Phys105A_W09_HW2.5sols

Phys105A_W09_HW2.5sols - ξ ( t ) = i η ω exp(-iωt + iθ...

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Physics 105A Homework # 3 Solutions Richard Eager Problem (1) Taylor 2.53 (a) A particle of charge q and mass m is moving in constant electric E and magnetic B Felds ± E = (0 , 0 ,E ) ± B = (0 , 0 ,B ) . The force is ± F = q ( ± E + ±v × ± B ) = q ( Bv y , - Bv x ,E ) Expanding Newton’s second law m d±v ddt = ± F in components yields the three equations m ˙ v x = qBv y m ˙ v y = - qBv x m ˙ v z = qE The last equation yields z ( t )= z 0 + v z 0 + 1 2 a z t 2 where we have deFned the acceleration coe±cient a z = qE m . The x and y equa- tions of motion are coupled and we can solve them as outlined in section 2.5. Let η = v x + iv y . Then the Frst two equations become ˙ η =( - ) η where ω = qB m is the cyclotron frequency. The solution of this Frst order linear di²erential equation is η ( t )= η 0 exp( - iωt ) . If the initial velocity has components in the x and y directions, e.g. η 0 = v x 0 + iv y 0 it may seem complicated to expand everything out. However if we write η 0 = | η 0 | e 0 then η ( t )= | η 0 | exp( - iωt + 0 ) which has components v x ( t )= | η 0 | cos( - ωt + θ ) v y ( t )= | η 0 | sin( - ωt + θ ) 1
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Call the complexifed position ξ = x + iy so ˙ ξ = η. When we integrate the velocity η to get position
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Unformatted text preview: ξ ( t ) = i η ω exp(-iωt + iθ ) we see that the position and velocity have phases that diFer by a ±actor o± i. Thus we’ve recovered the ±act that velocity is perpendicular to position ±or uni±orm circular motion. [N.B. this is not true when the ±requenct ω is a ±unction o± time. In ±act it is straight-±orward to fnd the answer in this case as long as you remember that dω dt is no longer 0.] ²inally absorbing some constants in r, θ we have the equations o± motion x ( t ) = r cos(-ωt + θ ) y ( t ) = r sin(-ωt + θ ) z ( t ) = z + v z + 1 2 a z t 2 This simply describes a helix with pitch that increases as a ±unction o± time. ± 1.0 ± 0.5 0.0 0.5 1.0 ± 1.0 ± 0.5 0.0 0.5 1.0 1 2 3 4 ²igure 1: Charged Particle in Constant aligned E and B felds 2...
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This note was uploaded on 12/27/2011 for the course PHYS 105a taught by Professor Van,d during the Fall '08 term at UCSB.

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Phys105A_W09_HW2.5sols - ξ ( t ) = i η ω exp(-iωt + iθ...

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