Particle with specific charge q/m moves in the region of space where there are uniformly mutually perpendicular electric and magnetic fields with strength Ej and induction Bk. At moment t = 0 velocity of charge particle at origin is voi. Find speed at (x, y)
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Answer:
The equation of motion is, So, the equation becomes, Here, The last equation is easy to integrate; vz = constant = 0, since vz is zero initially. Thus integrating again, z = constant = 0, and motion is confined to the x - y plane. We now multiply the second equation by i and add to the first equation. we get the equation, This equation after being multiplied by be rewritten as, and integrated at once to give, where C and α are two real constants. Taking real and imaginary parts. Since vy = 0, when t = 0, we can take α = 0, then vx = 0 at t = 0 gives,C = -(E/B) and we get, Integrating again and using x = y = 0, at t = 0, we get This is the equation of a cycloid. (b) The velocity is zero, when ωt - 2nπ. We see that The quantity inside the modulus is positive for 0 < ωt < 2 π. Thus we can drop the modulus and write for the distance traversed between two successive zeroes of velocity.
Explanation:
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