This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 6 Charged Particle in Electric
and Magnetic Fields According to the Lorentz Force Law the equation of motion of a charged
particle of charge 6 and mass m in an electric ﬁeld E and magnetic ﬁeld H is given by mi‘:eE+E’i‘xH, (6.1) Where c is the speed of light in vacuum.
We ﬁrst note that a change in the kinetic energy of the particle is produced by the electric ﬁeld E alone. This is demonstrated as follows: —d— lm2 —E2v 93—1) (mr)
dt 2 v ‘ 2 dt_ (62) :eE~v+§(va)v=eEv. If E is an electrostatic ﬁeld, then
E : ——V¢ , (6.3) where d) is the electrostatic potential. Eq.(6.2) then reads d 1 2 _ dr _ d¢
dt (577W ) — quﬁ dt — edt , (6.4)
or d 1
2 _
dt (imv + 6gb) _ 0 . (6.5)
The quantity
8 E émvri + 691) (56) can be identiﬁed as the total energy of the particle; and (6.5) is just a statement
of the conservation of energy. 27 28 CHAPTER 6. ELECTRIC AND JVIAGNETIC FIELDS Consider the special case of constant E and H , with E = 0 and H = H 63 =
(07 07 H) (see Fig. 6.1). Then
Q 61 62 63
P.
7"tz j; y g zyHel—theg. (6.7)
O 0 H The equation of motion (6.1) thus becomes
.. e . .. 6 . ..
majzng, myz—EmH, mz=0. (6.8)
The last of these equations imply
z(t) = z(0) + i(0) t. (6.9) So the charged particle moves along the z—direction with constant velocity 2(0).
To obtain the motion projected on the m — y plane we have to solve the ﬁrst two
coupled equations in (6.8). Deﬁning a; E g , (6.10)
mc
the coupled equations can be written
:1: = my, 37 2: —w:'c. (6.11)
These can be solved most readily by invoking the complex function ((75) : R ——> (3
deﬁned by
((75) E (C(15) +iy(t) . (6.12)
Then, from (6.11), :3
§[email protected]+¢(—m)=—m(1~+z‘y), (6.13)
or .u .
C: —i Lug. (6.14)
This equation can be integrated once to yield
at) = 6(0) e—m . (6.15) and then another time to obtain ((t)=C(0)+ /0 C(t)dt=€(0)+§(0) /0 eiwtdt. (6.16) 29 The ﬁnal result is at) = 4(0) + 52810 — 6....) , (6.17) where
6(0) = mm) + m0) , 6(0) = 93(0) + 2' 12(0) , (6.18)
[C(O) I = v (constant speed) . (6.19) Let us assume that the charged particle is an electron so that e < 0 and
thus to < 0. The complex numbers in (6.17) can be Viewed as twodimensional
vectors and the projected motion of the particle on the (It — y plane can be
obtained diagrammatically in a straightforward manner. In fact, deﬁning the
constant complex number 77 3 91(3) 7 (6.20)
so that .
ln= ¥ =12 Z—nH—c , (6.21) we see that the projected motion is uniform circular motion with angular speed
w = ([6 H)/(mc) (see Fig. 6.2). The radius of the circular orbit is 77 = v/[w. Fig. 6.2 LS; 396,2 30 CHAPTER 6. ELECTRIC AND MAGNETIC FIELDS ...
View
Full Document
 Spring '08
 LAM
 mechanics

Click to edit the document details