P19. Theory and Design for Determining the Charge-to-mass Ratio of Electrons

Introduction

The problems to follow are an application of what you've learned about electric and magnetic fields and forces.  Elementary particles are often characterized not by their charge or mass alone but rather by the ratio of the two.  That's because q/m shows up frequently in equations involving electric and magnetic forces.  For example, the acceleration of a charged particle in a uniform electric field is a = qE/m.  For another example, the radius of the path of a charged particle in a uniform magnetic field is r = mv/qB.  In both cases, the ratio q/m or its inverse appears.

The apparatus to be used is a classic one for measuring q/m for electrons.  A photo of the cathode-ray tube together with a simplified graphic are shown below.  A filament heated with a 6.3-V AC source "boils" off some electrons.  These electrons have very little energy initially.  However, some of them enter a uniform electric field between two plates maintained at a constant potential difference of V1.  The electrons are accelerated in the field and leave the plates through an aperture at Q.  The electrons then enter another electric field between two deflection plates a distance L long and separated by distance d.  The plates are maintained at a constant potential difference of V2.  Between the plates the electrons follow a parabolic path characteristic of charged particle motion in a uniform electric field.  This animation depicts the situation.

Photo of electron tube Graphic of tube

You'll show in Case 1 below that the apparatus just described can't, in and of itself, be used to determine the charge-to-mass ratio.  That is, a deflecting electric field alone is insufficient.  By adding a magnetic field, however, two different methods can be used to determine q/m.  You'll investigate those methods in Cases 2 and 3. In solving the problems below, you'll be using the three major problem-solving techniques of the first trimester. This is an opportunity to apply much of what you've learned this year to develop the theory for a laboratory situation.

General Instructions

  • Work individually in solving the problems. This is not a group assignment.

  • In your solutions, let q represent the charge in both sign and magnitude.  Let |q| represent only the magnitude. As always, it's important to be cognizant of signs in your solutions.

  • Present your solutions neatly and in an organized way. If your work is incomplete or difficult to follow, it may be returned to you for a rewrite and would then incur a grade penalty. Note the following:

    • Show successive steps in derivations one below the previous one. Don't show steps side-by-side, because that format is difficult to follow.

    • Label your work using the same labels as below (for example, Case 1, Problem 1)

    • Use word phrases to guide the reader through your solutions and to justify and amplify your mathematical statements. A list of equations will be insufficient.

    • If your penmanship is marginal or if a computer will help you to display your work better, submit a word-processed document on paper instead. Otherwise, a handwritten paper is acceptable.

Case 1.  Electric field only

Problem 1

Goal: Determine an equation for the velocity, vo, of electrons as they leave the accelerating plates and enter the deflection plates. Give your equation in terms of these parameters only: charge q of the particle, mass m of the particle, accelerating potential V1. Assume that the electrons have negligible kinetic energy as they enter the region between the accelerating plates at point P.

Setup: This is a conservation of energy problem. While you should automatically be doing each of the following when solving conservation of energy problems, we list the details below since some people have been leaving them out. Clearly identify each of the following:

  1. What is your system (objects and forces)?
  2. Are there external forces that need to be considered?
  3. Describe the initial state in a sentence. (Where is the particle and what is it doing?)
  4. Describe the final state in a phrase. (Where is the particle and what is it doing?)

Solution: Write the general conservation of energy equation. Substitute the initial and final energy expressions and solve for vo.

Checks: Show each of the following.

  1. Is the result real? Check that the quantity inside the radical is real. Don't overlook the sign of q.
  2. Do the units reduce to m/s? Show the units reduction.

Problem 2

Refer to the diagram labeled Case 1 to the right.  x- and y-axes have been drawn.  Electrons enter the field of the deflection plates at the origin and leave the field at the point (L,y).  We're ignoring fringing of the field beyond the plates. 

Goal: Determine the equation of the path and show that it's independent of the charge-to-mass ratio. By path, we mean to find an equation for y as a function of x. Here are some things you know from previous work that should help in solving this problem.

  • applying the definition of electric field and Newton's 2nd Law to determine the acceleration of a charged particle in the field
  • applying the relationship E = -DV/Ds to a uniform electric field
  • applying dvat equations in 2-dimensions

Solution: Use the above techniques and the result of Problem 1 to determine a formula for y(x) in terms of these quantities only:  x, d, V1, V2. The symbols t (for time) and vo must not appear in your result.

Check: Open this animation. Substitute values from the applet into your equation. Does your equation give the correct value of y for a given x? Show your substitutions and calculation. Show that the units reduce to the expected units.

Case 2.  Magnetic field only

Problem 3

Refer to the diagram labeled Case 2 to the right.  Note that the potential difference across the plates is now zero. Instead we have a magnetic field produced by a large coil encircling the plates. Actually, there are a pair of identical side-by-side coils one behind the other. The purpose of these coils is to produce a uniform magnetic field in the region between the plates.  (Coils in such an arrangement are termed Helmholtz coils.)  It's possible to determine the charge-to-mass ratio by determining the radius of the electron's circular path in this field.

Goal: Determine an equation for the charge-to-mass ratio of the electron in terms of the accelerating potential V1, the radius of the path R, and the magnetic field B. Your equation may have no other letter symbols.

Strategy: Use the result of Problem 1 (Case 1) above as well as the physics that you know for the motion of a charged particle in a uniform magnetic field, and show your work appropriately for this problem type. Draw your force diagram for the instant that an electron enters the magnetic field.

Checks: Open this animation. Substitute values from the applet into your equation. Does your equation give the correct value of q/m? Show your substitutions and calculation. Show that the units reduce to the expected units.

 

Case 3.  Crossed electric and magnetic fields

Problem 4

You've read about velocity selectors in the textbook and have done a problem with them on M10c.  Setting up a velocity selector is another way to measure q/m.  Refer to the diagram labeled Case 3 to the right.  The potential difference between the deflection plates has been restored.  In addition, there is current in the coil.  The effect is to balance the electric and magnetic forces so that the electrons travel at constant velocity.

Goal: Determine an expression for q/m in terms of V1, V2, d, and B only.

Strategy: We leave this up to you this time.

Checks: Check your equation by using this animation. First determine the values of the parameters necessary for the electron to move along the x-axis at constant velocity. Then substitute these values into your equation above. Does your equation give the correct value of q/m?  Show your check, including the values you substituted. Make sure the units reduce to the expected result.