Lab V, Problems 6 & 8: Magnetic Force on a Moving ChargeEmily YoungApril 1, 2015

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

AbstractThe deflection of an electron beam as a result of a constant applied magnetic field was predicted using Newton’s laws, the magnetic force law, kinematics, the right

hand rule and the Biot-Savart law. The results fell outside the error parameters, but confirmed the results predicted by the right hand rule.

Introduction

As part of a research group attempting to improve the mechanism guiding an electron beam across a sample in an electron microscope, the deflection of an electron as a result of a constant magnetic field was studied. After setting up Helmholtz coils so that the distance between the centers of the coils and the radii of the coils were equal and establishing that the magnetic force in between the coils was relatively constant, a cathode ray tube was placed between the two coils so that the magnetic field was pointing perpendicular to the direction of the electron beam. The magnetic field between the coils was both measured using a magnetic field sensor (Hall probe) as well as calculated using the known current and radii of the coils. The deflection of the electron beam was predicted using kinematics, Newton’s laws, the magnetic force law, the Biot-Savart law, and the right hand rule.

Prediction

The deflection of an electron beam can be predicted using Newton’s laws, the magnetic force law, the laws of kinematics, and the Biot-Savart law.

A moving charge placed in a magnetic field perpendicular to it’s direction of motion will follow a circular pathway. The pathway of the electron beam inside the cathode ray tube will follow the pathway of a circumference 2πR with a radius R, stopping when it hits the plate of the CRT. This deflection ∆y plus an unknown d equal the radius R, as shown in the following diagram:

Then, the deflection of the electron beam ∆y can be expressed in terms of R and d:

∆y= R- d (1)

Since the horizontal distance D of the cathode ray tube is known, the unknown d can be found using the pythagorean theorem:

d= (R2 - D2)½ (2)

Then the deflection ∆y can be rewritten as:∆y= R- (R2 - D2)½ (3)

R is unknown, but can be derived by setting the magnetic force and electrostatic force equal to each other:

Fm= Fe (4)

Using the expressions for magnetic force and Newton’s second law (F=ma) and the expression for angular acceleration v2/R, equation 4 can be rewritten as:

QvB= mv2/R (5)

Solving for R gives:R= mv/QB (6)

Where R is the radius of the circular electron pathway, v is the velocity, m is the mass of the electron, Q is the charge of the electron, and B is the magnetic field.

Now the deflection can be rewritten as:

∆y= mv/QB - ((mv/QB)2 - D2)½ (7)

There are two unknowns in this equation- v and B. To find the speed v of the electron, we use the laws of kinematics. Work is force times distance (W=F⋅ d). Kinetic energy, which is work, can be expressed as ½mv2. Setting the two expressions equal to each other gives:

F ⋅ d=½mv2 (8)

Since F=QE, equation 8 can be rewritten as:QEd= ½mv2 (9)

Plugging in the expression for potential difference V=∫❑

❑

❑Ed and simplifying gives:

VaccQ= ½mv2 (10)

Solving for v then gives v=√❑ (11)

Next, B can be derived using the Biot-Savart law.

dB= ∫❑

❑

❑k ' Idl⋅ rr 3

(12)

For one coil with N turns and radius R, the magnetic field B at a distance x from the center of the coil having angle between coil segment dl and vector r to point P is:

dB= ∫❑

❑

❑Nk ' Idlr sin θr 3

(13)

The integral of dl in this case is the circumference of a circle, which is 2πR. Since sinθ= x/r, this can be rewritten as:

dB= ∫❑

❑

❑Nk ' I 2 π Rx

r 3 (14)

The distance r from the coil to the point where the magnetic field is being solved for can be expressed in terms of the horizontal distance from the center of the coil x and the radius of the circle R using the pythagorean theorem:

r= (R2 + x2)½ (15)

Then the magnetic field can be rewritten as:

B= ∫❑

❑

❑Nk ' I 2 π Rx

(R 2+x2)3/2 (16)

Then, with two coils, if D is the distance between the two coils and x is the distance from the midpoint between the two coils, the total magnetic field at a point x is the sum of the magnetic field resulting from each coil:

Then the distance from the center of the coil x in equation 15 can be rewritten as (D/2 - x) for B1 and (D/2 + x) for B2:

Btot= B1 + B2= ∫❑

❑

❑Nk ' I 2 π Rx

((D /2−x )2+R 2)3 /2 + ∫❑

❑

❑Nk ' I 2 π Rx

((D /2+x)2+R 2)3/2 (17)

