Page:TolmanNon1.djvu/5

four field equations, for the case of a moving point charger,-

${\mathsf {E}}=\left(1-{\frac {v^{2}}{c^{2}}}\right){\frac {\epsilon {\mathsf {R}}}{{\mathsf {R}}^{3}\left(1-{\frac {v^{2}}{c^{2}}}\sin ^{2}\psi \right)^{\frac {3}{2}}}}$

(6)

${\mathsf {H}}={\frac {1}{c}}{\mathsf {v}}\times {\mathsf {E}},$

(7)

where R is the radius vector connecting the moving charge with the point in question and $\psi$ is the angle between R and v.

For the field acting on the test electron $t$, situated at the point $x=0$, $y=y$, we may substitute ${\mathsf {R}}=y{\mathsf {j}}$ and $\sin \psi =1$, giving us,

${\mathsf {E}}={\frac {\epsilon }{y^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)^{\frac {1}{2}}}}{\mathsf {j}}$

(8)

and

${\mathsf {H}}={\frac {v}{c}}{\frac {\epsilon }{y^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)^{\frac {1}{2}}}}{\mathsf {k}},$

(9)

substituting into the fifth fundamental equation of electromagnetic theory,

${\mathsf {F}}={\mathsf {E}}+{\frac {1}{c}}{\mathsf {v}}\times {\mathsf {H}},$

(10)

we obtain the force acting on the unit test electron $t$.

[Note in the above equation that v, the velocity of the electron, is for our case $v{\mathsf {i}}+u_{y}{\mathsf {j}}$.]

${\mathsf {F}}={\frac {\epsilon }{y^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)^{\frac {1}{2}}}}{\mathsf {j}}-{\frac {1}{c^{2}}}{\frac {v^{2}\epsilon }{y^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)^{\frac {1}{2}}}}{\mathsf {j}}+{\frac {1}{c^{2}}}{\frac {v^{2}u_{y}\epsilon }{y^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)^{\frac {1}{2}}}}i,$

(11)

or

${\mathsf {F}}_{x}={\frac {\epsilon }{y^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)^{\frac {1}{2}}}}{\frac {vu_{y}}{c^{2}}},$

(12)

${\mathsf {F}}_{y}={\frac {\epsilon }{y^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)^{\frac {1}{2}}}}\left(1-{\frac {v^{2}}{c^{2}}}\right)$

(13)

Under the action of the component force ${\mathsf {F}}_{x}$ we might at first sight expect the electron $t$ to aquire an acceleration in the X direction: Such condition, however, would not be in agreement with the principle of relativity, since from the