Module code: FC312 Module title: Physics Cohort and group: Assessment type: Lab report Project title: Force on a magnetic dipole in a spatially varying magnetic field Tutor's name: Student ID number: Date of submission: Word count: Student declaration: I confirm that this assignment is my own work. Where I have referred to academic sources, I have provided in-text citations and included the sources in the final reference list. 1 Force on a magnetic dipole in a spatially varying magnetic field Introduction One of the most fundamental concepts in electromagnetism is the magnetic dipole. When a magnetic dipole (which is a bar magnet in the case of this experiment) is put in a magnetic field, it experiences turning force (torque) and levels itself with the magnetic field direction (Griffiths, 2021). If a magnetic field varies with the position of the magnetic dipole placed there, then it is called a nonuniform field, and the bar magnet will experience a net force acting on it (Purcell and Morin, 2013). The general equation for a linear force acting on a magnetic dipole with moment μd is following: F = Ma (1) dB dz F is the force; ° Ma is the magnetic dipole moment; dB dz is the rate of change of magnetic field B with position z (magnetic field gradient). For this experiment, a pair of Helmholtz coils is used as a generator of a magnetic field B. A Helmholtz coil is an electromagnet of radius R, which is separated from another coil by distance R. There are two possible configurations of the Helmholtz pair: default and reversed. In the default configuration, the current flows in the same direction in the two coils, and the magnetic field produced between the coils is uniform, which means that it hardly changes depending on the position of the magnet bar (Pocketlab, 2017). In the case of a reversed Helmholtz pair, the current in the two coils flows in opposite directions, and the magnetic field between the coils changes linearly with distance z (Knoepfel, 2008). For this experiment, the reversed configuration is used. The rate of change of B with z in this configuration can be calculated using the following equation: 2 dB = 0.859μN (2) dz R2 dB is the rate of change of magnetic field B with position z (gradient); dz • μo is the permeability of free space constant (= 4π × 10¯7NA¯²); ° N is the number of turns in each coil (168); ° I is the current through the coils; ° R is the coils radius (= 7.0 ± 0.2 cm). The aim of this experiment was to study the effect of a magnetic field on a magnetic dipole, along with measuring the magnetic dipole moment and analysing its graph. Experimental methods were used in order to explore the behaviour of a magnetic dipole in a changing magnetic field. The results of this report were analysed based on the data obtained during the experiment, and the findings were discussed with possible inaccuracies taken into consideration. • Materials and methods The following equipment was used for this experiment: Helmholtz coils; • a brass rod; ° ° a spring which was suspended from the rod; a small neodymium-iron-boron magnetic disk which was suspended on the end of the spring; • five steel ball bearings; ° a power supply with connecting wires. The brass rod with the spring attached to it hanging the magnet were placed inside a tube that was set in the Helmholtz coils' bore. On the side of the tube, there was a z-axis scale with its origin at the centre between the two coils. This configuration is shown below in Figure 1. There was also a power supply with a built- in ammeter and voltmeter to provide current for the coils. Five steel ball bearings were used as weights to measure the spring constant k. 3 Figure 1: Experimental setup The following health and safety measures were followed during the experiment: ° Electrical hazards avoidance: the power supply was turned off when unnecessary, and was used with caution when needed. Risk of accidents reduction: the experiment was executed carefully, tidily, and under the supervision of an instructor at each stage. General safety: eating and drinking was avoided. = The first step of the experiment was to estimate spring constant k. The calculation of the spring constant k can be done using the formula mg = kx, where m is the mass; g is the acceleration due to gravity; k is the spring constant; x is the spring extension. This formula is derived from Hooke's law (F kx), but with the knowledge that the spring is vertical, the force F is substituted by mg (Young and Freedman, 2015). The mass and spring extension can be expressed as a delta, so the change in spring extension and mass can be measured by adding weights to the magnet and therefore changing spring extension. After the measurement had been conducted, the weights were removed. For the second part of the experiment, the system was connected in the reversed Helmholtz configuration to the power supply. To make sure everything is connected properly, the current was turned up from zero to three and then back to zero amperes as a test. The magnetic bar moved during this test, meaning the 4 system was connected correctly. The next step was conducting at least eight measurements with different currents from zero to three amperes and recording them in the table. Using the data received from the experiment, the magnetic dipole moment μа can