The magnetic field due to a straight conducting wire follows the relationship and using this as a linear function with B as the y value and I as the x value the gradient will be µ0/2πr. Then the gradient can simply multiplied by 2πr to find µ0. The relationship should be linear as all parts of the gradient are constants.
List of material
● magnetometer
● conductor
● power supply
● voltmeter
● ampere meter
Preparation
Before the experiment is conducted the scientist must set up the equipment like in the picture. The conductor that is going to be investigated must be set up far away from both the rest of the circuit, the power supply and the magnetometer in order to secure an as isolated system as possible. Notice that the power supply, magnetometer and every electric cord will induce a magnetic field. Also an ampere meter should be connected in series with the circuit and a voltmeter parallel to it to measure the independent variable. Notice that it is necessary to switch off cell phones and similar electronic equipment to decrease the magnetic interference.
Method
After that all equipment is prepared the laboration can start. The table that is prepared for the lab should include current and magnetic field. Notice that an uncertainty of half the smallest unit of the ruler or measurement equipment is apprpriate. I is going to be measured at approximately 1 A intervals from 13 A down to 3 A. At every interval B will be noted in mT. The inaccuracy of the current is thus ±0.005 A but the magnetometer is too sensitive for using half the smallest unit thus ±0.01 mT is used.
The procedure that is being followed is to change the current with constant intervals by changing the voltage of the power supply. After approximately every Ampere a reading should be conducted on the magnetometer. As the resistance is constant there will be an ohmic relationship between the voltage and the current. The voltage can be noted but only used to find the resistance and not to find the current as this will create additional uncertainties. The current should be measured by the ampere meter.
The data collection must be performed quickly or else the chords might get warm and impose a health hazard as a high current is led through a low resistance. The high current is necessary to decrease the relative influence on the magnetic field by external factors.
The sensor of the magnetometer should be attached to the conductor by a clamp in order to keep the radius between the sensor and the conductor constant. The radius should be 1 cm.
Analysis
A value for the relationship between I and B can be found through processing the raw data found and creating a graph of it.
Processing the data into the universal formula for a linear graph (y=mx+c) B can represent the y value and I can represent the x value, making the m value represent m=µ0/2πr. (see next page for graph) solving this formula for µ0 makes:
µ0 = maverage2πr
µ0max = mmax slope2πr
µ0min = mmin slope2πr
Notice that the real value of m is only 1/1000 of the value presented in the graph as mT is used as unit in the data table. Furthermore r is kept constant at 1 cm.
Trial #1
Thus m = 2.14*10-5 TA-1, mmax slope = 2.39*10-5 TA-1 and mmin slope = 1.99*10-5 TA-1
µ0 = (2.14*10-5)*2π*0.01 = 4.28π*10-7 TmA-1
µ0max = (2.39*10-5)*2π*0.01 = 4.78π*10-7 TmA-1
µ0min = (1.99*10-5)*2π*0.01 = 3.98π*10-7 TmA-1
inaccuracy = (max-min)/2
(4.78π*10-7-3.98π*10-7)/2 = 4π*10-8 TmA-1
Thus the calculated value for µ0 is:
4.28π*10-7 TmA-1 ±4π*10-8 TmA-1
The V/I relationship also known as the resistance of the circuit is controlled by the table to the right. As is implied in this table the resistance remained ohmic as it should.
Trial #2
In Trial #2 the radius was kept constant at 1.3 cm instead of 1 thus changing the m values slightly while keeping the final permeability of space value constant.
m = 1.68*10-5 TA-1, mmax slope = 1.90*10-5 TA-1 and mmin slope = 1.50*10-5 TA-1
µ0 = (1.68*10-5)*2π*0.013 = 4.368π*10-7 TmA-1
µ0max = (1.90*10-5)*2π*0.013 = 4.94π*10-7 TmA-1
µ0min = (1.50*10-5)*2π*0.013 = 3.90π*10-7 TmA-1
inaccuracy = (max-min)/2
(4.94π * 10-7 - 3.90π * 10-7) / 2 ≈ 5π*10-8 TmA-1
Thus the calculated value for µ0 is:
4.368π*10-7 TmA-1 ±5π*10-8 TmA-1
The resistance was kept constant in Trial #2 to.
Trial #3
In Trial #2 the radius was kept constant at 1.3 cm instead of 1 thus changing the m values slightly while keeping the final permeability of space value constant.
m = 1.56*10-5 TA-1, mmax slope = 1.87*10-5 TA-1 and mmin slope = 1.43*10-5 TA-1
µ0 = (1.56*10-5)*2π*0.013 = 4.05π*10-7 TmA-1
µ0max = (1.87*10-5)*2π*0.013 = 4.86π*10-7 TmA-1
µ0min = (1.43*10-5)*2π*0.013 = 3.718π*10-7 TmA-1
inaccuracy = (max-min)/2
(4.86π*10-7-3.718π*10-7)/2 ≈ 6π*10-8 TmA-1
Thus the calculated value for µ0 is:
4.05π*10-7 TmA-1 ±6π*10-8 TmA-1
The resistance was kept constant in Trial #3.
Having found the average value for µ0 the greatest inaccuracy of the three trials which is ±6π*10-8 TmA-1 should be used. This gives an overall average of:
4.233π*10-7 TmA-1 ±0.6π*10-7 TmA-1
Conclusion
The found value for the relationship between I and B is close to the real value of 4π*10-7 TmA-1. The deviation of the calculated values for the permeability of free space in the first two trials implies that there is a systematic error. However the third trial is very precise without such a systematic error.
Both the fact that the graph is linear and that the calculated value supports the hypothesis as the real value lies within the inaccuracy. Also the fact that the repeated data collection is accurate supports known theory and the hypothesis.
Evaluation
The evaluation seems reasonable as the results were close to the real value all implying that the lab was well designed and executed. However the magnetometer was very sensitive to external factors and made several resets of the scale were necessary before a data collection could be performed. The inaccuracy of the readings of the magnetic field made the error bars in the graph unnecessarily large.
The fact that the y intercept is not zero in the first two graphs suggests a minor systematic error however the error remains unidentified. The repeated measurement and calculation of average removes the unavoidable difficulties in making accurate measurements every time.
There are not many ways this laboration can be improved, except for finding the systematic error, as it gave a clear and correct value.
