DianaSooknauth_LabReportExp19

Report for Experiment #19 Magnetic Force and Lorentz’s Law Diana Sooknauth Lab Partner: Nola Hallemeier TA: Cort Thoreson November 14 th , 2023 Abstract The purpose of this experiment was to explore electromagnetism and magnetic fields by gaining an understanding of one of the main laws of electromagnetism: Lorentz ’ s Law. We were able to explore this by studying the relationship between the magnetic force and the current in the wire of a current balance, in Investigation 1. We then saw how the length of the wire immersed in the magnetic field affects the magnetic force in Investigation 2. Lastly, we observed the relationship between the magnetic force and the magnetic field in Investigation 3. We calculated the magnetic field value for Investigations 1 and 2 and obtained a result of .092 ± 7.58E-06 T and 081 ± 2.20E-05 T, respectfully. However, these values were not within the calculated error for the typical magnetic field value of horseshoe magnets, which is 0.050 Tesla. For Investigation 3, saw that there was a relatively linearly proportional relationship between the magnetic force and the magnetic field based on our plot which had a slope of y= 0.013x ± .001.

Introduction Electromagnetism is the interaction between electricity and magnetism. Electromagnetism is the reason why we have advanced to the modern-day technologies we have. The laws of electromagnetism are Lorentz’s Law, Ampere’s Law, and Faraday's Law. The law that is focused on for this experiment is Lorentz’s Law. Lorentz’s Law states that a charge moving with a velocity in a magnetic field experiences a force that can be expressed as: 𝐹 ⃗ = 𝑞𝑣 ⃗ × 𝐵 ⃗⃗ (1) The velocity, magnetic field, and force on the charge are all proportional to each other. The faster the velocity of the charge and the bigger that the magnetic field is, then the stronger the force on the charge will be. Lorentz’s law is in three dimensions, where velocity and the magnetic field both lie in a plane together, and the force on the charge is perpendicular to the both of them. The magnitude of the force on the charge also depends on the angle between the velocity and the magnetic field as can be seen in the expression: 𝐹 = |𝑞|𝑣𝐵𝑠𝑖?𝜃 (2) When dealing with currents flowing through a straight wire of length L, we find that the magnitude of the force can be simplified to: 𝐹 = 𝐼𝐿𝐵 (3) The objectives of this investigation were to study Lorentz’s law and determine how the magnetic force on a current-carrying wire depends on the current flowing in the wire, the magnetic field surrounding the wire, and the length of the wire that is immersed in the field. In order to complete this experiment, we used an apparatus called a current balance. The current balance has an armature that is a stiff brass wire shaped into a loop. The ends of this wire are inserted into a plastic bar which has an arrangement that allows the armature to pivot. There are counterweights on the threaded rods that are also attached to the plastic bar and mechanically balance the armature. There is a small mass on the wire loop that can slide and can change the balance. The short straight section of the wire dips into the air gap of an assembly of horseshoe magnets. The steel bars and the horseshoe magnets create a reasonably uniform magnetic field pointing across the gap from one steel bar to the other. Initially, there is a small iron strip that sits on top of the gap between the two steel bars which short circuits the magnetic flux across the gap. A visual of the current balance can be seen as Figure 1.

DC current within a range of 20 A maximum. Not using the high current in out and scale risks blowing a fuse and damaging the meter. Before turning on the power supply, we adjusted its controls to operate in the current control mode. We then turned the power supply on, then the meter, and we slowly increased the current. The armature was moving up, indicating that the current was flowing in the right direction. We recorded the length of wire inside the magnetic field which is also given by the length of the magnet assembly. We found the length to be .040 ± .001 m. We then slid the rider along from its leftmost position until it reached a major unit marking on the scale which in for the Griffin model was 10 mm. The armature moved as the torque on the balance changes, so we increased the current so that the armature could go back to being level. We then recorded the current I and the rider’s displacement x from its origin. We then continued to move the rider to every major marking and increased the current to level the armature; we did this until the ride got to the end of the scale. Once we got to the end of the scale, we went back down the scale in steps of 10 mm, balancing the armature and measuring and recording the current and rider displacement at each step. Once we finished collecting data, we turned the current off. We used the two measurements of current at each major marking from our way going up the scale and going down the scale. We obtained an average I and estimated the error of that average by using the formula: 𝛿𝐼 𝑎𝑣𝑔 = √(𝛿𝐼) 2 + (𝛿𝑥) 2 (5) We then plotted our data and made a straight line fit for the data which can be seen as Figure 2.

