Mapping of the Magnetic Field from Helmholtz Coils

 

Mapping of the Magnetic Field from Helmholtz Coils

Name: John Gibson

Course code: 1440L

Section: 809

Lab partner name: (no partner)

Date of experiment: Apr 4, 2024

Objective

Measure magnetic field strength with distances of Helmholtz Coils to verify the theoretical correctness of uniform field strength at optimal position.

Introduction

According to Ampere’s law, a magnetic field B can be created with a moving charge or, equivalently, an electric current. 

Using Biot-Savart’s law, we can calculate the magnetic field by integrating a dB over a dl distance of the said moving charge or equivalent current, as follows,

dB=0I/4dl/r3, 0=410-7Tm/A

B=dBdl

.

When the current goes around a circle, the integration yields

B=dBdl=0I/42R2/(R2+z2)3/2 Equation 1

.

The Helmholtz coils produce a magnetic field by superpositioning 2 circular coils of N turns with current I in the same direction. So the magnetic field strength is  

B=0NI/42R2/(R2+z12)3/2+0NI/42R2/(R2+z22)3/2

=0NIR2/2(R2+z12)3/2+0NIR2/2(R2+z22)3/2 Equation 2.

, where z1 + z2 = h , z2 = h - z1 , or if z = z1, z2 = (h-z).

To derive the local minimum or local maximum B, 

dB/dz=0.

Solving the differential equation yields z = h/2 . Equation 3

To obtain the optimal h , dBz (z=h/2) / dh = 0 . Solving the differential equation yields h = R.

So, when z1=z2=R/2 , the optimal spacing of the 2 coils gives

B=80NNI/R(125)1/2 Equation 5

at the center point between the 2 coils.

Steps

There are 2 sections of the experiment, Section A to Section B. Before starting each of the 2 sections, calibrate the detector by 

1. Powering off the power supply.

2. Restart the computer Helmholtz program.

3. Select the axial configuration or radial configuration on the switch of the detector.

4. Click on the tare button of the detector. 

5. Click on the Zero button of the Helmholtz program.

For the 2 experiment sections,

1. Measure axial configuration Helmholtz coil magnetic fields with spacing 1.5R, 1.0R, and 0.5R

1. Space the Helmholtz coils at 1.5R , where R is the radius of the coil.

2. Place the movement arm through Helmholtz coils’ center, making a protrusion of about 30cm.

3. Connect the coils to the power supply at 1.00A. In the computer program Helmholtz, click on the Acquisition option for Axial. Enter R value, spacing value and number of coil N.

4. Click Helmholtz program’s Start button, and pull the string of the detector cart from one side of the coil through the coil to the other side for about 20cm travel on each side of the coil.

5. Click Helmholtz program’s Stop button.

6. Click on Analysis option for Axial in Helmholtz program.

7. Click on the Position button in the position tool pane.

8. Change the offset number to best align the theoretical magnetic field curve with measured magnetic field strength value data points.

9. Place the cursor at the optimal center data point and photograph the Helmholtz program screen.

10. Repeat from step 1 but change spacing to 1.0R and 

11. Estimate at 3cm to the right of the optimal center the magnetic field strength.

12. Estimate at 2cm to the right of the optimal center the magnetic field strength.

13. Estimate at 1cm to the right of the optimal center the magnetic field strength.

14. Estimate at 1cm to the left of the optimal center the magnetic field strength.

15. Estimate at 2cm to the left of the optimal center the magnetic field strength.

16. Estimate at 3cm to the left of the optimal center the magnetic field strength.

17. Perform calculation for deviation of measured data with theoretical data.

18. Repeat from step 1 but without estimate magnetic field strength between 3cm and center and with h=0.5R.

2. Measure radial, transverse configuration Helmholtz coil magnetic fields with spacing 1.0R

1. Space the Helmholtz coils at 1.0R , where R is the radius of the coil.

2. Place the movement arm through the space between Helmholtz coils, making a protrusion of about 30cm.

3. Connect the coils to the power supply at 1.00A. In the computer program Helmholtz, click on the Acquisition option for Transverse. Enter R value, spacing value and number of coil N.

