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: We investigate the effect of the Biot-Savart law by measuring the magnetic field produced by systems of current carrying coils through a central axis as a function of distance and comparing these physically measured results with theoretical results calculated using Matlab. We find that the predicted results calculated with matlab match up very closely to the measured results showing clear evidence of the Biot-Savart law.

### References

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

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

3. "Magnetic constant", Fundamental Physical Constants. (Committee on Data for Science and Technology, 2006)

4. H. Young, R. Freedman, University Physics with Modern Physics, 13th Ed. (Addison-Wesley, 2015)

5. D. Halliday, R. Resnick, K. Krane, Physics, Volume 1, 5th Ed. (Wiley, 2001)

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 Biot-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’ [2]

We find that according to the Biot-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 [3]

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. [2,4]

Figure 2. Magnetic field on the axis of a circular loop. [4]

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.[4,5,1]

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 Biot-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.

Here is an example of the code used for our first plot for a single clockwise coil.

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 clockwise current, centred at 0.63m.

In plot a we can see that the B-field peaks at around 0.63m, which is where the centre of the coil is located. This is exactly what we expect according to the Biot-Savart law. We also see that the magnetic field decreases as you move away from the centre of the coil. This is another expected result that arises due to the Biot-Savart law. The theoretical results closely match the measured results. This was a simple set-up so I expected the results to be close to the theoretical ones.

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

In plot b, we look at the same set up as in plot a, but with the current flow reversed. As we would expect, the magnetic field is negative, or is emanating in the opposite direction to the magnetic field observed in plot a. We once again see the peak B-field value at the location of the centre of the coil, just like in plot a, and we see the B-field decrease as a function of distance from the centre of the coil as we would expect due to the Biot-Savart law. The experimental values closely match the theoretical values, which I would have expected due to the simple set up.

Plot c. Theoretical and measured values of B-field as a function of distance along the x-axis for two coils with clockwise current, centred at 0.43m and 0.63m respectively.

In plot c we have now added a second coil into our experimental setup. The two coils are separated by 20cm, with the coils placed at 0.43m and 0.63m along the track. We now see 2 peaks on the graph, one for each coil, and a trough in between the two peaks. As expected, we see the B-field decreasing outside of the two coils. Between the two coils the B-field also begins to decrease, but it does not decrease as much as outside of the two coils. This is because the magnetic field adds between the two coils, but does not add as much outside of the coils. The bottom of the trough between the two peaks is present at the mid point between the two coils, which we expect, as this is the farthest distance from each coil, between them. The peak values at the coils are higher than in the set ups with just one coil, as would be expected. Our measured values are close to the theoretical values, but not entirely accurate. This could be due to inaccuracy in the distance measurements and misalignment of the two coils, i.e. they might not have been completely perpendicular or at the exact same height.

Plot d. Theoretical and measured values of B-field as a function of distance along the x-axis for two coils, one with clockwise current, and the other with anti-clockwise current, centred at 0.43m and 0.63m respectively.

In plot d we have a similar setup to the previous arrangement, although now one of the coils has anti-clockwise current. We see the peak values of B-field at the centre of each coil, as expected with a maximum value of B-field at the centre of the clockwise coil and a minimum and the centre of the anti-clockwise coil. We see outside the two coils and in between the two coils, the B-field tends towards zero. This is because the coils are each producing a magnetic field of the same magnitude, but in opposite directions, so they are cancelling each other out due to superposition. The measure and theoretical values are very close with some slight inaccuracy possibly due to error measuring the distance and some misalignment of the two coils.

Plot e. Theoretical and measured values of B-field as a function of distance along the x-axis for three coils with clockwise current, centred at 0.23m, 0.43m and 0.63m respectively.

In plot e we examine the case where we have 3 coils containing the same current that are equally spaced apart. We see three peaks on the graph, each located at the centre of one of the coils, as expected. The peak value at the centre coil is higher than the other two peaks, because the magnetic fields of all three coils superpose most at this point. Between the coils the B-field decreases as a function of distance, towards the centre point between the coils. The measured values closely follow the theoretical values with some slight error. In the centre of the graph, there are no measured values because the base of the hall probe could not fit in between the bases of the coils.

5. Discussion

The experimental values recorded for the most part closely followed the theoretical values calculated in Matlab. The small amount of error in these measurements could possibly be attributed to a number of different reasons. For example, errors in the measurement of distance along the x-axis would cause values to be shifted to the side on the graphs. This is a human error which could be solved by measuring the distances with a digital distance meter or by simply being more careful with reading the measurements. Differences in the number of turns in the coils could also affect the measurements. This could be avoided by using coils that were manufactured with a high level of accuracy. Differences in the currents in the coils could affect the readings and this could be solved by using a power source that deliver current to a high degree of accuracy. The coils not being perpendicular to one another and the coils not having their centres aligned in the x axis could be solved by using some kind of laser alignment to ensure the coils are perfectly aligned and perpendicular.

The experimental procedure was a fairly simple process. It consisted of taking a distance and B-field measurement, moving the hall probe, and repeating until an incremental path had been taking on both sides of the coil system. One problem that I faced was that I was unable to fit the base of the hall probe in between two of the coil bases when there were three coils on the track. This could be solved by having smaller bases for the apparatus.

Matlab was a useful tool that enabled us to create a visual model of the Biot-Savart law for our system with coils of wire. The code allowed manipulation of the variables including current, number of turns in the coils, coil radius, number of coils, spacing of coils which could create infinite possible arrangements and this code could be used for more complex systems consisting of more coils with different sizes and charges.

6. Conclusion

In conclusion, I think this investigation showed clear evidence of the Biot-Savart law in action and by comparison of theoretical data and manually measured results, we saw the magnetic field behaviour that we expected to see and that is predicted by the Biot-Savart law. The data from our model that we created in Matlab and the data from our manual measurement were almost identical, so this leaves no doubt that the magnetic fields are produced by the coils of wire exactly as we would expect. There was some small error in our physically measured results, but these could probably be nearly completely eliminated with more careful measurement.

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