In an experiment with cosmic rays, a vertical beam of...

In an experiment with cosmic rays, a vertical beam of particles that have charge of magnitude 3$e$ and mass 12 times the proton mass enters a uniform horizontal magnetic field of 0.250 $\mathrm{T}$ and is bent in a semicircle of diameter $95.0 \mathrm{cm},$ as shown in Fig. E27.22. (a) Find the speed of the particles and the sign of their charge. (b) Is it reasonable to ignore the gravity force on the particles? (c) How does the speed of the particles as they enter the field compare to their speed as they exit the field?

Key Concepts

Gravitational versus Magnetic Forces

In many experiments involving charged particles in magnetic fields, the gravitational force can often be ignored if it is much smaller than the magnetic force. Comparing the magnitudes of the forces involves assessing the acceleration or deflection produced by gravity relative to the centripetal deflection induced by the magnetic field. This helps in simplifying the analysis by neglecting forces that do not significantly affect the particle's motion.

Lorentz Force and Circular Motion

When a charged particle enters a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field known as the Lorentz force. This force acts as the centripetal force required to keep the particle moving in a circular path. By equating the magnetic force to the centripetal force, one can determine properties like the speed of the particle or the radius of its path.

Right-Hand Rule and Charge Sign Determination

The right-hand rule is used to determine the direction of the magnetic force acting on a charged particle in a magnetic field. By pointing the fingers of the right hand in the direction of the particle's velocity and curling them towards the magnetic field, the thumb indicates the direction of the force for a positive charge; a negative charge would experience force in the opposite direction. This principle allows one to deduce the sign of a particle's charge based on the observed curvature of its trajectory.

Conservation of Kinetic Energy in a Magnetic Field

A magnetic field does no work on a charged particle because the magnetic force is always perpendicular to the particle's velocity. As a result, the particle's speed remains constant as it moves through the field, even though its direction changes. This conservation of kinetic energy is a crucial concept when comparing the speeds of the particle at different points along its path.

Transcript

00:01 A circle or in an arc of a circle, the force is always perpendicular to the tangential velocity and is always towards the center of the circle.

00:16 And the value of force is given by mv square over r, where r is the radius of this semicircle.

00:34 Now, since we have a magnetic field over here, so this force f is caused by the magnetic field or this force is the magnetic force which is given by charge times velocity cross the magnetic field vector.

01:00 Now the direction of this force is then given by the right hand rule.

01:09 And so where phi is the angle between velocity and the magnetic field.

01:20 So here, phi is zero because magnetic field is outwards from the screen and velocity is on the plane.

01:29 So, phi is 90 degrees.

01:31 So this is equal to q times v times b.

01:35 Now, going with the right -hand rule, we see that the velocity, the force, if the charge is positive, the force will be in this direction.

01:51 And if the charge is negative, the force will be in this direction, which is same as the centripetal force over here.

02:03 Now since these two forces are the same, so we confirm that this charge q should be negative.

02:13 So let's write this down.

02:16 So this is the same.

02:16 So this is.

02:16 Is actually q should be negative.

02:24 Now we can combine these two equations and since we are just concerned with the magnitude of force now because we already know the direction.

02:34 So we'll say mv square over r is equal to q mod times v over b.

02:47 So here we found that q is negative so basically q should be equal to negative times 3 times 1 .6 times 10 to minus 19 columns from here.

03:06 So the mod of q will just have the positive part...

Please give Ace some feedback