- #1
JHans
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The question:
A particle in a cyclotron gaines energy
[tex]
q \Delta V
[/tex]
from the alternating power supply each time it passes from one dee to the other. The time interval for each full orbit is
[tex]
T = \frac{2 \pi}{\omega} = \frac{2 \pi m}{q B}
[/tex]
so the particle's average rate of increase in energy is
[tex]
T = \frac{2 q \Delta V}{T} = \frac{q^2 B \Delta V}{\pi m}
[/tex]
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. Show that the rate of increase in the radius r of the particle's path is given by
[tex]
\frac{d r}{d t} = \frac{1}{r} \frac{\delta V}{\pi B}
[/tex]My thoughts:
I'm completely stuck on how to accomplish this. I have a general idea, but I don't know how to go about it. My thought is: if the radius of motion increases, then the velocity must increase. For the velocity to increase, the kinetic energy must increase. That increase in energy comes from the electric field between the dees. I should be able to model a function of the kinetic energy at some time t and then put it into an equation that shows how the radius depends on the velocity of the particle. From there, I can just differentiate the radius equation with respect to time and prove the law. How, though, can I go about showing that the energy of the particle depends on time?
A particle in a cyclotron gaines energy
[tex]
q \Delta V
[/tex]
from the alternating power supply each time it passes from one dee to the other. The time interval for each full orbit is
[tex]
T = \frac{2 \pi}{\omega} = \frac{2 \pi m}{q B}
[/tex]
so the particle's average rate of increase in energy is
[tex]
T = \frac{2 q \Delta V}{T} = \frac{q^2 B \Delta V}{\pi m}
[/tex]
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. Show that the rate of increase in the radius r of the particle's path is given by
[tex]
\frac{d r}{d t} = \frac{1}{r} \frac{\delta V}{\pi B}
[/tex]My thoughts:
I'm completely stuck on how to accomplish this. I have a general idea, but I don't know how to go about it. My thought is: if the radius of motion increases, then the velocity must increase. For the velocity to increase, the kinetic energy must increase. That increase in energy comes from the electric field between the dees. I should be able to model a function of the kinetic energy at some time t and then put it into an equation that shows how the radius depends on the velocity of the particle. From there, I can just differentiate the radius equation with respect to time and prove the law. How, though, can I go about showing that the energy of the particle depends on time?