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# CMChap6 - Chapter 6 Charged Particle in Electric and...

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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=eE-v. 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) (5-6) 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 e-iwtdt. (6.16) 29 The ﬁnal result is at) = 4(0) + 5-2810 — 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 two-dimensional 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 ...
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CMChap6 - Chapter 6 Charged Particle in Electric and...

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