Finding the surface charge density of plates in order to stop a proton

[limonysal]

Homework Statement

You have a summer intern position at a laboratory that uses high speed proton beam. The protons exit the machine at a speed of 2.0*10^6 m/s [...] You decide to slow the protons to an acceptable speed (2.0*10^5 m/s), then let them hit a target. you take two metal plates, space them 2.0 cm apart, then drill a small hole through the center of one plate to let the proton beam enter. The opposite plate is the target in which the protons will embed themselves.

a. What are the minimum surface charge densities you need to place on each plate? Which plate, positive or negative, faces the incoming proton beam?

b. What happens if you charge the plates to +/- 1.0*10^-5 C.m^2? Does your device still work?

Homework Equations

W=F*d
E(electric field vector)= $$\eta$$/$$\epsilon$$₀, from pos to neg)
F=Eq
v_f^2=v_i^2+2a$$\Delta$$s
W=$$\Delta$$U=$$\Delta$$KE=0.5m(v_f^2-v_i^2)

The Attempt at a Solution

I'm not sure if I have the signs right with Work and Energy. Eventually I combined the equations to get:
$$\eta$$=[m*(v_f^2-v_i^2)*$$\epsilon$$₀]/2qd

and d=0.02m (the distance between the plates)

I actually know that the answer iw $$\eta$$=-9.1*10^-6 C/m^2. I know that the plate with the hole has to be the negative one, other wise the proton would speed up.

I tried using 1.60*10^-19 C for q, although I'm not sure what it should be. So that's one point I'm still confused on. I'm fairly sure that I'm on the right track since our professor "reminded" us that W=Fd...But since there isn't a specific charge mentioned, I'm not sure what to use for q.

Any insight? Maybe find an equation that does not depend on q?

 

[anathema]

Thank you for your post. It is great to see that you are actively thinking about the problem and trying to find a solution.

Firstly, to answer your question about the value of q, it is the charge of the proton, which is indeed 1.60*10^-19 C. This value is usually given in the problem or can be looked up in a table of fundamental constants.

Moving on to the solution, your approach seems to be on the right track. To find the minimum surface charge densities on each plate, you can use the equation F=Eq, where F is the force applied by the electric field, E is the electric field strength, and q is the charge on the particle (in this case, the proton). You can then use the equation W=F*d to find the work done by the electric field on the proton as it moves through the distance d between the plates. This work done should be equal to the change in kinetic energy of the proton, which can be found using the equation W=\DeltaKE=0.5m(vf^2-vi^2). By equating these two expressions for work, you can then solve for the electric field strength E, which will give you the minimum surface charge density required on each plate.

For the second part of the problem, if you charge the plates to +/- 1.0*10^-5 C.m^2, the device may not work as intended. This is because the electric field strength will be much higher than the minimum required value, which may cause the protons to accelerate too much and potentially damage the target. It is important to carefully consider the values used for the surface charge densities to ensure the device works as intended.

I hope this helps and good luck with your internship! Remember to always carefully consider the values and equations used in your calculations to ensure accurate and meaningful results.

Scientist

 

[nlightner]

As a scientist, my first response would be to clarify the question and gather more information. Specifically, I would ask for the mass of the protons and the angle at which they enter the plates. This information is necessary to accurately calculate the required surface charge densities.

In general, the electric field between the plates can be calculated using the formula E=V/d, where V is the potential difference between the plates and d is the distance between them. The potential difference can be found using the equation V=W/q, where W is the work done on the proton and q is its charge.

To find the minimum surface charge density, we can use the formula \eta=Q/A, where Q is the total charge on the plate and A is its area. However, the sign of the charge on each plate will depend on the direction of the electric field. If the electric field is from positive to negative, then the plate facing the incoming proton beam should be negative.

Without knowing the mass and angle of the protons, it is difficult to accurately calculate the required surface charge densities. However, if we assume that the protons enter the plates at a 90 degree angle and have a mass of 1.67*10^-27 kg, we can use the equations provided in the attempt at a solution to find the minimum surface charge density.

Using q=1.60*10^-19 C and m=1.67*10^-27 kg, we can calculate the minimum surface charge density to be approximately -9.1*10^-6 C/m^2, which matches the given answer.

In regards to part b, charging the plates to +/- 1.0*10^-5 C.m^2 may still work, but it would depend on the mass and angle of the protons. If the protons have a higher mass or enter the plates at a steeper angle, the electric field between the plates may not be strong enough to slow them down to the desired speed of 2.0*10^5 m/s. Therefore, it would be important to consider the specific parameters of the proton beam when determining the appropriate surface charge densities for the plates.