Investigating the Biot-Savart Law of Magnetostatics

Luke Joseph Mulvey,1

1School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK

Abstract: In this project report, I intend to explore the Biot-Savart law of Magnetostatics and use it to predict the magnetic field strength created along the centre axis of a number of current carrying coils. This shall initially be carried out using Matlab, with these theoretical predictions experimentally validated by using a Hall probe and three coils. These results provide a strong and convincing link between the theoretical and experimental results. This is further reinforced by comparing plots of both data sets overlaid and noting the compelling resemblance. Overall this experiment provides compelling evidence to support the Biot-Savart Law.

### References

1. D. Griffiths, Introduction to Electrodynamics. (Pearson Education Inc, San Francisco, 2008)

2. Heriot Watt University, Physics Undergraduate Courses Level 3 Physics Laboratory. (Heriot Watt, 2015)

3. Practical Physics, G.L.Squires (McGraw-Hill, 1968)

4. The Biot-Savart Operator and Electrodynamics on Bonded Subdomains of the Three Sphere, Robert Jason Parsley ( University of Pennsylvania, 2004)

5. Magnetic Field Along The Axis of A Circular Coil Carrying Current, (vlab.amrita.edu, 2004)

6. "Magnetic constant". Fundamental Physical Constants. Committee on Data for Science and Technology. 2006

7. University Physics with Modern Physics, 13th Edition

1. Introduction

In magnetostatics, we consider the case where we have a steady electrical current flowing through an electrical conductor, resulting in a magnetic field surrounding the conductor. We find that the magnetic field measured varies depending upon where it is measured; the field depends on the shape of the current carrying conductor and the distance from the conductor where the field is measured, which we find as a result of the Bio-Savart law, which we will discuss later. In this investigation we consider a system where there are coils of wire that have steady currents running through them. The fields produced by these coils of wire are dependant on the amount of current flowing through them and the point where the field is measured, as well as on the number of turns in the coils of wire. Using the Matlab software package, we can create a theoretical simulation of the magnitude of the B-field generated by these coils of wire as a function of distance, for different configurations of coils, and we can then compare these theoretical plots to actual measured results from our experiment set-up.

2. Background

We find that a wire carrying a steady current will produce a magnetic field. We now consider a loop of wire of arbitrary shape. We find the magnetic field dB produced by a small section dl’ of the wire, measured at a point r’, will be proportional to the amount of current I flowing in the wire and inversely proportional to the distance r between the section of wire and the point r’ where the magnetic field is measured.

Figure 1. Magnetic field at point r due to current I in wire segment dl’

We find that according to the Bio-Savart law, the magnetic field at point r’, due to the current I, flowing though the wire segment dl’ is equal to:

Where is known as the permeability of free space, which is a physical constant defining the value of magnetic permeability in a classical vacuum and is equal to 4π×10−7 N/A2 [5]

In vector form, using the unit vector , we have:

Which is equal to the magnetic field of a current field element.

Thus we can then use this law to find the total magnetic field B at any point in space due to a complete circuit by integrating over all segments of dl’ that carry current:

This expression is applicable to any shape of wire and for the magnetic field in all space, but this is only easy to calculate in the case of the wire having a simple geometry, so often it is necessary to numerically integrate this value.

In our investigation, we consider a simple geometry, in which we will investigate the magnetic field as a function of a distance along an axis which coincides with the centre of one (or more) of the coils. This simplification allows us to analytically solve the integral expression for the B field. [2]

Figure 2. Magnetic field on the axis of a circular loop. [7]

In figure 2, we see a loop of wire with radius a and current I. The magnetic fields produced by the section of wire leading in and out of the loop nearly cancel each other. Using the law of Biot-Savart, we can find the magnetic field at a point P along the central axis of the loop at a distance x from the centre of the loop. We see that dl and r are perpendicular to each other, so the B-field lies in the x-y axis. Since , we find that the magnetic field dB due to a segment of the loop dl will be:

The x and y components will be equal to:

Although we know that there will only be an x-axis contribution because for every y axis contribution of magnetic field, there will be a corresponding opposite contribution of magnetic field on the other side of the loop that cancels it out, but all the x axis elements will add in the same direction. Thus to find the magnetic field in the x-axis, we integrate around the loop:

So the B-field in the x-axis of a circular loop is equal to:

For a coil of N number of turns, and radius a, we can multiply this equation by N to find the total field produced by the coil at a distance x from the centre of the coil. From this equation we also find a maximum field strength at the centre of the coil (i.e. x=0). We will also find the B-field to be zero when measured perpendicular to the x-axis.

3. Numerical Modelling

Before beginning the experimental procedure, we want to develop a numerical model that allows us to predict the outcome of an experiment carried out using any configuration of coils. We will use the Matlab software package to accomplish this as this will allow us to write code than can easily be adapted to the different combinations and alignments of coils of wire. In our numerical modelling we will assume that all of the coils we are using are exactly the same and that they are aligned horizontally, are perpendicular to each other and have a common axis through the centre of them. Our model will predict the magnitude and direction of the magnetic field along the x-axis for any combination of coils with any number of turns per coil and with any amount of current in the coils. Our model will be based on the Bio-Savart law, which for a coil of wire, radius a and with N number of turns is:

With this equation we will construct code that will allow us to accurately predict the magnetic field along the x axis for the 5 different experimental set-ups that we will discuss later. We will set up out model with the coils situated at 20cm, 40cm and 60cm along the track (as we will do in the experimental procedure as well). We can set the currents for the 2nd or 3rd coils to be zero if there are only one or two currents in that particular set-up and we will be able to change the direction of current in the coils by inputting a negative value of current into our code. We also want to be able to compare the theoretical results with the experimentally measured results so we will need to overlay these values onto our graphs as well

4. Experimental Method

Figure 3. Arrangement of experimental apparatus: 3 equally sized coils of wire arranged on a track with a Hall probe aligned with the centre of the coils.

Our experimental apparatus consists of 3 coils of equal size, each comprised of 500 turns of wire and a radius of 9.8cm with a steady current of 0.8 amps flowing through each of the coils. The coils of wire are each held up by a pole and are placed along a track with a Hall probe that is aligned with the centre of the coils. The hall probe can move backwards and forwards along the track incrementally. We take readings of the B-field as a function of its position on the x-axis with regards to the coils.

There are five different configurations of coils that we will be investigating, which are as follows:

a) A single coil with clockwise current

b) A single coil with counter-clockwise current

c) Two coils with clockwise current

d) Two coils with clockwise current in one coil and counter-clockwise in the other coil

e) Three coils all with clockwise current flow

To measure the distance along the x axis, we will take the centre of the coil or system of coils as the zero point and measure distance positively and negatively along the axis from there. We will want to reset the zero position of the hall probe by shielding the Hall probe sensor and clicking the reset button on the attached probe monitor. We then can carry out the measurements, which will involve recording the B-field displayed on the probe monitor with the distance from the centre point. We take these measurements incrementally from both side of the coil arrangement to map out the shape of the B-field around the coil along the x-axis.

5. Results

Plot a. Theoretical and measured values of B-field as a function of distance along the x-axis for a coil with with the coil centred at 0.63m.

From plot 1 it can be seen that the B-field is strongest directly in the centre of the wire which validates the requirement that the magnetic field at any given point has to be directly proportional to the strength of the current. So as the distance between the centre of the wire and the hall probe is increased, the current can be expected to have a weaker impact on the magnetic field leading to a decrease in B-field reading. Further inspection of the graph leads to validation of another requirement, as the probe is moved further left or right from the centre of the coil, the drop in the magnetic field follows a dependency.

Plot 2. Theoretical and experimental data recorded when B-field strength is examined compared to the position of a Hall probe along a central axis. Experimental set up comprises of one coil with an anti-clockwise current.