Figure 2 : Scatterplot for Investigation 1. The slope that we found was y= 46.0x ± 0.088 A/m which was found by using the data and putting it in the IPL Straight Fit Line calculator. The straight line fit is a good fit because the data points are close to the line, showing that there is proportionality between the magnetic force and the current in the wire. The slight variance could be due to systematic or experimental error, since we are measuring everything by hand and relying on our own judgement for when determining if the wire is balanced properly or not. We then used Eq. 4 to find the value of magnetic field B in the gap of the magnet assembly. We arranged Eq. 4 so that we were solving for B : 𝐵 = 𝑚𝑔𝑥 𝑑𝐿𝐼 (5) The slope is giving us the current per meter, which we took the inverse of to use in Eq. 5 for x/I . The mass of the rider was given to us as .0019 kg. The length L of the wire was found earlier in the investigation to be .040 ± .001 m. The distance d of the apparatus was measured to be .11 ± .001 m We also found the uncertainty of the magnetic field by using the formula: 𝛿𝐵 = √(−1 𝛿𝐼 𝐼 ) 2 + (−1 𝛿𝑥 𝑥 ) 2 × 𝐵 (6) The value for the magnetic field that we found was .092 ± 7.58E-06 T. The magnetic field for horseshoe magnets are around 500 Gauss which are 0.050 Tesla. This result is somewhat consistent with the value we found for the magnetic field because the magnetic field we calculated is .092 Tesla. It is not within our error calculated, however, it is not too far off. The inconsistency may be due to instrumental errors y = 46.0x ± 0.088 R² = 0.9947 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 I (A) x (m) Rider Position (m) vs. Current I (A)

found was .081 ± 2.20E-05 T. In comparison to the magnetic field value we calculated in the first investigation, and the typical magnetic field for horseshoe magnets, these two results are not consistent within their calculated uncertainties. This could be due to random error or instrumental error. The data from this investigation can be seen in the appendix. Investigation 3 The setup for this investigation was once again the same as in the previous two investigations. For this investigation, we were exploring the relationship between the magnetic force and the magnetic field by removing the horseshoe magnets and seeing how it affects the magnetic field. In this investigation, the current and the length of wire in the magnetic field remained constant. First, we set up the current balance as we did initially in Investigation 1, where the magnet assembly was back in its original position. We then set the rider to .07 m and adjusted the current until the part of the loop carrying the slider was horizontal. The rider’s displacement was 0.070 ± 0.001 m and had a current of 3.08 ± .031 A. We then turned off the power supply and removed one of the magnets from the assembly. We spread out the remaining magnets on the assembly and reinstalled the assembly in its previous position. We then turned on the power supply and set the current back to the current we initially recorded. We then moved the armature until it was leveled. We then recorded the rider’s displacement . We repeated this for 3, 2, 1, and 0 horseshoe magnets. We then plotted the displacement vs. the number of magnets and the graph can be seen in Figure 4.

Figure 4: Plot of Numbers of Magnets vs. Rider Position (m) for Investigation 3. The graph is relatively straight and there is a linear there is some curvature. Reasons for this curvature could be due to the magnets not being uniformly arranged, or there being error in the data collection. The magnets that we used were older and could have lost their magnetism over time. The curve does not pass through the origin and is slightly above it because the rider was never able to balance while being positioned at the origin. The way the current balance was set up was also inhibiting the rider from reaching the origin on the scale because of the bar above the pivots. Conclusion This experiment was focused on studying Lorentz ’ s Law and how the magnetic force on a current- carrying wire is affected by the current flowing in the wire, the magnetic field surrounding the wire, and the length of the wire that is immersed in the magnetic field. In order to do this experiment, we had to use an apparatus called a current balance. In Investigation 1, we studied the relationship between the magnetic force and the current in the wire by sliding the rider on the current balance along the scale and adjusting the current so that the armature was balanced. By doing this, we were able to see how the current of the wire affects the magnetic field that was produced in the magnet assembly. In Investigation 2, we saw how the length of the wire immersed in the magnetic field affects the magnetic force. By doing this, we were able to see that the magnetic force and the length of the wire were proportional. We also observed a linear relationship between the position of the rider on the wire and the position of the magnet assembly. In Investigation 3, we observed the relationship between the magnetic force and the magnetic field by observing the magnetic force on the rider when different amounts of magnets were presented. We saw y = 0.013x + 0.001 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 1 2 3 4 5 6 Rider Position (m) # of magnets Number of Magnets vs. Rider Position (m)

5. Note that the zero of the scale is a few mm away from the actual pivot point. Do you have to take that into account? Explain! a. Overall, we did not have to take this into account since it is just a few mm and we were still able to see the relationships between the variables in the Lorentz Law, however there are a few imperfections which may be due to this. The displacement values were affected slightly because the ride is never able to be at a true origin. We had to take this into account only when we saw that the relationships between the two variables in each investigation were not perfectly linear, which would ’ ve be ideal. Acknowledgments Thank you to my TA Cort and my lab partner Nola. Appendix The data for this experiment: PhysLabExp19.xlsx