Accelerating Ions through semicircular plates

In summary, the problem involves ions being accelerated through a potential difference V_0 and entering the space between semicylindrical electrodes A and B. It is shown that an ion will follow a semicircular path of radius r_0 if the potentials of the outer and inner electrodes are maintained at 2V_0*ln(b/r_0) and 2V_0*ln(a/r_0) respectively. The solution involves using the equations for kinetic and rotational motion, and integrating for a variable r from b to r_0 and a to r_0.
  • #1
tmvphil
4
0

Homework Statement


Hey guys, this is from Purcell 3.19
"Ions are accelerated through a potential difference [tex]V_0[/tex] and then enter the space between the semicylindrical electrodes A and B. Show that an ion will follow the semicircular path of radius [tex]r_0[/tex] if the potentials of the outer and inner electrodes are maintained, respectively, at [tex]2V_0*ln(b/r_0)[/tex] and [tex]2V_0*ln(a/r_0)[/tex]."


Homework Equations





The Attempt at a Solution


Well i started with
[tex]q*V_0 = KE = (mv^2)/2[/tex]
so
[tex]V^2 = (2qV_0)/m[/tex]
which combined with the rotational motion eq
[tex]E = F/q = (mv^2)/(qr_0) = (2V_0)/r_0[/tex]
but i can't think of any way to pin down what the field is between the two cylinders.
just looking at the solution provided it seems like i should integrate the final equation for a variable r from b to r-nought and a to r-nought. I just don't know why. Any way to elucidate this for me?
 

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  • #2
Attached a diagram (sorry for the paint) if it helps make it clearer
 
  • #3



Hi there,

It seems like you are on the right track with your approach. To find the electric field between the two semicircular plates, you can use the equation E = V/d, where d is the distance between the plates. Since the plates are semicircular, you can use the radius of the semicircle as the distance d.

Now, to find the electric potential at any point between the plates, you can use the equation V = -Ed, where d is the distance from the point to the negative plate. In this case, since the negative plate is at the center of the semicircle, you can use the distance from the center to the point as d.

Combining these two equations, you can find the potential at any point between the plates. Then, using the given potentials at the outer and inner electrodes, you can set up an equation and solve for the radius r_0. This will give you the radius of the semicircular path that the ion will follow.

I hope this helps. Let me know if you need further clarification.
 

Related to Accelerating Ions through semicircular plates

1. What is the purpose of accelerating ions through semicircular plates?

The purpose of accelerating ions through semicircular plates is to increase their kinetic energy and velocity, allowing them to travel at higher speeds.

2. How do semicircular plates accelerate ions?

Semicircular plates use an electric field to accelerate ions. When a positive charge is placed between the plates, it experiences a force in the direction of the electric field, causing it to gain kinetic energy and accelerate.

3. What are the benefits of accelerating ions through semicircular plates?

Accelerating ions through semicircular plates allows for the manipulation and control of ion beams, which has many practical applications in fields such as medicine, materials science, and imaging technology.

4. Are there any limitations to accelerating ions through semicircular plates?

One limitation is that the acceleration is dependent on the charge of the ion and the strength of the electric field, which may not be suitable for all types of ions. Additionally, the ions may experience collisions with other particles, reducing their acceleration.

5. Can semicircular plates be used to accelerate ions to extremely high speeds?

Yes, semicircular plates can be used to accelerate ions to very high speeds, but there may be practical limitations such as the size and cost of the equipment required to generate strong enough electric fields.

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