A stream of charged particles accelerates through a potential difference V as sh

Question

A stream of charged particles accelerates through a potential difference V as shown and enters a region of uniform magnetic field, B, directed along the z–axis (out of page). The beam consists of two types of particles having the same charge, –q, but having different masses, m1 and m2 with m1 < m2.

a) What is the ratio of kinetic energies of the two types of particles just before they enter the magnetic field?

Suppose that one of each type of particle enters the field at the origin at t = 0.

b) Which particle crosses the y-axis first?
c) Which particle crosses the y-axis farthest from the origin?
d) Suppose m1 crosses the y-axis at y1 and m2 crosses at y2, what is the distance between y1 and y2?

Explanation / Answer

(a) Since the potential energy given by the electric field is U = q*V, it doesn't depend on the mass of the particle.

The kinetic energy(KE) of both particles is given by the electric field, meaning that
U = KE

Thus, both particles have the same KE. The ratio of KE = 1:1 = 1

(b) KE = 0.5*m*v2

since m1 < m2 and they have the same KE, v1 must be greater than v2.

(v1 > v2)

Thus, m1 crosses the y axis first.

(c) When the particles go into the Magnetic field, they will start to do circular motion.

Since F = q*v*B and v1 > v2, F1 > F2 when q and B are the same as stated in the question.

Note that F1 and F2 are centrifugal forces.

F1 = q*v1*B = m*v12/R1 = 0.5*m*v12 / (0.5*R1) = KE/(0.5*R1)

R1 = 2KE/(q*v1*B)

R2 = 2KE/(q*v2*B)

Since v1>v2, R1 < R2

As a result, m2 would travel furthest from the origin.

(d) y1 = 2*R1 and y2 = 2*R2

Thus, y2 - y1 = 2*(R2 - R1)

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