Accelerating Ions through semicircular plates

[tmvphil]

Homework Statement

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

Homework Equations

The Attempt at a Solution

Well i started with
$$q*V_0 = KE = (mv^2)/2$$
so
$$V^2 = (2qV_0)/m$$
which combined with the rotational motion eq
$$E = F/q = (mv^2)/(qr_0) = (2V_0)/r_0$$
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?

 

[tmvphil]

Attached a diagram (sorry for the paint) if it helps make it clearer

 

[thomate1]

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.