogous to a current.
Since
d
ξ
1
d
x
+
d
η
1
d
y
+
d
ζ
1
d
z
=
d
d
x
(
f
d
u
d
x
+
g
d
u
d
y
+
h
d
u
d
z
)
+
d
d
y
(
f
d
v
d
x
+
g
d
v
d
y
+
h
d
v
d
z
)
+
d
d
z
(
f
d
w
d
x
+
g
d
w
d
y
+
h
d
w
d
z
)
,
{\displaystyle {\begin{array}{l}{\dfrac {d\xi _{1}}{dx}}+{\dfrac {d\eta _{1}}{dy}}+{\dfrac {d\zeta _{1}}{dz}}={\dfrac {d}{dx}}\left(f{\dfrac {du}{dx}}+g{\dfrac {du}{dy}}+h{\dfrac {du}{dz}}\right)\\\\\qquad +{\dfrac {d}{dy}}\left(f{\dfrac {dv}{dx}}+g{\dfrac {dv}{dy}}+h{\dfrac {dv}{dz}}\right)+{\dfrac {d}{dz}}\left(f{\dfrac {dw}{dx}}+g{\dfrac {dw}{dy}}+h{\dfrac {dw}{dz}}\right),\end{array}}}
we see that the currents will be closed if we add on to the components
ξ
1
,
η
1
,
ζ
1
{\displaystyle \xi _{1},\eta _{1},\zeta _{1}}
the components
ξ
0
,
ζ
0
,
ρ
0
{\displaystyle \xi _{0},\zeta _{0},\rho _{0}}
, where
−
ξ
0
=
f
d
u
d
x
+
g
d
u
d
y
+
h
d
u
d
z
,
−
η
0
=
f
d
v
d
x
+
g
d
v
d
y
+
h
d
v
d
z
,
−
ζ
0
=
f
d
w
d
x
+
g
d
w
d
y
+
h
d
w
d
z
.
{\displaystyle {\begin{array}{c}-\xi _{0}=f{\dfrac {du}{dx}}+g{\dfrac {du}{dy}}+h{\dfrac {du}{dz}},\\\\-\eta _{0}=f{\dfrac {dv}{dx}}+g{\dfrac {dv}{dy}}+h{\dfrac {dv}{dz}},\\\\-\zeta _{0}=f{\dfrac {dw}{dx}}+g{\dfrac {dw}{dy}}+h{\dfrac {dw}{dz}}.\end{array}}}
The medium is assumed to be incompressible, so that
d
u
d
x
+
d
v
d
y
+
d
w
d
z
=
0.
{\displaystyle {\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0.}
Hence the components of the total effective currents are