Plot 2 looks at the same scenario as plot 1, however this time the current direction has been reversed. The theoretical results prove the intuitive guess that the data plot should follow the same shape but simply inverted around the central axis, as the Hall probe is now picking up a negative reading. Just as before, the B-field follows a drop in magnetic field as the current strength decreases, and this creates a curve that follows a dependency. The graph also peaks at the same point along the central axis as before which is to be expected.

Plot 3. Theoretical and experimental data recorded when B-field strength is examined compared to the position of a Hall probe along a central axis. Experimental set up comprises of two coils with a clockwise current.

Plot three incorporates the first addition of a coil into the experimental set up. It now comprises of two coils, each with a clockwise current, separated by a distance of 0.2 metres along the central axis. As expected the initial curve and final curve of the graph follows the same shape as the previous graphs. However in the centre of the gap between the two coils there is now a superimposed B-field. The curve does not drop all the way at 0.5 m as it is not completely outside the influence of the coils. As expected however the point of weakest B-field falls at the midpoint between the two coils. For this particular set up the two peaks are slightly higher than the single peak of the one loop experiment, as would be expected due to superposition

Plot 4. Theoretical and experimental data recorded when B-field strength is examined compared to the position of a Hall probe along a central axis. Experimental set up comprises of two coils with an anti-clockwise and clockwise current.

Plot 4 examines the same scenario as plot 3, however the second coil has now had the current direction reversed. The graph still has turning points at the same position along the central axis as before, however they are now a maximum and minimum as opposed to two maximums. This creates a steep drop in the centre. This can be explained with the principle of superposition. At the midpoint, where the Tesla reading is 0 we have two identical readings acting in different directions. Therefore at any point along this steep incline there is basically the addition of two separate B-field readings, one of which is positive and one of which is negative. For this particular set up the turning points are slightly lower than the single peak of the one loop experiment, as would be expected due to superposition.

Plot 5. Theoretical and experimental data recorded when B-field strength is examined compared to the position of a Hall probe along a central axis. Experimental set up comprises of one coil with an anti-clockwise current.

Plot 5 examines the final scenario where we have three coils all with a clockwise current of similar magnitude, each separated by a distance of 0.2 metres. There is now three maximum turning points on the graph, the middle of which is higher than the two on the outside. At this point all three peaks are higher than the single peak generated by the single coil set up, as each peak is contributing to its neighbours B-field through the principle of superposition. This scenario provided experimental data that did fit the general shape of the theoretical data points, but unfortunately achieved weaker magnetic field readings than predicted. This could be down to a number of factors such as the combinations of background readings creating strong errors in the measurements, the Hall probe not being properly reset to zero, the intricate nature of taking readings at set intervals with three coils in play may also have led to human error in the readings.

5. Discussion

The use of Matlab to create a modelling tool capable of creating a visual representation of the Biot-Savart law for different scenarios worked very well. The code was well designed, easy to manipulate and allowed a wide range of scenarios to be considered. Although in the scope of this experiment only five relatively simple configurations were examined, the program could in theory be used to create graphs for a wide range of situations. For example a scenario involving 5 loops of varying radius, each with a different number of coils, each with a different current passing through them could be examined. This turns a relatively simple Matlab program into quite a powerful analytical tool.

The process of collecting the experimental data was quite a simple task when it conferenced only the first two scenarios with a single coil. After multiple coils were introduced to the system, the process became slightly more tricky.as the probe was not long enough to reach the centre of the multi-loop scenario’s, it had to be taken off the track and moved into the space in-between the loops. Moving the Hall probe off the track is almost an open invitation to add errors and great care has to be taken to ensure that everything is lined up as it was previously, as any deviation from the central axis could lead to discrepancies in the B-field reading. The introduction of a longer probe and the use of a Vernier scale would have greatly improved the accuracy of the readings and helped to remove any human error form the experiment. This having been said, if the proper time is given to the task and great care taken when making manual measurements, it is possible to remove a vast amount of the human error to the point where one can count them to be almost insignificant, something that can be backed up by the accuracy of the experimental data compared to the theoretical data.

