Lab IV, Problem 5: Deflection of an Electron Beam and Velocity

Emily YoungFebruary 6, 2015

Physics 1202W, Professor: Thomas Walsh, TA: Sergey Monin

AbstractThe vertical displacement of an electron passing through a horizontal set of oppositely charged plates was calculated

using kinematics, Newton’s laws, and the properties of electric forces. The results were confirmed to within expected

experimental error.

Introduction

As a part of a team designing a particle accelerator used on malignant tumors, the aim of the charged particles was studied using a cathode ray tube (CRT). It is necessary to accurately predict the vertical displacement of the electron as a result of the potential difference between two charged plates. The properties of electric forces, along with kinematics and Newton’s laws, were used to predict the vertical displacement.

Prediction

The vertical displacement of the electron beam can be predicted using the properties of electric forces along with kinematics and Newton’s laws.

First, a potential difference Vacc is applied in the positive x direction. The expression for electric potential gives us the electric field Ex over a displacement ∆x,

Ex=V acc∆ x (1)

To find the speed Vx as the electron (mass m) enters the charged plates, we use the laws of kinematics.

F∆x=½mVx2 (2)

Since F=qE and q in this case is the charge of the electron e, equations 1 and 2 give us

Vacce= ½mVx2 (3)

Therefore,

Vx=√❑ (4)

As the electron enters the plates, the downward pointing electric field subjects it to an upward vertical force. The expression for electric potential Vd with a plate separation s gives for the vertical electric field Ey:

Ey=Vd/s (5)

The electron experiences vertical acceleration for the time period it is between the plates. Newton’s second law F=ma and F=qE where q=e combined with equation 5 give for the vertical acceleration a,

a=eV dsm (6)

Since the initial vertical speed of the electron =0, equation 6 gives us the vertical speed of the electron as it leaves the charged plates,

Vy=t1eV dsm (7)

where t1 is the time the electron spends between the plates. t1 can be calculated using the known horizontal distance the electron travels through the plates L and equation 4,

t1=L

√❑ (8)

Then Vy as the electron leaves the plates is

Vy=eV dsm

⋅ L√❑

(9)

The total vertical distance displaced ∆ytot can be found by adding the vertical distance displaced within the charged plates ∆y1 (while it is experiencing vertical acceleration) and the vertical distance displaced after the electron

leaves the plates ∆y2 (while the vertical speed of the electron is constant). Since the initial vertical speed is zero, using equations 6 and 8 and simplifying gives

∆y1= ¼L2❑V dV acc S

(10)

Using equation 9 for the period of constant vertical speed (zero acceleration), we geteVdsm

∆y2 =eVdsm

⋅ L√❑⋅t2 (11)

t2 can be found using equation 4 and the known horizontal distance traveled D (since the horizontal speed is constant),

t2=D√❑

(12)

That gives the vertical distance traveled during the period of constant vertical speed,

∆y2= VdLD

2VaccS (13)

Adding equations 10 and 13 and simplifying gives the total vertical distance displaced deltaYtot,

∆ytot= VdL

2VaccS⋅(D + L/2) (14)

Equation 14 predicts the vertical displacement of the electron.

Procedure

Figure 1: The mechanism used in this experiment

Figure 1 illustrates the cathode ray tube (CRT) used in this experiment. The potential difference Vacc was kept constant along with the the length of the plates L, the plate separation S, and the horizontal distance traveled past the charged plates D. Various voltages were Vd were placed on

the Vy set of plates, and the vertical deflection of the electrons ∆ytot was recorded over 18 trials.

Data

Constants:D= 0.074m L= 0.02m S= 0.003m Vacc= 250V

Voltage versus vertical displacement:

Trial Applied voltage Vd (V)

Experimental vertical displacement (mm)

Theoretical vertical displacement (mm)

1 18.97 11 21

2 13.77 8 15

3 9.90 6 11

4 6.35 4 7

5 2.09 2 2

6 0.00 0 0

7 18.99 11 21

8 13.78 8 15

9 9.89 6 11

10 6.38 4 7

11 2.79 2 3

12 0.00 0 0

13 19.03 11 21

14 13.84 8 15

15 9.91 6 11

16 6.11 4 6

17 2.76 2 3

18 0.00 0 0

Average percent error: 34.1%Standard Deviation: +4.22mmUncertainty: ±13.2mm (predicted) ±4.0mm (measured)Expected range of error: ±17.2mm

Graph 1: Voltage Vs. Displacement (Measured and Theoretical)

Analysis

The uncertainty for both the experimental (measured) vertical displacement and theoretical (predicted) vertical displacement was determined by calculating the maximum possible variation in either direction by accounting for any uncertainty in the measurements used to calculate the angles. The expected range of error was calculated by combining both uncertainties to create parameters for a “worst case scenario” for both the theoretical and experimental angles. Although the average percent error was considerably high at 34.1%, both the standard deviation (4.22mm) and maximum deviation (10mm) fell within the expected range of error (±17.2mm).

The random error in this experiment is reflected in the uncertainty for the predicted vertical displacement and comes primarily from the voltage reading, which jumped around considerably during the experiment. The consistency of the measured vertical displacement over several trials per voltage suggests that there wasn’t significant random error in the measured displacement- but systematic errors led to considerable inaccuracy in the measured displacement. Systematic error exists in the measured vertical displacement, as the measurements toward the outer edges of the

screen are considerably less accurate. This is clearly reflected in the results, as the deviation jumps from around zero millimeters at low voltages to ten millimeters at high voltages, consistently (measurements were made for six different voltages, three times each).

Conclusion

A cathode ray tube (CRT) was used to model particle accelerators used on malignant tumors. The aim of the charged particles was studied using the vertical deflection of an electron beam through a CRT as a result of a vertical electric field. The properties of electric fields, along with Newton’s laws and kinematics, were used to predict the vertical displacement of the electron beam. Using a constant electric potential Vacc and varying applied voltages Vd, the vertical displacement was measured and compared to the predicted values. With a standard deviation of +4.22mm, the predicted and measured vertical displacements fell within each other’s error parameters. The consistent trend of increasing deviation with larger applied voltages/vertical displacements reflects the systematic error in the reading of the deviation towards the outer edges of the screen. A more accurate cathode ray tube and measurements kept within the inner portion of the screen would have yielded more accurate results.