4. Click Helmholtz program’s Start button, and pull the string of the detector cart from one side of the coil through the coil to the other side for about 20cm travel on each side of the coil.

5. Click Helmholtz program’s Stop button.

6. Click on Analysis option for Transverse in Helmholtz program.

7. Click on the Position button in the position tool pane.

8. Change the offset number to best align the measured magnetic field strength value data points near the center of the scope pane.

9. Place the cursor at the optimal center on the data point and photograph the program screen.

10. Place the cursor at 3cm to the right of the optimal center on the data point and photograph again.

11. Place the cursor at 2cm to the right of the optimal center on the data point and photograph again.

12. Place the cursor at 1cm to the right of the optimal center on the data point and photograph again.

13. Place the cursor at 1cm to the left of the optimal center on the data point and photograph again.

14. Place the cursor at 2cm to the left of the optimal center on the data point and photograph again.

15. Place the cursor at 3cm to the left of the optimal center on the data point and photograph again.

16. Perform calculation for deviation of measured data with theoretical data.

Apparatus and Procedure

Complete list of equipment

Helmholtz coils with 200 turns with a stand Ruler Computer acquisition unit and Helmholtz program

Hall sensor detector  Rail about 1 m with a pulley on either end. Power supply up to 1A Cart with a string looped to the pulley

Labeled block diagram of each part of the experiment

Describe the experiment

The power of the power supply is stoped before each section experiment. Restart the computer Helmholtz program for each section. Axial configuration then radial configuration on the switch of the detector was set. The tare button of the detector was pushed before running Acquisition. Zero button of the Helmholtz program was used to calibration zero magnetic field.

For the axial configuration Helmholtz coil magnetic fields measurement, space the Helmholtz coils at 1.5R , where R is the radius of the coil. Place the movement arm through Helmholtz coils’ center, making a protrusion of about 30cm. Connect the coils to the power supply at 1.00A. In the computer program Helmholtz, click on the Acquisition option for Axial. Enter R value, spacing value and number of coil N. Click Helmholtz program’s Start button, and pull the string of the detector cart from one side of the coil through the coil to the other side for about 20cm travel on each side of the coil. Click Helmholtz program’s Stop button. Click on Analysis option for Axial in Helmholtz program. Click on the Position button in the position tool pane. Change the offset number to best align the theoretical magnetic field curve with measured magnetic field strength value data points. Place the cursor at the optimal center data point and photograph the Helmholtz program screen. Perform calculation for deviation of measured data with theoretical data. Repeat from spacing the coils twice but change spacing to 1.0R and then 0.5R.

For the radial, transverse configuration Helmholtz coil magnetic fields with spacing 1.0R, Space the Helmholtz coils at 1.0R , where R is the radius of the coil. Place the movement arm through the space between Helmholtz coils, making a protrusion of about 30cm. Connect the coils to the power supply at 1.00A. In the computer program Helmholtz, click on the Acquisition option for Transverse. Enter R value, spacing value and number of coil N. Click Helmholtz program’s Start button, and pull the string of the detector cart from one side of the coil through the coil to the other side for about 20cm travel on each side of the coil. Click Helmholtz program’s Stop button. Click on Analysis option for Transverse in Helmholtz program. Click on the Position button in the position tool pane. Change the offset number to best align the measured magnetic field strength value data points near the center of the scope pane. Place the cursor at the optimal center on the data point and photograph the program screen. The the cursor is moved to 3cm to the right of the optimal center to obtain magnetic field strength data, then moved 1cm at a time to the left to obtain 7 data points. Then perform calculation for deviation of measured data with theoretical data

Results and Analysis

For Section A, The measured magnetic field is shown in the following Figure 1, Figure 2, and Figure 3, for spacing of the 2 coils of 1.5R, 1.0R, and 0.5R, respectively. The solid line in each figure is the theoretical magnetic field strength by calculation.

Figure 1. Magnetic Field with Axiel Coils Spaced 1.5R

 

Figure 2. Magnetic Field with Axiel Coils Spaced 1.0R

Figure 3. Magnetic Field with Axiel Coils Spaced 0.5R

Analysis

As shown in Figure 1 with h=1.5R, The measured center magnetic field strength is 11.6 Gauss.

