A particle of mass m and charge +q starts from rest at the origin. There is a un

Question

A particle of mass m and charge +q starts from rest at the origin. There is a uniform electrostatic field E in the positive Y direction and a uniform magnetic field B out of the page. In more advanced courses, the path of the particle is shown to be a cycloid whose radius of curvature at the top points is twice the y coordinate at that level.

a) Explain why the path has this general shape and why it is repetitve.

c) Prove that the speed at the top point is 2E/B. (You can use the result in Part b) and the relation stated about radius and y.)

A particle of mass m and charge +q starts from rest at the origin. There is a uniform electrostatic field E in the positive Y direction and a uniform magnetic field B out of the page. In more advanced courses, the path of the particle is shown to be a cycloid whose radius of curvature at the top points is twice the y coordinate at that level. a) Explain why the path has this general shape and why it is repetitve. b) Prove that the speed of the particle at any point is equal to c) Prove that the speed at the top point is 2E/B. (You can use the result in Part b) and the relation stated about radius and y.)

Explanation / Answer

Initially particle is at rest so no force will act on it due to magnetic field

only force will act on the particle is electric force that will be

F = qE

as the particle will start moving upward due to electric field then magnetic field will also start acting as it act on moving charge only

the direction on magnetic force will be perpendicular to both magnetic field and velocity

so the direction of velocity due to magnetic field will be +ve x axis

now particle will start to deflect

as the particle will deflect and the direction of velocity will change so will the direction of the magnetic force will change

since no force is acting along z axis so motion will be in the x-y plane

this will result in the cycloid motion of the particle

magnitude of velocity is dependent only on electric force

so at any point of time its centrifugal force = electric force

m * v^2 / r = qE

v = sqrt(qEr/m)

here r = 2y so,

v = sqrt(2qyE/m)

speed of the particle at any point = sqrt(2qyE/m)

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