The main problem is that the experimental data consistently falls just short of the points predicted by the Matlab code. With more time greater attention could be paid to the points that form the peak. By choosing a smaller increment of movement along the central axis it would be possible to pinpoint the exact location of the peak, and eliminate the gap between the data sets.

6. Conclusion

In conclusion this experiment produced strong and compelling evidence to support the basis of the Biot-Savart Law. It has been clearly demonstrated that a current passing through a coil of wire produces a magnetic field with a measurable field and direction. The striking resemblance between the theoretical data and experimental data, eradicates any doubts that the experiment could have produced inaccurate answers. The dual approach of using Matlab and conventional hand-recorded measurements complemented each other well and helped to prove beyond a doubt the application of the Biot-Savart law. It can clearly be shown that for n-number of coils of similar current magnitude, the Biot-Savart law can be used to predict the direction and magnitude of the B-field.

7. Appendix

Below is an example of the code used to predict the B-field for figure 1.

clear all

close all

k=(1.25e-6)/2; %This is the Permeability constant /2.

r=0.098; %Radius in meters.

c=500; %Number of coils.

I=[0.5 0 0]; %Current values.

d=1; %Distance in meters.

z=[0.4 0.6 0.8]; %Distance of each coil.

dz=d/100; %Steps to take in the z axis.

Z=(0:dz:d); %limits along the z axis.

B=zeros(size(Z)); %Defines a maxtrix of all zeros.

for i=1:length(z) %Starts a for loop for vales of I

x=sqrt((Z-z(i)).^2); %Calculates the distance difference between Z and z(i).

h=sqrt(r^2+x.^2); %This calculates the hypotanuse.

B=B+c.*(k.*I(i).*(r^2))./h.^3; %This calculates out the magnetic field and adds the new magnetic field value to the old one.

end

ZPos = (0.25:0.01:0.55);

ManualData = [0.0275e-3 0.315e-3 0.352e-3 0.416e-3 0.504e-3 0.566e-3 0.653e-3 0.776e-3 0.857e-3 1.003e-3 1.128e-3 1.256e-3 1.325e-3 1.4e-3 1.479e-3 1.480e-3 1.449e-3 1.376e-3 1.272e-3 1.178e-3 1.061e-3 0.942e-3 0.85e-3 0.744e-3 0.644e-3 0.566e-3 0.494e-3 0.425e-3 0.365e-3 0.310e-3 0.274e-3];

figure, plot(Z,B, '.')

hold on

plot(ZPos, ManualData, 'x')

xlabel(' Distance (metres)')

ylabel('Tesla Reading')

title('B field Strength around a Clockwise Coil') %the above three lines annotate the graph

clear all;

clc;

k=(1.25e-6)/6; %This is the Permeability constant /2.

r=0.1; %Radius in meters.

c=500; %Number of coils.

I=[2.5 0 0]; %Current values.

d=1; %Distance in meters.

z=[0.63 0.5 0.7]; %Distance of each coil.

dz=d/100; %Steps to take in the z axis.

Z=(0:dz:d); %limits along the z axis.

B=zeros(size(Z)); %Defines a maxtrix of all zeros.

for i=1:length(z) %Starts a for loop for vales of I

x=sqrt((Z-z(i)).^2); %Calculates the distance difference between Z and z(i).

h=sqrt(r^2+x.^2); %This calculates the hypotanuse.

B=B+c.*(k.*I(i).*(r^2))./h.^3; %This calculates out the magnetic field and adds the new magnetic field value to the old one.

end

Distance = [0.77:-0.02:0.49];

Field = [0.529e-3 0.705e-3 0.94e-3 1.238e-3 1.610e-3 2.02e-3 2.345e-3 2.47e-3 2.24e-3 1.87e-3 1.485e-3 1.3e-3 0.85e-3 0.640e-3 0.48e-3 ];

figure, plot(Z, B)

hold on

plot(Distance, Field)

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