A particle in the cyclotron shown in Figure 29.16a gains energy q ΔV from the alternating power

A particle in the cyclotron shown in Figure 29.16a gains...

A particle in the cyclotron shown in Figure 29.16a gains energy $q \Delta V$ from the alternating power supply each time it passes from one dee to the other. The time interval for each full orbit is $$T=\frac{2 \pi}{\omega}=\frac{2 \pi m}{q B}$$ so the particle’s average rate of increase in energy is $$\frac{2 q \Delta V}{T}=\frac{q^{2} B \Delta V}{\pi m}$$ Notice that this power input is constant in time. On the other hand, the rate of increase in the radius $r$ of its path is not constant. (a) Show that the rate of increase in the radius $r$ of the particle's path is given by $$\frac{d r}{d t}=\frac{1}{r} \frac{\Delta V}{\pi B}$$ (b) Describe how the path of the particles in Figure 29.16a is consistent with the result of part (a). (c) At what rate is the radial position of the protons in a cyclotron increasing immediately before the protons leave the cyclotron? Assume the cyclotron has an outer radius of 0.350 m, an accelerating voltage of $\Delta V=600 \mathrm{V},$ and a magnetic field of magnitude 0.800 $\mathrm{T}$ . (d) By how much does the radius of the protons' path increase during their last full revolution?

A particle in the cyclotron shown in Figure 29.16a gains energy $q \Delta V$ from the alternating power supply each time it passes from one dee to the other. The time interval for each full orbit is
$$T=\frac{2 \pi}{\omega}=\frac{2 \pi m}{q B}$$
so the particle’s average rate of increase in energy is
$$\frac{2 q \Delta V}{T}=\frac{q^{2} B \Delta V}{\pi m}$$
Notice that this power input is constant in time. On the other hand, the rate of increase in the radius $r$ of its path is not constant. (a) Show that the rate of increase in the radius $r$ of the particle's path is given by
$$\frac{d r}{d t}=\frac{1}{r} \frac{\Delta V}{\pi B}$$
(b) Describe how the path of the particles in Figure 29.16a is consistent with the result of part (a). (c) At what rate is the radial position of the protons in a cyclotron increasing immediately before the protons leave the cyclotron? Assume the cyclotron has an outer radius of 0.350 m, an accelerating voltage of $\Delta V=600 \mathrm{V},$ and a magnetic field of magnitude 0.800 $\mathrm{T}$ . (d) By how much does the radius of the protons' path increase during their last full revolution?

Key Concepts

Differential Radial Increase

The change in the orbital radius over time, d(r)/dt, reflects how the energy input translates into an expanding spiral path. By relating the particle’s constant energy gain with its circular motion parameters (e.g., radius and cyclotron frequency), one can derive expressions that describe the instantaneous rate of radial increase, which is crucial to understanding the particle's trajectory inside the cyclotron.

Radius-Energy Relationship

The radius of a charged particle’s circular path in a magnetic field is linked to its momentum through the equation r = mv/(qB). As the particle gains energy via the accelerating voltage, its speed increases, resulting in a larger circular orbit. Understanding this relationship is vital for linking the energy input to the spatial motion of the particle in the cyclotron.

Energy Gain per Crossing

In a cyclotron, particles gain kinetic energy each time they cross the gap between the dees due to the applied accelerating voltage. The energy increment per crossing is quantified as q?V. This periodic and discrete energy gain, when averaged over time, helps determine the overall rate at which the particle’s energy increases.

Cyclotron

A cyclotron is a type of particle accelerator that uses a constant magnetic field to bend charged particles into circular paths while an alternating electric field accelerates them each time they cross the gap between the electrodes, or 'dees.' This combination allows the particles to gain energy in a stepwise manner and spiral outward as their speed increases.

Lorentz Force and Circular Motion

Charged particles moving in a magnetic field experience the Lorentz force, which acts perpendicular to their velocity, causing them to follow a circular path. This force, given by F = qvB, provides the necessary centripetal force for circular motion and is central to understanding how the particle's trajectory and radius are determined in a cyclotron.

Cyclotron Frequency

The cyclotron frequency refers to the constant angular frequency at which a charged particle orbits in a constant magnetic field. It is given by ? = qB/m, implying that the orbital frequency is independent of the particle’s speed, an essential property that allows the cyclotron to effectively accelerate particles over multiple cycles.

Transcript

00:01 We're told to refer to a figure of a cyclotron, and we're given these equations that are related to the cyclotron.

00:09 And this first equation here is the time interval that it takes for one full orbit of a particle in the cyclotron.

00:17 And the second equation is for the average rate of increase in the energy of a particle traveling in the cyclotron.

00:26 So this problem has multiple parts to it, and we're going to start with part a that asks us to prove, that this dr over dt is equal to 1 over r times the potential difference divided by pi times b.

00:44 And we're going to be using these two equations here to help solve this problem.

00:52 So to start, i'm going to first rewrite the first equation and then i'm also going to rewrite the second equation.

01:32 And what we can do with this equation here is that since we know that this is a rate increase, an average rate increase, we can write it in terms of a derivative.

01:47 So we can rewrite this as a derivative with respect to time, and we're going to rewrite the left side as 1 over 2 times m times v squared, and we're going to leave the right side unchanged.

02:16 So next thing i'm going to write the equation for the velocity of a particle traveling in the cyclotron, and that's going to be b times q times the radius of the cyclotron r divided by m and what we can do is we can take this term here for v and we can plug it into this term here so i'm now going to rewrite this equation here with the derivative as d over d t times one over two times m times the new value for b which will now become b times q times r divided by m squared and then the right side is going to remain unchanged and now we're going to simplify this equation further so simplifying we get b squared times d over d t times one -a -half r squared is equal to b times a change of v divided by pi and simplifying further, we get 2r divided by 2 times dr over dt is equal to the change in b divided by pi times b.

04:28 And then we can finally simplify it as dr over dt is equal to 1 over r times times v divided by pi times b.

04:46 And this is the final equation that we're looking for.

04:56 So now part b, part b is asking us to find or to describe the path of the particles with respect to this equation that we calculated from part a.

05:18 So looking at the figure and also at this equation, we can say that the rate of increase, we're going to say that from the equation, from the equation, from the equation, in part a, the rate of increase, we're going to say that from the equation, from the equation, the rate of increase.

05:56 Increase in the radius is inversely proportional to the radius r...

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