According to Equation 1, the theoretical best-fitting curve shows that the center magnetic field strength should be

B=0NIR2/2(R2+z12))3/2+0NIR2/2(R2+z22))3/2

= 4*3.14159e-7*200*0.1025**2 / (2 * (0.1025**2+0.076875**2)**1.5) * 2 = (6.277*2)10e-4 Tesla

=12.55(Gauss) , notice that 1 Tesla = 10000 Gauss. 

Notice that that calculating string is of Python program language.

Also, as shown in Figure 1, when h=1.5R, when the spacing is larger than the optimal spacing for Helmholtz coils, there is a local minimum value, with the concave shape, for the center position’s magnetic field strength.   

Also, as shown in Figure 1, the theoretical best-fitting curve is always slightly higher than the actual measured values.

As shown in Figure 2 with h=1.0R, The measured maximum magnetic field strength is 16.8 Gauss.

According to Equation 1, the theoretical best-fitting curve shows that the center magnetic field strength should be

B=0NIR2/2(R2+z12))3/2+0NIR2/2(R2+z22))3/2

= 4*3.14159e-7*200*0.1025**2 / (2 * (0.1025**2+0.05125**2)**1.5) * 10000 * 2 = 8.772*2=17.54(Gauss)

And, according to Equation 5, the theoretical best-fitting curve shows that the center magnetic field strength should be B=80NI/R(125)1/2=0.001754 Tesla=17.54 Gauss.

Also, as shown in Figure 2, when h=1.0R, when the spacing is the optimal spacing for Helmholtz coils, there is a flat top curve around the center position’s magnetic field strength.

Also, as shown in Figure 2, the theoretical best-fitting curve is always slightly higher than the actual measured values.

The magnetic field strength deviation within 3cm of the center position can be estimated, namely,

3 cm to the right of center position, field strength deviation from center = |16.9 Gauss - 16.8 Gauss| = 0.1 Gauss,

2 cm to the right of center position, field strength deviation from center = |16.8 Gauss - 16.8 Gauss| = 0 Gauss,

1 cm to the right of center position, field strength deviation from center = |16.8 Gauss - 16.8 Gauss| = 0 Gauss,

1 cm to the left of center position, field strength deviation from center = |16.8 Gauss - 16.8 Gauss| = 0 Gauss,

2 cm to the left of center position, field strength deviation from center = |16.8 Gauss - 16.8 Gauss| = 0 Gauss,

3 cm to the left of center position, field strength deviation from center = |16.7 Gauss - 16.8 Gauss| = 0.1 Gauss.

3 cm to the right of center position, field strength percent deviation = 0.1 Gauss / 16.8 Gauss * 100% = 0%,

2 cm to the right of center position, field strength percent deviation = 0 Gauss / 16.8 Gauss * 100% = 0%,

1 cm to the right of center position, field strength percent deviation = 0 Gauss / 16.8 Gauss * 100% = 0.595%,

1 cm to the left of center position, field strength percent deviation = 0 Gauss / 16.8 Gauss * 100% = 0%,

2 cm to the left of center position, field strength percent deviation = 0 Gauss / 16.8 Gauss * 100% = 0%,

3 cm to the left of center position, field strength percent deviation = 0.1 Gauss / 16.8 Gauss * 100% = 0.595%,

Average magnetic field strength of the top curve is (16.9+16.8+16.8+16.8+16.8+16.7)/6 = 16.8 Gauss.

Standard deviation is (0.1 + 0 + 0 + 0 + 0 + 0.1) / 6 = 0.0333 Gauss. Percent deviation is 0.0333/16.8*100%=0.198%

As shown in Figure 3 with h=0.5R, The measured maximum magnetic field strength is 21.1 Gauss

According to Equation 1, the theoretical best-fitting curve shows that the center magnetic field strength should be

B=0NIR2/2(R2+z12))3/2+0NIR2/2(R2+z22))3/2

= 4*3.14159e-7*200*0.1025**2 / (2 * (0.1025**2+0.05125**2)**1.5) * 10000 * 2 = 11.19*2=22.39(Gauss)

Also, as shown in Figure 2, when h=0.5R, when the spacing is the optimal spacing for Helmholtz coils, the top curve is sharp around the center position’s magnetic field strength.

