A proton follows a spiral path through a gas in a magnetic...

A proton follows a spiral path through a gas in a magnetic field of 0.018 $\mathrm{T}$ , perpendicular to the plane of the spiral, as shown in Fig. $54 .$ In two successive loops, at points $P$ and $O,$ the radii are 10.0 $\mathrm{mm}$ and 8.5 $\mathrm{mm}$ , respectively. Calculate the change in the kinetic energy of the proton as it travels from $\mathrm{P}$ to $\mathrm{Q}$ .

A proton follows a spiral path through a gas in a magnetic field
of 0.018 $\mathrm{T}$ , perpendicular to the plane of the spiral, as shown in Fig. $54 .$ In two successive loops, at
points $P$ and $O,$ the radii are
10.0 $\mathrm{mm}$ and 8.5 $\mathrm{mm}$ , respectively.
Calculate the change in the kinetic
energy of the proton as it travels
from $\mathrm{P}$ to $\mathrm{Q}$ .

Key Concepts

Charged Particle Motion in Magnetic Fields

When a charged particle moves in a magnetic field, it experiences a force (the Lorentz force) that is perpendicular to both its velocity and the field. This force does not change the speed of the particle by itself, but it continuously redirects the particle, causing it to follow a curved, typically circular or helical, path.

Centripetal Force and the Radius-Velocity Relationship

In a uniform magnetic field, the magnetic force acts as the centripetal force necessary for circular motion. The balance between the magnetic force (qBv) and the required centripetal force (mv²/r) leads to a relation where the product of the particle's speed and mass equals the product of its charge, the magnetic field, and the radius of curvature (m*v = q*B*r). This relationship shows that the radius of the circular path is directly proportional to the particle’s speed.

Kinetic Energy Dependence on Speed

The kinetic energy of a particle is given by ½mv². Using the relationship between speed and radius in a magnetic field, kinetic energy can be seen to be proportional to the square of the orbit radius (KE ? r²) if the magnetic field and particle charge remain constant. Thus, any change in the radius of the particle's path can be directly related to a change in its kinetic energy.

Energy Dissipation in a Medium

While the magnetic force itself does no work on the particle, a charged particle moving through a medium such as a gas can lose energy through collisions or other forms of friction. These energy losses manifest as a reduction in the speed (and thus the kinetic energy) of the particle, which in the context of circular motion will lead to a decrease in the orbit radius.

Transcript

00:01 Okay, so during chapter 27, problem 73.

00:04 It says a proton follows a spiral path through a gas and a magnetic field of b, 0 .018 tesla, perpendicular to the plane of the spiral.

00:16 In two successive loops at points p and q in the picture, i'll go ahead and draw what this looks like here.

00:28 So this just spirals in, and the points they showed were here and here.

00:37 So this is p, this is q.

00:42 It says the radii are at point p is 10 millimeter, and r at point q is 8 .5 kilometer.

00:55 So it says calculate the change in kinetic energy of the proton as it travels from point p to q.

01:03 Okay, so we can calculate the speed by relating the forces together because we know that the magnetic force q, q, is going to be equal to the m times the centripetal acceleration.

01:17 This is r not two sorry.

01:20 M v squared over r.

01:21 So we can relate this to being v is given by qbr over the mass.

01:32 And from this we know that the kinetic energy is one half in b squared so we can plug that in.

01:38 So one half and now we have the q squared, b squared, r squared over m squared.

01:47 One of these ms is going to cancel out and we're left with q squared, b squared, r squared over 2m.

01:56 Okay, so now if we're looking for the change in kinetic energy, this is going to be the kinetic energy final minus the kinetic energy initial, or the kinetic energy at point q minus the kinetic energy up point p...

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