Please solve (a) and (b) and (c) A particle of mass m and charge +q starts from

Question

Please solve (a) and (b) and (c)

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. Explain why the path has this general shape and why it is repetitive. Prove that the speed of the particle at any point is equal to squareroot 2qyE/m 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

Given,

mass of the particle = m , its charge = q , electric field = E and magnetic field = B

(a) When the moving particle enters the region of uniform electrostatic and magnetic field. it experiences forces due to the present of both the fields.

Due to the electric field the particle gets accelerated upwards. Then under the influenece of magnetic field, the particle experieneces a magnetic force which is perpendicular to both the velocity and magnetic field, and hence the particle follows a circular path. At the top point E and B fields apposes each other and due to this the particle begins to move downwards and this keep on repeating.

b) From energy conservation we have,

gain in kinetic energy = loss in electrostatic energy

1/2 m v2 = q E x y

v = sqrt [ 2 q y E / m ]

c)There are three forces acting on the particle, such that:

Fmagnetic - Felectrostatic = Fcentripital

q v B - q E = m v2 / r

using the above expression derived for v we get

q v B - q E = m x ( 2 q y E / m ) / 2 y

q v B = 2 q E

v B = 2 E

v = 2E/B

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