Theory and Design for L23

Introduction

The problems to follow are an application of what you've learned about electric and magnetic fields and forces.  The problems also serve as a prelab to a lab in which you'll take measurements from photographs in order to determine the charge-to-mass ratio of the electron. 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 the following as well as in your solutions, let q represent the charge in both sign and magnitude.  Let |q| represent only the magnitude. Whether you use q or |q|, maintain consistency in signs and notation throughout your work.

Case 1.  Electric field only

1.  Begin this problem by determining an equation for the velocity, v0, of electrons as they leave the accelerating plates and enter the deflection plates. This is a conservation of energy problem. Clearly indicate your system and initial and final states and start with a correct statement for the law of conservation of energy. Assume that the electrons have negligible kinetic energy as they enter the region between the accelerating plates at point P.  Give your result for v0 in terms of V1 and q/m. Mentally check that your result has the correct units and doesn't give an imaginary number.

2.  Now 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.  The goal is to show that the path of electrons in the field is independent of their 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:

  • The magnitude of the acceleration of a charged particle in a uniform field is a = |q|E/m. (E represents the magnitude of the field.)
  • The magnitude of the uniform electric field between parallel plates separated by distance d and maintained at potential difference V is E = V/d.
  • The horizontal velocity and vertical acceleration of an electron are both constant. Thus, you can apply dvat equations along the axes.

Use the above facts (including the result of step 1) to determine a formula for y(x) in terms of these quantities only:  x, d, V1, V2. You should find that the q/m ratio will divide out. You'll also need to eliminate time, t, from the equation.

3.  Check your equation by using this animation. Substitute values from the applet and calculate y for a value of x that you select. Does your equation give the correct value of y for a given x? Show your check, including the values you substituted. Make sure the units reduce to the expected result.

Case 2.  Magnetic field only

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.  Do the following.

4.  The magnetic field is produced by current which travels the same direction in both coils. In order for the electrons to curve downward as shown, what must be the direction of the current in the coils, clockwise or counterclockwise?  Tell how you found the answer. (Hint:  The direction of the magnetic field is determined the same way as for a current loop.)

5.  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.  You'll need to use the result of step 1 and do a net force problem for the motion of a charged particle in a uniform magnetic field.

6. Check your equation by using this animation. Substitute values from the applet. Tell how you determined the radius of the path. 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.

7.  In order to actually use the apparatus to determine q/m, one has to have a way to measure the radius of the path.  The diagram below completes the circular path as a dashed line in order to show the center and radius of the circle.  A background grid has also been drawn between the plates as a way to take measurements of position coordinates.  Devise a way that you could use position coordinates read from the grid in order to calculate the radius of the circle. Place yourself in the position of the experimenter.  This means that you can only view the apparatus and the electron beam from the outside, because the apparatus is enclosed in a vacuum in a large glass vessel.  (A good vacuum is needed to keep collisions between electrons and gas molecules to a minimum.  Collisions would change the direction and magnitude of the electron's velocity.)  By the way, you're able to view the path of the electrons, because they're made to skim along beside a phosphorescent screen on which the grid is drawn.  As some of the electrons strike the phosphor, they produce a glow.  (This works similar to the way the electron beams in your TV set cause the phosphors on the inside surface of the screen to glow different colors.)  Here's a photograph showing the phosphorescent glow.

Describe your method in enough detail that the reader could follow your instructions to determine the radius of the path. You'll need to define symbols, draw a diagram, and derive any equations that you'll need in algebraic form.

Important note: While one can visually estimate the radius by extrapolating the circular arc to an angle of p/2, this method is too crude to achieve the accuracy expected for this problem.

Case 3.  Crossed electric and magnetic fields

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 still current in the coil.  The effect is to balance the electric and magnetic forces so that the electrons travel at constant velocity.

8.  Determine an expression for q/m in terms of V1, V2, d, and B. Show your work completely.

9. 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 from step 8. 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.