Lab Report 1: Deflection of a Electron Beam By An Electric Field
Carter Bauman
February 17, 2022
Physics 1302W, Professor Sudhakar Prasad, TA: Hongyu Wang
Abstract
In the lab a cathode ray tube (CRT) was used to determine the deflection of an electron beam due
to the electric field. We used these findings to determine the relationship between deflection and
the voltage applied. Using the theoretical value of .003 m for the separation of the plates we
increased the voltage to get larger deflections to create a Deflection vs. Voltage graph and a
Deflection vs. Electric Field Graph. From the data we gather we were able to determine a value
of .41 mm/(N/C) as well as the fact that deflection is a linear function of the electric field.
Introduction
Using a CRT, we are trying to determine how an electron deflects as it goes through an electric
field based on varying voltages. We then measured each deflection as the voltage varied
throughout the experiment.
Prediction
The change in position of the electron can be found using the electric field produced by the
electric field as it passes through the deflection plates in the CRT.
To begin solving for the deflection caused by the electric field as voltage increases, we must split
the equation into two parts, since it follows a different trajectory when between the deflection
plates vs. outside.
Δy = Δy1 + Δy 2
Δy1 is for when the electron is between the deflection plates, meaning it is under the force of the
electric field, and therefore accelerating:
∆𝑦1 = ∆𝑦𝑜 + 𝑉𝑦𝑜 +
1
2
2
𝑎𝑡
We can simplify this equation to get rid of the first two terms since the initial y position as well
as the initial y velocity is zero. The acceleration can be replaced by qEx/me for the acceleration of
an electron due to the electric field.
Δ𝑦1 =
1 𝑞𝐸𝑥 2
t
2 𝑚𝑒 1
Since t1 is an unknown value, but in doing so it also produces the term vz which needs to be
calculated as well.
t1=
𝐿
𝑣𝑧
and 𝑣𝑧 =
Using these values, we can simplify Δy1 even further:
2𝑞𝑉𝑎𝑐𝑐
𝑚𝑒
∆𝑦1 =
1 ∆𝑉𝐿
4 𝑆𝑉𝑎𝑐𝑐
The second part of the deflection, ∆y2, happens when the electron is passed the deflection plates,
meaning it does not undergo any acceleration and follows a linear path.
∆𝑦2 = 𝑣𝑦𝑡2
With t2 being equal to distance over velocity, and vy converting we get this equation:
∆𝑦2 =
𝑞𝐸𝐿 𝐷
𝑚𝑒𝑣𝑧 𝑣𝑧
This can be simplified when subbing in for vz:
∆𝑦2 =
∆𝑉 𝐿𝐷
𝑆 2𝑉𝑎𝑐𝑐
Now that we have a way to find both Δy1 and Δy2 we can combine them to get the overall
deflection of the electron beam. Where ΔV is potential difference, D is the distance travelled, L
is the length of the plates, S is the distance between plates, Vacc is the acceleration between the
plates.
Δy = Δy1 + Δy 2 =
1 ∆𝑉𝐿
4 𝑆𝑉𝑎𝑐𝑐
+
∆𝑉 𝐿𝐷
𝑆 2𝑉𝑎𝑐𝑐
Produce
Starting with the power off, we connected our power supply to the CRT using the AC 6.3V to
connect to the Heather port using the wires. After this we plugged our other wires into the -250V
and 250V anode and cathode to have a potential difference of 500. To start the experiment we
turned the CRT power supply on and continually increased the voltage until its maximum: 19
volts. After each increase in voltage the deflection was measured and then plotted to determine
the relationship between voltage and deflection
Data/Analysis
The measurement for the slope of the deflection vs. voltage graph came out to be 4.54 * 10-4 +/1.18 * 10-6 m/volts at 500 V. With this error we can conclude that this is likely accurate with
99.6% certainty.
A potential source of error during the experiment may have been that of a faulty CRT. When we
looked at the placement of the beam when no voltage was applied, the beam was not centered at
zero, meaning that the beam may have been travelling at an angle. By travelling at an angle, our
assumption when calculating the deflection becomes unjust, and our values wouldn’t match up
comparably with the theoretical. The beam also disappeared at some stages of the experiment
when the voltage increased, giving random deflection.
At the beginning of the experiment some values describing the measurements of the CRT were
given, which enabled us to find theoretical values of what we should get close to when
measuring the deflection. These values were:
𝐷 = 7.4 cm, 𝐿 = 2.0 cm, 𝑆 = 0.30 cm, Dtot= 9.6 cm
These values allowed us to see what the predicted values would be and that gave us a equation
for the predicted values of 5.6 * 10-4 m/volts for the slope which is higher than what we
calculated in the lab. The overall of the equation came out to Δy = 5.6 * 10-4ΔV + 2.65 * 10-7 m.
Using the two equations that we now have we can calculate the percent error between the two.
To do this we plug the values into this equation:
Percent Error =
Percent Error =
𝑆𝑙𝑜𝑝𝑒𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙− 𝑆𝑙𝑜𝑝𝑒𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙
𝑆𝑙𝑜𝑝𝑒𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
.00056− .000454
.00056
× 100%
× 100% = 1.89%
With such a small percent error it is assumed that our prediction stands and that our equation can be used
to determine the deflection of an electric beam when differing the voltage.
Fig. 1 graph presenting the relationship between deflection and voltage as a linear function
Conclusion
In the lab, a cathode ray tube (CRT) was used to determine the deflection as voltage, and therefore electric
field, was increased at a potential difference of 500V. The equation we found in lab for this was y = (4.54
* 10-4 +/- 1.18 * 10-6)V+1.82 * 10-4 m which was very close to the predicted values from the
equations that were derived prior to the experiment being done. The overall results determined
that the deflection and voltage are directly proportional, meaning that deflection and electric field
strength are also directly proportional since electric field is just a factor of the applied voltage.