Two particles that have the same mass and same magnitude charge are following co

Question

Two particles that have the same mass and same magnitude charge are following concentric circular orbits in a uniform magnetic field in the directions shown below. One particle has a radius of 13.5 cm and the other has a radius of 3.0 cm. Each makes a complete trip around the circular orbit in 5.1 ms (1 ms = 10^-3 s). The strength of the magnetic field is 1.6 times 10^-4 T. Each particle has mass of 24 u (1 u = 1.66 times 10-27 kg). (Ignore the interaction between the particles. That is, assume one particle does not affect the interaction other.) a) Do the particles have the same type of charge (positive or negative)? Determine the type of charge for each particle. b) What is the speed of each particle?

Explanation / Answer

part a:

let direction to the right be +ve x axis, direction in upward direction is +y axis and direction along out of the page is +ve z axis.

force on a charged particle in a magnetic field is given by

force=charge*cross product of velocity and magnetic field

for the two particles, the velocity direction is different.

but at every point,direction of force is same .

so their charges are different.

for the particle with larger radius:

at the top most point , velocity is along -ve x axis, field is along +ve z axis.

force acting on it is along -ve y axis.

as force=charge * cross product of velocity and field

cross product of velocity and field is along +ve y axis

so the charge is negative in nature.

for the particle with smaller radius:

at the top most point of its trajectory, velocity is along +ve x axis, field is along +ve z axis

cross product of velocity and field is along -ve y axis.

force is along -ve y axis.

hence charge is positive in nature.

part b:

speed=circumference of the trajectory/time taken

=2*pi*radius/time taken

for the particle with radius of 13.5 cm:

speed=2*pi*0.135/(5.1*0.001)

=166.32 m/s

for the particle with radius of 3 cm:

speed=2*pi*0.03/(5.1*0.001)=36.96 m/s

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