Also, as shown in Figure 3, the theoretical best-fitting curve is always slightly higher than the actual measured values.

For Section B, The measured magnetic field is shown in the following Figure 4 with numeric mark at the center.

Figure 4. The magnetic field strength at the center of the radial setup is 16.3 Gauss

As shown in below Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, and Figure 10, for measured magnetic field at 3cm to the right of the center, 2cm to the right of the center, 1cm to the right of the center, 1cm to the left of the center, 2cm to the left of the center, and 3cm to the left of the center, respectively.

Figure 5. Magnetic field at 3cm to the right of center	Figure 6. Magnetic field at 2cm to the right of center
Figure 7. Magnetic field at 1cm to the right of center	Figure 8. Magnetic field at 1cm to the left of center
Figure 9. Magnetic field at 2cm to the left of center	Figure 10. Magnetic field at 3cm to the left of center

Analysis

As shown in above Figure 4, The measured center magnetic field strength is 16.3 Gauss.

The theoretical best-fitting curve shows that the center magnetic field strength should be

B=0NIR2/2(R2+z12))3/2+0NIR2/2(R2+z22))3/2

= 4*3.14159e-7*200*0.1025**2 / (2 * (0.1025**2+0.05125**2)**1.5) * 10000 * 2 = 8.772*2=17.54(Gauss)

Also, as shown in Figure 4, when h=1.0R, when the spacing is the optimal spacing for Helmholtz coils, there is a flat top curve around the center position’s magnetic field strength.

Also, as shown in Figure 4, the theoretical best-fitting curve is always slightly higher than the actual measured values.

The magnetic field strength deviation within 3cm of the center position can be seen in Figures 5-10, namely,

3 cm to the right of center position, field strength deviation from center = |16.3 Gauss - 16.3 Gauss| = 0 Gauss,

2 cm to the right of center position, field strength deviation from center = |16.3 Gauss - 16.3 Gauss| = 0 Gauss,

1 cm to the right of center position, field strength deviation from center = |16.4 Gauss - 16.3 Gauss| = 0.1 Gauss,

1 cm to the left of center position, field strength deviation from center = |16.3 Gauss - 16.3 Gauss| = 0 Gauss,

2 cm to the left of center position, field strength deviation from center = |16.3 Gauss - 16.3 Gauss| = 0 Gauss,

3 cm to the left of center position, field strength deviation from center = |16.2 Gauss - 16.3 Gauss| = 0.1 Gauss.

3 cm to the right of center position, field strength percent deviation = 0 Gauss / 16.3 Gauss * 100% = 0%,

2 cm to the right of center position, field strength percent deviation = 0 Gauss / 16.3 Gauss * 100% = 0%,

1 cm to the right of center position, field strength percent deviation = 0.1 Gauss / 16.3 Gauss * 100% = 0.613%,

1 cm to the left of center position, field strength percent deviation = 0 Gauss / 16.3 Gauss * 100% = 0%,

2 cm to the left of center position, field strength percent deviation = 0 Gauss / 16.3 Gauss * 100% = 0%,

3 cm to the left of center position, field strength percent deviation = 0.1 Gauss / 16.3 Gauss * 100% = 0.613%,

Average magnetic field strength of the top curve is (16.3+16.3+16.4+16.3+16.3+16.2)/6 = 16.3 Gauss.

Standard deviation is (0 + 0 + 0.1 + 0 + 0 + 0.1) / 6 = 0.0333 Gauss. Percent deviation is 0.0333/16.3*100%=0.204%

Discussion

Compared to theory

For Section A, 

As shown in Figure 1, the measured values in data dot points closely track the theoretical value curve.

The center points’s theoretical value is B=0NIR2/2(R2+(h-z)2))3/2+0NIR2/2(R2+(h-z)2))3/2=(6.277+6.277)10-4Tesla

=12.55(Gauss)

, while the measured value is 11.6 Gauss. 

