What is the ratio of the kinetic energies for an alpha particle and a beta particle if both make

What is the ratio of the kinetic energies for an alpha...

Key Concepts

Kinetic Energy Expression in a Magnetic Field

Using the expression for the radius and the relation between velocity, charge, mass, and magnetic field, the kinetic energy of a charged particle can be expressed as KE = ½ m (qBr/m)² = ½ (q²B²r²)/m. This shows that for particles following the same curved path in a magnetic field, the kinetic energy is proportional to the square of the charge divided by the mass.

Influence of Particle Mass and Charge in Energy Comparison

When comparing different particles such as alpha and beta particles in a magnetic field, differences in their masses and charges are crucial. The kinetic energy ratio depends on the factor (q²/m) for each particle. Hence, even with the same radius of curvature, a particle with a higher charge-to-mass ratio will have a different kinetic energy. This concept is essential in deducing the relationship between the kinetic energies of particles of different types under identical magnetic deflection conditions.

Motion of Charged Particles in Magnetic Fields

Charged particles moving perpendicular to a magnetic field experience a Lorentz force that causes them to travel in curved paths. The fundamental relationship is given by equating the magnetic Lorentz force (qvB) to the centripetal force (mv²/r), which leads to the formula r = mv/(qB). This equation is central to analyzing particle trajectories in magnetic fields.

Relationship Between Momentum, Charge and Radius

The radius of curvature in a magnetic field depends on the momentum of the particle and its charge, as seen in the equation r = p/(qB). Since momentum (p) equals the product of mass and velocity (mv), this relation illustrates how differences in mass and charge between particles affect their paths. It forms the basis for comparing particles like alpha and beta particles when they have the same curvature in a magnetic field.

Transcript

00:01 Okay, so if we're trying to compare the kinetic energies for an alpha and a beta particle, they both have the same radius.

00:07 We want to figure out what determines the radius.

00:10 So basically, this is probably a review, but in general, why do these particles move in a curved path? it's that their centripetal force, which their centripetal force is the lorentz force, which is qvb, and then for circular motion.

00:29 You can set the mass times the acceleration, the net 4 is equal to mv squared over r.

00:34 So if we're looking at their velocity, actually, this is kind of nice because i did this differently in my notes, but now i'm seeing a nice way to do it.

00:47 So mv squared is equal to rqvb.

00:55 No, that's, oh yeah, that's right.

00:57 And so, and then mv squared is proportional to the kinetic energy.

01:03 And so to find the ratio of the kinetic energies, oh, weird, this is giving it.

01:13 Oh, oh, oh, i can't do it in this clever way.

01:17 Okay, rewind, unclever way coming.

01:23 I forgot these vs canceled.

01:25 Okay, so basically let's get a formula for the velocity.

01:30 So v is equal to rqvb.

01:35 Oh, and then these vs cancel.

01:37 So sorry, let me cancel that.

01:39 So sat over these v's canceling.