be calculated using the formula µа = F. dz (derived from Eqn. (1)). dB Results The following measurements were made by adding extra weights to the spring and therefore changing its extension and overall mass: Mass, m (kg) Force, F (N) mg = Spring Extension, z (m) 0.001 0.00981 0.008 0.002 0.01962 0.017 0.003 0.02943 0.025 0.004 0.03924 0.034 0.005 0.04905 0.041 Table 1: Spring measurements Using the data from Table 1, the spring constant k was calculated Using the Hooke's law in the following way: mg Amg F = kz => mg = kz => k = => k = Z Az - (0.005 0.001)(kg) × (9.81)(ms-2) k (0.041 0.008)(m) (0.004)(kg) x (9.81) (ms-2) (0.033)(m) The values used for the equation are minimum and maximum from the data (Table 1). Taking the difference between the largest and smallest values ensures high level of accuracy of calculations. 5/n Lab Skills Guide to Writing a Lab Report Submission Checklist • • • Ensure your lab report has a title, your name, your student ID number, your Lab Group number (if appropriate). Please include the module title (e.g. Physics) and date of submission. Use of visual data as well as text will be expected: diagrams; tables; and graphs where appropriate. The lab report should be 1500 (+/- 20%) words in length. Lab reports should contain all the necessary sections and should be written with enough detail and clarity to enable repetition of your experiment exactly. You may be penalised if: • You hand in your assignment later than the stated deadline. You must submit work for assessment by the stated deadline. If you do not, the following penalties for written work will apply: (Unless there are valid reasons for the lateness (e.g. illness) and this is supported with an EEC form and evidence) Penalty Awarded • 1 Number of Working Days Late 85% of original mark 80% of original mark 75% of original mark Zero mark awarded 2 3 More than 3 Any part of your work is found to have been directly copied from a source (e.g. a book, another student's essay, an internet website) without the appropriate referencing convention Please be reminded that this must be written by you. If the level and sophistication of your language is considerably higher than other pieces of written work you have produced, it will be assumed that you were given unfair assistance and you will either be penalised for plagiarism or receive no credit for clarity of expression. Please note that the following examples of collusion are considered as academic misconduct and, in the absence of hard evidence, you may be asked to attend a meeting to discuss your understanding of the work. If you do not attend such a meeting, tutors may use academic judgement to determine whether or not an offence has taken place. Collusion Examples include a situation where a student: a. intentionally submits as entirely his or her own work, an essay or report written by another person. b. permits another candidate to copy all or part of their own work, knowing it is to be submitted as that other candidate's own work; C. allows another person to re-write large sections of the students' work. Lab Report Structure Abstract The abstract should start with the objectives of the experiment, followed by a brief summary of the results and conclusions. The purpose of an abstract is to provide a short description of the contents of your report, so that a potential reader can decide whether it is of interest. An abstract should be a single paragraph of less than 150 words, with no pictures or equations. It is best to write this section last. Introduction The introduction sets the scene for the rest of your report. It gives the background to the experiment or study, explaining why it was important to do it. The objectives of the work should be clearly stated here, but do not discuss your results – leave the good bits until later. In a proper technical paper, you would use the introduction section to review and summarise previous relevant work. Reports should be written in the third person and the passive voice. In other words, never use ‘l' or ‘we' in technical writing. Generally, the past tense is used when giving details of what was done. Occasionally, the present tense is used when giving details of the background to the work or inferring general relationships from the results. It is good practice to use simple words and short sentences. Use of informal/conversational expressions must be avoided. Make sure that you write in complete sentences, and use lists only when absolutely essential. Opinions vary on paragraph layout. You are advised to use justified text (although left justified is permissible), with no indent at the start of the paragraph and a line space between paragraphs. Use reasonable margins for feedback (and for binding) purposes, and do not let figures or tables stray into the margins. Put page numbers and your name etc in a footer. Do not be tempted to reduce text size to fit more on a page. A 12 point font size is recommended, with a serif font such as Times New Roman making for ease of reading (and a more aesthetic look). For longer reports you may wish to number sections and sub-sections, but this is not generally necessary for lab reports. Bold text is only used in titles, never in the text. If you wish to emphasise a statement, use italics. This document provides a set of consistent