q b r a − r b
E ab =
.
(2.272)
4πf 0 |r a − r b | 3
The magnetic field is given by
µ 0 q b v b × (r a − r b )
B ab =
.
(2.273)
4π
|r a − r b | 3
This field points into the plane of the page for positive charge q b .
The Lorentz force on particle a is then
d
dt
(γ a m a v a ) =
q a q b
4πf 0 |r a − r b | 3
1
·
(r a − r b ) + c 2 v a × [v b × (r a − r b )] ,
(2.274)
96
Chapter 2. Geometrical optics
Figure 2.15: Lorentz force on a particle due to a second particle.
where E ab is the electric field at particle a due to particle b, and
B ab is the magnetic field at particle a due to particle b. We can
assume without loss of generality that the vectors v b and r a − r b
determine a plane, and this plane coincides with the plane of the
page in Figure 2.15. The electric field at the position of particle a
due to particle b is given by
E ab =
.
(2.272)
4πf 0 |r a − r b | 3
The magnetic field is given by
µ 0 q b v b × (r a − r b )
B ab =
.
(2.273)
4π
|r a − r b | 3
This field points into the plane of the page for positive charge q b .
The Lorentz force on particle a is then
d
dt
(γ a m a v a ) =
q a q b
4πf 0 |r a − r b | 3
1
·
(r a − r b ) + c 2 v a × [v b × (r a − r b )] ,
(2.274)
96
Chapter 2. Geometrical optics
Figure 2.15: Lorentz force on a particle due to a second particle.
where E ab is the electric field at particle a due to particle b, and
B ab is the magnetic field at particle a due to particle b. We can
assume without loss of generality that the vectors v b and r a − r b
determine a plane, and this plane coincides with the plane of the
page in Figure 2.15. The electric field at the position of particle a
due to particle b is given by