Then, if the coil separation D is equal to the radii of the coils R, then solving for the magnetic field at the midpoint between the coils where x=0 gives:

B= ∫❑

❑

❑Nk ' I 2π Rx

((R/2)2+R 2)3 /2 + ∫❑

❑

❑Nk ' I 2π Rx

((R/2)2+R 2)3 /2 (18)

Simplifying this expression gives

B= ∫❑

❑

❑32 π k '∋ ¿R(125)1/2

¿ (19)

Finally, the solutions for B and v can be plugged into equation 7 and simplified to give:

∆y=Rm (2 QVacc/m)1/2

8.99 k ' NQI - ((

Rm (2 QVacc/m)1/28.99 k ' NQI

)2 - D2)½ (20)

Equation 20 predicts the vertical deflection of the electron beam, where is the deflection, R is the radius of the Helmholtz coils, Q is the charge of the electron, Vacc is the potential differeance, I is the current, and D is the horizontal distance. If the magnetic field is pointing perpendicular to the electron beam and out of the page, the right hand rule states that the electron will be deflected upwards.

Procedure

Figure 1: The mechanism used to create a magnetic field

Figure 1 depicts the mechanism used to create a magnetic field in this experiment. Two Helmholtz coils with N turns and radii R separated by R both have a current I running through them. A DC power supply connected directly across the coils provides the current as well as measures it. A Hall Probe (magnetic field sensor) placed in the center of the coils (lying horizontally) is attached to logger pro, recording the magnetic field between the coils.

Figure 2: Cathode Ray Tube (CRT)

Figure 2 depicts the cathode ray tube (CRT), which was placed horizontally between the two coils in Figure 1 so that the magnetic field pointed perpendicular to the beam of electrons and in the positive z direction, as shown. The electrons were deflected upwards a distance ∆y from the center as a result of the magnetic field.

Data

Radius R= 0.1m Number of coils N= 200 Horizontal distance D= 0.096m Electron charge Q= 1.6x10-

19C Electron mass m= 9.11x10-31kg Potential Difference Vacc= 375 VProportionality constant k’= 10-7 N/A2

The Magnetic Field at Different Points Along the X Axis

Graph 1: The magnetic field at different points along the x axisGraph 1 depicts the magnetic field (in Gauss) resulting from the Helmholtz coils at different points along the x axis. The midpoint between the coils was at 10cm, which was also where the peak magnitude of magnetic field occurred, as shown in the graph.

Experimental Data and Theoretical Values from CRT Electron Deflection:

Trial current I (A) magnetic field B (Exp.) (T)

magnetic field B (Theor.) (T)

deflection ∆y (Exp.)

deflection ∆y (Theor.)