paragraph styles for you to use as a template. Keep a separate copy as a reference, and then replace text and figures with your own work. Finally, please do not forget to check your spelling and grammar! Word will do this for you, from the Tools menu. There is no excuse for spelling mistakes in a word-processed report. Theory There is no need for a full derivation of a result in a lab report; only the main equations used in the data reduction or analysis are needed. As instructed in the lab class, you may need to do additional background reading to complete this section. In Word, use the MS Equation Editor add-on. Do not hand-write equations, or attempt to type equations in your text; this is never satisfactory. Equations are numbered in the order in which they appear in the report, and then referred to by number. For example, Eqn. (1) is one form of the well-known Bernoulli's Equation p+ ½ pv² = Po (1) where p is the static pressure, p the density of air, V the true airspeed and po the total (or stagnation) pressure. The Equation Editor object is centred in the page, with the number to the left in brackets. Note the use of tabs in the ‘equation' paragraph style to get the formatting right. All symbols used in equations should be defined in the text immediately afterwards. Use italics for variables in the text, and either the 'symbol' font or the 'Insert Symbol' command to create symbols. Use text descriptions to link equations in a sequence. Experimental Procedure Use this section to describe the lab equipment and the experimental procedure. Line drawings of the experimental setup and diagrams of equipment can be neatly hand drawn, although computer generated sketches are preferred. If photos are used to show equipment, they should be labelled properly. If you have used hand-drawn figures it is not necessary to scan them into the file leave a space and attach to your hard-copy submission. - It is not acceptable to use pictures and drawings downloaded from the internet. Write in complete sentences, and do not use bullet points. It is not necessary to reproduce the lab instructions in full – summarise them briefly. Instructions on figure layout and captions will be given in the next section. Results The results section is where you present your processed data. The significance and/or implications of the results should be discussed in the next section; however, it is good practice to briefly comment on them as they are presented here. For example, any trends or patterns which emerge from data, or results which do not meet expectations, should be pointed out in this section. Raw or unprocessed data (for example from your lab log book) should be included in an appendix, along with at least one specimen data reduction calculation. If you have used an Excel spreadsheet (or similar program) to analyse your data then this should also be included in tabulated form in an appendix. An electronic copy of the code must be submitted along with your report. - Experimental results to be discussed should be presented here, in either graphical or tabulated form – but not both. Data tables used to produce graphs should be included in an appendix, since you should not present the same data twice in the main body of a report. In general, use graphs to present processed experimental data and tables to compare or summarise specific aspects of the data; for example percentage differences from theoretical values. All graphs, pictures and drawings should be numbered as 'Figures' in the order they appear in the report. Tables are numbered separately. If you are an experienced Word user, you can use the 'Insert Cross-Reference' command to automatically number figures, but this is not essential. Each figure should have a caption centrally aligned under it, as shown below for Figure 1, and have a brief explanation in the text. Leave any detailed discussion to the next section. smoothed curves for data redundant title points on theoretical curve aspect ratio border 1.5 Cp vs theta 0.5- 0 -50 -0.50 laminar 50 100 150 200 250 1 - -1.5 turbulent theoretical label -2 rotated -2.5 -3 -3.5 기 theta grey background axis label away filled symbols horizontal from axis grid units ? legend away from graph Figure 1: An example of a poor Excel graph Figure titles should be brief and to the point. It is not necessary to write 'A graph of Cp vs theta' - readers can see that for themselves. Axes must be labelled to show quantity and units. Experimental data points must be marked with open symbols of a reasonable size. Different symbols and line-types are used to denote different conditions on the same graph no colour should be necessary. Legends must be clear and informative. Theoretical lines have no data-points, and should be differentiated from experimental curves by using a different line-type. Avoid background grids – they are only necessary if you expect the reader to read values directly from the graph. Be careful with the 'smoothed line' (or - spline) option in Excel – it can introduce spurious ‘wiggles' in your plots as it attempts to join the dots with a continuous curve. - Figures do not need to fill an entire page – 2 figures on an A4 page is acceptable. Make sure all figures are upright, so that the reader does not have to keep turning the report to read them. It may