Physics 4BL: Experiment 3

Laboratory 3

Magnetism

Max Woo
11/5/14
Lab 11, Michael Ip
Partners: Zoe Zhu

Introduction

The purpose of this lab is to investigate the magnetic fields produced by steady state sources. For this particular lab, we will be using a Hall probe to measure magnetic field strength. More specifically, in part 1 of the lab, we will be measuring the field strength at various distances through a toroidal coil. In part 2, we will be measuring the field strength at various distances from a permanent magnet. For part 3, we will measure the force between two permanent magnets at different distances. For each of these parts, we will try to confirm the predicted relationships and equations by using linear regression and finding the error and the correlation coefficient.

Experimental Data

Part 1

In this part, we measure the magnetic field inside and outside of a toroidal coil, shown in Fig. 1:

Fig. 1: A toroidal coil used in Part 1

We first power the toroidal coil with a 15V power supply. We can calculate the magnetic field strength $B$ along a path $l$ around the central axis using Ampere’s law, which states that

$\oint B\cdot dl=\mu_0I_{enclosed}$ (Eq. 1)

Because we know that, for a circular path around the toroid, the magnetic field strength along the path will always be the same (due to symmetry), we can simplify this to

$2\pi rB = \mu_0 I_{enclosed}$ (Eq. 2)

where $r$ is the distance from the central axis of the toroid. Rearranging the equation to get magnetic field strength in terms of radius, we get:

$B = \frac{\mu_0 I_{enclosed}}{2\pi} \frac{1}{r}$ (Eq. 3)

where it becomes apparent that the field strength $B$ is directly related to the enclosed current $I_{enclosed}$ and the inverse of the radius $1/r$. This also tells us that the magnetic field does not change as long as radius is kept constant. We test this by measuring the magnetic field at different points around the toroid, but at the same distance from the central axis, noting that the magnetic field strength stays the same.

If we now look at the toroid from a top view, labeling the inner radius with $a$, the outer radius with $b$, and the distance at which we are measuring the magnetic field with $r$:

Fig 2: Top view cross-section of the toroid

we can see that, for $r < a$, there is no enclosed current, and for $r > b$, the current going into the page is equal to the current going out of the page, making the total enclosed current equal to 0. Thus, for $r < a$ and $r > b$, the magnetic field should be 0.

To confirm these calculations, we measure the magnetic field at various distances from the central axis. The results are shown in the graph below:

Graph 1: The magnetic field at different distances away from the center axis of the toroid

The graph confirms our predictions, starting at a magnetic field of zero when $r < a$, and then jumping to a certain value when $a < r < b$, and then decreasing at an inverse relationship to $r$ until $r > b$, where the magnetic field abruptly drops back to 0.

Part 2

For this part, we start with a cylindrical permanent magnet with its central axis pointing upwards, which we call the z-axis. We will be measuring the magnetic field at different angles away from the z-axis, as shown in Fig. 3:

Fig 3: Diagram showing how we will be measuring the magnetic field around a cylindrical permanent magnet

Assuming that the magnetization is uniform through the magnet, we can approximate the magnetic field at large distances away using the equations

$B_z=\frac{\mu_0}{4\pi}\frac{m}{r^3}\left(3\cos^2\theta-1\right)$ (Eq. 4)
$B_{\rho}=\frac{\mu_0}{4\pi}\frac{3m}{2r^3}\sin2\theta$ (Eq. 5)

where $z$ represents the z-axis, $\rho$ represents the unit vector of the radius $r$, and $m$ is the magnetic moment, and $\theta$ is the angle between $\rho$ and $z$. Because we will only be measuring the magnetic field at angles $\theta = 0, \pi / 2$, $B_{\rho} = 0$ for both cases and we only have to measure $B_z$, the magnetic field directed vertically. In addition, from the equation we can see that $B_z$ should vary directly with $r^{-3}$.

To see if our prediction is correct, we use the Hall probe to measure the magnetic field at different distances along two axes, $\theta = 0$ and $\theta = \pi /2$, making sure to start a few centimeters away from the permanent magnet before moving outwards. The results are shown below:

Graph 2: The magnetic field at increasing distances measured at an angle $\theta = 0$

Graph 3: The magnetic field at increasing distances measured at an angle $\theta = \pi /2$

Because we measured with the Hall probe facing upwards, we can see from the graph that the magnetic field seems to behave as expected, with the field directed upwards and the strength seeming to vary with $r^{-3}$.

Part 3

For Part 3 of the experiment, we measure the repulsive force between 2 cylindrical permanent magnets at different distances from each other. We place one of the magnets on a mass balance, and then place another on a vertical track with distance markings, as shown in Fig. 4:

Fig. 4: The setup for measuring the repulsion between two permanent magnets

From Eq. 4 and 5, we know that the magnetic field along the central axis of a cylindrical magnet is directly related to $r^{-3}$. Thus, the force $F$ between the two dipoles should be directly related to $r^{-4}$. To verify this, we vary the distance between the two magnets and use the mass measured by the mass balance to calculate the force (factoring in earth’s gravitational constant). The results are shown below:

Graph 4: The force between 2 repelling magnets at different distances

The graph seems to confirm our prediction, with the force between the magnets seeming to vary directly with $r^{-4}$.

Analysis

Part 1

We know from Eq. 3 that the magnetic field for $a < r < b$ should be directly related to $r^{-1}$. Thus, we first crop our data to the values where $a < r < b$, find $r^{-1}$ for each data point, and then graph the results against the magnetic field for each point:

Graph 5: Graph of the relationship between 1/r and the magnetic field

The linear regression yields a correlation coefficient of 0.9965, which is very close to 1, confirming that magnetic field inside a toroid is directly related to $1/r$ and verifying our Eq. 3.

Part 2

For this part, we need to verify Eq. 4 for the angle $\theta = 0$, and show that the magnetic field upwards $B_z$ is directly related to $r^{-3}$. To do that, we use our data for $\theta = 0$, and calculate $r^{-3}$ for each data point, graphing it against $B_z$:

Graph 6: Graph of the relationship between $r^{-3}$ and $B_z$

The linear regression for this graph yields a correlation coefficient of 0.9992, which is extremely close to 1, conclusively proving that $B_z$ and $r^{-3}$ are directly related and verifying Eq. 4.

Part 3

For the repulsive force between 2 magnets, we predicted in the “Experimental Data” section that the force should be directly related to $r^{-4}$. Following the same procedure as with Part 1 and 2, we calculate $r^{-4}$ for each data point and graph the results against force $F$, giving us:

Graph 7: Graph of the relationship between $r^{-4}$ and force

The linear regression for this graph gives us a correlation coefficient of 0.9984, which is again extremely close to 1, confirming our prediction that the force is directly related to $r^{-4}$.

Conclusion

For this lab, we investigated the effects of magnetic fields over varying distances, using correlation coefficients to judge the validity of our predictions. For part 1, we were able to verify that the magnetic field inside a toroid ($a < r < b$) varied inversely with radius $r$. For part 2, we measured the magnetic field parallel and perpendicular to the central axis of a permanent magnet, confirming that the magnetic field varied directly with $r^{-3}$. For the last part, we measured the repulsive force between 2 magnets at different distances, and verified that the force varied directly with $r^{-4}$.