1 0.01 1.25x10-5 8.99x10-6 0.001 0.00088

2 0.02 2.10x10-5 1.8x10-5 0.002 0.0015

3 0.03 2.95x10-5 2.70x10-5 0.003 0.0021

4 0.04 3.80x10-5 3.60x10-5 0.004 0.003

5 0.05 4.65x10-5 4.50x10-5 0.005 0.003

6 0.06 5.50x10-5 5.40x10-5 0.006 0.004

7 0.07 6.35x10-5 6.29x10-5 0.007 0.005

8 0.08 7.20x10-5 7.18x10-5 0.008 0.006

9 0.09 8.05x10-5 8.08x10-5 0.009 0.006

10 0.10 8.50x10-5 8.98x10-5 0.0105 0.006

11 0.11 9.75x10-5 9.88x10-5 0.0115 0.007

12 0.12 1.06x10-4 1.07x10-4 0.013 0.008

13 0.13 1.145x10-4 1.16x10-4 0.014 0.009

14 0.14 1.23x10-4 1.25x10-4 0.015 0.012

15 0.01 1.25x10-5 8.99x10-9 0.0005 0.00088

16 0.02 2.10x10-5 1.8x10-5 0.0015 0.0015

17 0.03 2.95x10-5 2.70x10-5 0.0025 0.0021

18 0.04 3.80x10-5 3.60x10-5 0.0035 0.003

19 0.05 4.65x10-5 4.50x10-5 0.005 0.003

20 0.06 5.50x10-5 5.40x10-5 0.006 0.004

21 0.07 6.35x10-5 6.29x10-5 0.007 0.005

22 0.08 7.20x10-5 7.18x10-5 0.008 0.006

23 0.09 8.05x10-5 8.08x10-5 0.0095 0.006

24 0.10 8.50x10-5 8.98x10-5 0.0105 0.006

25 0.11 9.75x10-5 9.88x10-5 0.012 0.007

26 0.12 1.06x10-4 1.07x10-4 0.013 0.008

27 0.13 1.145x10-4 1.16x10-4 0.014 0.009

28 0.14 1.23x10-4 1.25x10-4 0.015 0.012

Uncertainty of Magnetic Field (Exp.): ±8x10-7TUncertainty of Magnetic Field (Theor.): ±1x10-4TExpected Range of Error of Magnetic Field: ±1.01x10-4TStandard Deviation of Magnetic Field: ±3.5x10-6TAverage Percentage Error of Magnetic Field: ±4.53%Uncertainty of Deflection (Exp.): ±0.002mUncertainty of Deflection (Theor.): ±0.0006mExpected Range of Error of Deflection: ±0.0026mStandard Deviation of Deflection: ±0.0045mAverage Percentage Error of Deflection: 29.5%

Electron Deflection by a Magnetic Field, Experimental Avg. vs. Theoretical

Graph 2: Electron Deflection (mm), Experimental Avg. versus Theoretical Graph 1 shows the deflection of the electron beam (in millimeters) as a result of an applied magnetic field (in Gauss). The green line represents the experimental deflection values, while the purple line represents the theoretical (calculated) deflection values.

Analysis

The theoretical and experimental uncertainties for both the magnetic field and the deflection of the electron beam were determined by calculating the maximum possible variation in either direction by accounting for any uncertainty in the measurements used to calculate the figures. The expected range of error for both the magnetic field and the deflection was calculated by combining the uncertainty parameters for the theoretical and experimental values to calculate a “worst case scenario” or maximum possible expected error.

The standard deviation of the magnetic field, at ±3.5x10-6T (with an average percent error of ±4.53%), was found to fall within the uncertainty parameters of ±1.01x10-4T. The standard deviation of deflection, at ±0.0045m (with an average percent error of 29.5%), was found to fall outside the uncertainty parameters of ±0.0026m.

Several sources of consistent human error existed in this experiment, which may explain the the high percentage of error in the deflection of the electron. The first is in the centering of the cathode ray tube (CRT) between the Helmholtz coils. Any deviation from the center (which was determined by eye) would result in a change in the magnetic field, which would impact the deflection reading.

As Graph 1 shows, the magnetic field increased from one coil to the other, reaching its maximum at the midpoint between the two coils and decreasing again beyond that point. This is consistent with the prediction made in equation 17. The magnetic field increases with proximity to the coil, so the magnetic field between two coils is at its maximum at the midpoint between the two coils, where the contribution to the magnetic field from both coils is equal.

As Graph 2 depicts, the experimental deflection was consistently higher in value than the theoretical deflection. Both a smaller potential difference Vacc and greater magnetic field B would result in a greater deflection, which suggests that the Vacc value used to calculate B may have been smaller than 375 volts, and the magnetic field was greater than the values recorded.

In this experiment, the Helmholtz coils were oriented so that the magnetic field was pointing out of the page (positive z direction), with the cathode ray tube placed between the coils so that the screen points towards the right (the electrons traveling in the positive x direction). Since the electrons are negatively charged, the right hand rule states that an upward force is exerted on the electrons, meaning they will be deflected upwards- which was confirmed experimentally.

A more thorough and accurate experiment would include a mechanism to ensure that the cathode ray tube is perfectly centered, as well as accurate placement of the magnetic field sensor.

Conclusion

To model the path of an electron beam in an electron microscope, a cathode ray tube was placed so that a constant magnetic field was perpendicular to the electron beam. The deflection of the beam was then predicted and experimentally determined. A system of Helmholtz coils was used to produce a magnetic field pointing out of the page, with the cathode ray tube placed in between the coils so that the beam would be deflected upwards. The deflection of the beam was predicted using kinematics, Newton’s laws, the magnetic force law, and the Biot-Savart law. The magnetic field prediction was confirmed to within experimental error, with an average percent error of ±4.53%. The standard deviation in deflection, however, fell well outside the expected parameters given by the experimental and theoretical uncertainty values. This can be explained by possible error in the value of the potential difference Vacc, as well as in the centering of the cathode ray tube (which would impact the magnitude of the magnetic field). In general, however, the electrons were deflected upwards as expected by the right hand rule.