Investigation of magnetic flux density with Hall probe

Theory

The equation for the magnetic flux density inside a long solenoid is as follows: \(B=nI\mu_0\)

Where \(B\) is magnetic flux density (measured in T), \(n\) is the number of turns of the solenoid per metre (measured in m\(^{-1}\)), \(I\) passed through the solenoid (measured in A), and \(\mu_0\) is the permeability of free space (a constant). Our aim with this practical is to obtain a measurement for the permeability of free space.

Apparatus

  • Hall probe
  • Power pack
  • Spring
  • Wires
  • Ruler
  • Magnetic flux density meter

Method

We attached the spring to the end of the Hall probe. We attached the Hall probe to the magnetic flux density meter. We attached wires to power pack and either end of the spring. We limited the current output of the power pack to 1 Amp. We varied the number turns per metre of the spring by holding the end of the spring and extending it. Each time we counted the number of turns in a given range (1.5cm) and read the measurement from the flux density meter. This gave us a series of measurements. We adjusted these measurements to the correct scale by multiplying the flux densities by 10\({^3}\) (since our initial readings were in mT), and multiplied the number of turns by \(100/1.5\). We then plotted these results and performed linear regression to find the gradient (which should be approximately equal to \(4\pi*10^{-7}\))

import matplotlib.pyplot as plt
import numpy as np

turns = [22, 16, 13, 10, 8, 7, 6, 5]
turns_per_m = [n*100/1.5 for n in turns]
field_strength = [1.234, 1, 0.808, 0.661, 0.582, 0.516, 0.464, 0.351]
field_strength = [f*10**-3 for f in field_strength]
plt.scatter(turns_per_m, field_strength, label="Experimental results")

# Linear regression
gradient, intercept = np.polyfit(turns_per_m, field_strength, 1)
regression_field_strength = [gradient*i + intercept for i in range(int(min(turns_per_m)), int(max(turns_per_m)))]
plt.plot([i for i in range(int(min(turns_per_m)), int(max(turns_per_m)))], regression_field_strength, label="Regression line", color="darkorange")

print(f"Measured permeability of free space (gradient) = {gradient*10**7}*10^-7")
print(f"Actual permeability of free space = 4π*10^-7 = 1.25663706*10^-6")

plt.xlabel("Turns per metre/m^-1)")
plt.ylabel("Field strength/T")
plt.legend()
plt.show()
Measured permeability of free space (gradient) = 7.595778364116089*10^-7
Actual permeability of free space = 4π*10^-7 = 1.25663706*10^-6
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Conclusion

The value we obtained for permeability of free space was the correct order of magnitude, which we judged to be a success given the nature of the method and equipment we used. Our results were certainly enough to demonstrate the linear relationship between magnetic flux density in a solenoid and the turn density of said solenoid.