ELECTRO-MAGNETIC THEORY OF MOVING CHARGES.
185
=
x
F
1
+
y
F
2
+
k
ζ
F
3
F
1
2
+
F
2
2
+
k
2
F
3
2
{\displaystyle ={\frac {xF_{1}+yF_{2}+k\zeta F_{3}}{\sqrt {F_{1}^{2}+F_{2}^{2}+k^{2}F_{3}^{2}}}}}
=
x
F
1
+
y
F
2
+
z
F
3
F
1
2
+
F
2
2
+
k
2
F
3
2
{\displaystyle ={\frac {xF_{1}+yF_{2}+zF_{3}}{\sqrt {F_{1}^{2}+F_{2}^{2}+k^{2}F_{3}^{2}}}}}
;
∴
σ
σ
′
=
A
p
p
′
{\displaystyle \therefore {\frac {\sigma }{\sigma '}}={\frac {Ap}{p'}}}
.
If now F(x y z ) = C is an ellipsoid, then we know that
σ
∝
p
{\displaystyle \sigma \propto p}
, therefore also
σ
′
∝
p
′
{\displaystyle \sigma '\propto p'}
, that is the arrangement of charge on the moving ellipsoid is the same as if it were at rest.
7. Applying the above to the ellipsoid (a b c ), we find that φ as a function of (x y ζ ) is the potential of a free distribution on the ellipsoid
x
2
a
2
+
y
2
b
2
+
k
2
ζ
2
c
2
=
1
{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {k^{2}\zeta ^{2}}{c^{2}}}=1}
;
∴
ϕ
=
C
∫
μ
∞
d
λ
(
a
2
+
λ
)
(
b
2
+
λ
)
(
c
2
k
2
+
λ
)
{\displaystyle \therefore \ \phi =C\int _{\mu }^{\infty }{\frac {d\lambda }{\sqrt {(a^{2}+\lambda )(b^{2}+\lambda )\left({\frac {c^{2}}{k^{2}}}+\lambda \right)}}}}
=
C
k
∫
μ
∞
d
λ
(
a
2
+
λ
)
(
b
2
+
λ
)
(
c
2
+
k
2
λ
)
{\displaystyle =Ck\int _{\mu }^{\infty }{\frac {d\lambda }{\sqrt {(a^{2}+\lambda )(b^{2}+\lambda )\left(c^{2}+k^{2}\lambda \right)}}}}
,
where μ is given by
x
2
a
2
+
μ
+
y
2
b
2
+
μ
+
ζ
2
c
2
k
2
+
μ
=
1
{\displaystyle {\frac {x^{2}}{a^{2}+\mu }}+{\frac {y^{2}}{b^{2}+\mu }}+{\frac {\zeta ^{2}}{{\frac {c^{2}}{k^{2}}}+\mu }}=1}
,
or
x
2
a
2
+
μ
+
y
2
b
2
+
μ
+
z
2
c
2
+
k
2
μ
=
1
{\displaystyle {\frac {x^{2}}{a^{2}+\mu }}+{\frac {y^{2}}{b^{2}+\mu }}+{\frac {z^{2}}{c^{2}+k^{2}\mu }}=1}
.
Deteimining the value of the constant C so that the density at a point shall be
e
p
4
π
a
b
c
{\displaystyle {\frac {ep}{4\pi abc}}}
, we get
ϕ
=
e
8
π
∫
μ
∞
d
λ
(
a
2
+
λ
)
(
b
2
+
λ
)
(
c
2
+
k
2
λ
)
{\displaystyle \phi ={\frac {e}{8\pi }}\int _{\mu }^{\infty }{\frac {d\lambda }{\sqrt {(a^{2}+\lambda )(b^{2}+\lambda )\left(c^{2}+k^{2}\lambda \right)}}}}
.
Putting b=a, c=ka , we get
ϕ
=
e
4
π
k
2
(
x
2
+
y
2
)
+
z
2
{\displaystyle \phi ={\frac {e}{4\pi {\sqrt {k^{2}(x^{2}+y^{2})+z^{2}}}}}}
.