The difference is 12.55-11.6=0.95 Gauss , percent difference 0.95/12.55 * 100%=7.56% .

This lower measured value is likely due to the non-infinitely-small coil thickness because Equation 2 is derived based on the assumption that the current coil is infinitely narrow. Part of the current I running on the outer rim is more distant from the measuring sensor and expected to have a weaker magnetic field.

As shown in Figure 2, the measured values in data dot points closely track the theoretical value curve.

The center points’s theoretical value is   

B=0NIR2/2(R2+(h-z)2))3/2+0NIR2/2(R2+(h-z)2))3/2=8.772+8.772=17.54(Gauss).

, while the measured value is 16.8 Gauss. 

The difference is 17.54 - 16.8 = 0.74 Gauss , percent difference 0.74/17.54 * 100%=4.22% .

Again, this lower measured value is likely due to the non-infinitely-small coil thickness because Equation 2 is derived based on the assumption that the current coil is infinitely narrow. Part of the current I running on the outer rim is more distant from the measuring sensor and expected to have a weaker magnetic field.

The flat top curve has a standard deviation of 0.0333 Gauss. Percent deviation is 0.198% , which is very good.

As shown in Figure 3, the measured values in data dot points closely track the theoretical value curve.

The center points’s theoretical value is 

B=0NIR2/2(R2+(h-z)2))3/2+0NIR2/2(R2+(h-z)2))3/2=11.19+11.19=22.39(Gauss).

, while the measured value is 21.1 Gauss. 

The difference is 22.39-21.1= 1.29 Gauss , percent difference 1.29/22.39 * 100%=5.76% .

Again, this lower measured value is likely due to the non-infinitely-small coil thickness because Equation 2 is derived based on the assumption that the current coil is infinitely narrow. Part of the current I running on the outer rim is more distant from the measuring sensor and expected to have a weaker magnetic field.

For Section B

As shown in Figure 4, the measured values in data dot points closely track the theoretical value curve.

The center points’s theoretical value is   

B=0NIR2/2(R2+(h-z)2))3/2+0NIR2/2(R2+(h-z)2))3/2=8.772+8.772=17.54(Gauss).

, while the measured value is 16.3 Gauss. 

The difference is 17.54 - 16.3 = 0.79 Gauss , percent difference 0.79/17.54 * 100%=4.50% .

Again, this lower measured value is likely due to the non-infinitely-small coil thickness because Equation 2 is derived based on the assumption that the current coil is infinitely narrow. Part of the current I running on the outer rim is more distant from the measuring sensor and expected to have a weaker magnetic field.

The flat top curve has a standard deviation of 0.0333 Gauss. Percent deviation is 0.204% , which is very good.

Uncertainty

The electrical current coil’s current has a precision of 0.01Amp. 

The magnetic field for all data points have a precision of 3 significant figures with Gauss as the unit. 1 Tesla is 10,000 Gauss.

The position values for all data points have a precision of millimeters because the ruler is graduated every milimeter. 

Difficulties

There is no particular difficulties in this experiment

Conclusion

My Section A’s result magnetic field with axial Helmholtz coil configuration is within 8% error from the theoretical calculation, and the slightly weaker value is expected by the non-infinitely-small coil thickness construction. The optimal spacing between Helmholtz coils produced a flat top curve as expected.

My Section B’s result magnetic field with transverse Helmholtz coil configuration is within 5% error from the theoretical calculation, and the slightly weaker value is expected by the non-infinitely-small coil thickness construction. The optimal spacing between Helmholtz coils produced a flat top curve as expected, with a very small standard deviation at 0.204% within a 6cm range around the center of the optimal uniform magnetic field position.

Overall, all measured values are within 8% error of theoretical with reasonable explanations.

Restatement of the objection of this experiment is to measure magnetic field strength with distances of Helmholtz Coils to verify theoretical correctness of uniform field strength at optimal position. The measured results match the theoretical predictions within a reasonable error margin. The experiment is a success.

Questions

(none)