P. Piot, PHYS 571 – Fall 2007Particle in a static E-field• Consider a charge particle q interacting with a E-field E=Ex.• Particle’s initial conditions p(t=0)=p0y. • The Lorentz force is:• integrate• So that • total energy a time t is:P. Piot, PHYS 571 – Fall 2007Particle in a static E-field• The velocity is then• Integrate• Similarly for y axis we have• IntegrateP. Piot, PHYS 571 – Fall 2007Particle in a static E-field•Explicit t as a function of y• And insert in x(t)• Note the NR limitP. Piot, PHYS 571 – Fall 2007Particle in a static E-fieldP. Piot, PHYS 571 – Fall 2007Particle in a static B-field• Consider a charge particle q interacting with a B-field B=Ez.• Particle’s initial conditions p(t=0)=p0y. • The Lorentz force is:•From• We get the 3 equationsP. Piot, PHYS 571 – Fall 2007Particle in a static B-field• These are a system of 3 coupled ODEs of the formwith • We can cast the transverse equations into one equation• Whose solution is so finally ( )P. Piot, PHYS 571 – Fall 2007Particle in a static B-field• This is the equation of an helix with axis along z and radius R• R is the gyro-radius•ωis the gyro-frequency • In SI
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