Two charged particles are projected (in the same
plane as the paper) in opposite directions at the
same level into a region of uniform magnetic field B
as shown in the figure (Assume the region covered by
B extends vertically for very long distance).
i. Determine if the two particles are likely to collide
or not? Why?
ii. Sketch the paths of both particles.
iii. Determine the length of the track for each particle
and the time spent inside the region covered by B.
To determine if the two particles are likely to collide or not, we need to analyze their motion in the presence of the magnetic field. The interaction between a charged particle and a magnetic field creates a magnetic force, known as the Lorentz force, which acts perpendicular to both the velocity of the particle and the magnetic field. The Lorentz force is given by the equation:
F = q(v x B)
where F is the magnetic force, q is the charge of the particle, v is its velocity, and B is the magnetic field.
i. To determine if the two particles will collide, we need to consider the direction of the magnetic force acting on each particle. If the magnetic forces on the particles are opposite to each other in direction, then the particles will experience a net force towards each other, which can cause them to collide. On the other hand, if the magnetic forces on the particles have the same direction, then the particles will experience a net force away from each other, and they will not collide.
ii. To sketch the paths of both particles, we need to consider the motion of each particle in the presence of the magnetic field. The Lorentz force causes the particles to move in circular paths, with a radius determined by the magnitude of the velocity and the magnetic field. The direction of the circular path is given by the right-hand rule. If we let the velocity of the particles point in the x-direction and the magnetic field point in the z-direction (out of the plane of the paper), then applying the right-hand rule, we find that the positive charge particle will move in a counterclockwise circle, while the negative charge particle will move in a clockwise circle.
iii. To determine the length of the track for each particle and the time spent inside the region covered by B, we need to consider the speed and the radius of the circular path. The magnitude of the velocity remains constant for each particle, but the direction changes due to the circular motion. The radius of the circular path can be calculated using the equation:
r = mv / (qB)
where r is the radius, m is the mass of the particle, v is its velocity, q is the charge, and B is the magnetic field.
The length of the track can be calculated using the formula for the circumference of a circle:
length = 2πr
The time spent inside the region covered by B can be determined by dividing the length of the track by the speed:
time = length / speed
where speed is the magnitude of the velocity of the particle.