occasionally be necessary to rotate a wide table to fit it on a page, but you should not have to do this. Think carefully about the layout of your figures. These are the primary communication tool of an engineer, so they must be clear and unambiguous. If you have more than 5 or 6 curves on a single graph it will be difficult to interpret – reconsider how you are going to present your data. formatted labels 1 Ср 0 no grid 309 -1 fewer ticks no colour -3 aspect ratio 60° 90° A A ☑ dashed smooth line for theory 120° 150° 0 180° 中 中 laminar -turbulent theoretical units on axis open symbols straight lines for data legend on graph Figure 1: Variation of pressure coefficient Cp angle 0 with separate numbered title Figure 2: An example of a good Excel graph When inserting tables and figures from other software packages, try to avoid inserting them as objects. Embedded objects can be edited by the reader, leading to inadvertent changes in formatting and potential plagiarism. For example, ‘cut-and-paste' from Excel inserts data as a Word table (Table 1), or you can use the 'Paste Special' command to insert as an Extended Metafile (.emf) object which cannot be easily edited. In order to prevent random formatting changes, format pictures and figures so that the layout is set to 'In-line with text'. Do not attempt to use frames. For tables, units must be given in row/column headings. All numerical values should have the correct number of significant figures. When listing numerical data, use a period '.' for the decimal point, never a comma ‘,' – for example 201.5, not 201,5. This rather old- fashioned European usage is not standard in engineering and causes much confusion. For the same reason, never use a comma in a number to separate thousands for example 201500 or 201 500, not 201,500 or 20,1500. If you really must do this, then at least put the comma in the correct place./nSpring constant measurement data Mass, m/kg Force, F/N Spring Extension, = mg z/m 0,001 9.81×103 0.008 0.002 0.01962 0.016 0.024 0.003 0.02943 0.0021 0.03924 0.032 0.005 0.04905 0.04 Magnetic Dipole Moment measurement data Current, I/A dB/dz Spring Extension, Force, F/N z/m 0.25 9.2×10 -3 0275 1.15×10 0.50 0.018 6,365 2.26 x 10. 0.75 0.027 0575 3.39×108 1.00 0.037 018525 46.458 1.25 0.046 1120 578×108 1.50 0.055 1.402 6.91x10 1.75 01064 1.6775 8.04 × 108 x 2,00 0.074 1,95 9.29×10 -8 2.25 01083 2.2 1.04×157 2.50 0.092 2.5 1.15×167/n Force on a magnetic dipole in a spatially varying magnetic field Background In this experiment you will investigate the effect of a magnetic field on a small magnet (magnetic dipole) and measure its magnetic dipole moment. This is a number which characterises the strength of the magnet. Figure 1 shows the equipment you will use. The magnet is suspended by a spring within a magnetic field generated by a Helmholtz coil. In Physics terms you can think of a bar magnet as a magnetic dipole... two magnetic poles (North and South) separated by a short distance. If a magnetic dipole is placed in a magnetic field it will experience a torque (turning force) and try to align itself to the magnetic field. This is exactly what happens in a magnetic compass. The needle, which is a magnet/magnetic dipole, feels a torque and turns to align with the Earth's magnetic field. Figure 1 Magnetic force measurement equipment. If a magnetic dipole sits in a magnetic field which is varying with position it will experience a linear (pulling or pushing) force. You will use this second effect to determine the magnetic dipole moment, Ma, of a small magnet. In general, a magnet with dipole moment, μa, experiences a linear force given by, F=Ma dB dz' [1] where dB/dz is the gradient of the applied magnetic field, B. That is, the rate of change of B with position, z (see below). Equation [1] assumes that the dipole direction (defined by the line joining the two poles) is aligned with the magnetic field, which will be the case during your experiment. The magnetic field B is generated by a Helmholtz coil, a pair of electromagnets of radius R separated by distance R (see below). When the current flows in the same direction in the two coils they produce a magnetic field which is very uniform (doesn't vary much with position) in the region between the coils. In the reversed configuration, where the current in the two coils flows in opposite directions, the magnetic field in the centre between the coils varies linearly with z. Upper coil Lower coil B 2R Helmholz pair Reversed Helmholz pair The rate of change of B with z in the reversed configuration is given by the equation, dB = 0.859μN dz R2 , [2] where I is the current through the coils, μ (= 4л × 107 NA²) is the permeability of free space, N is the number of turns in each coil (168) and R is the coil radius (= 7.0 ± 0.2 cm). Experiment You are given a brass rod upon which a spring is suspended, and on the end of that is suspended a small neodymium-iron-boron magnetic disk permanently magnetized along its cylindrical axis. You will measure the magnetic dipole moment of the disk. The disk is suspended in a tube that sits in the bore of the Helmholtz coil. There is a z-axis scale on the side of the tube and the origin of the z-axis is at the centre of the two coils. I. II. Make sure the power supply to the coils is turned off. Use the five 1.00 ± 0.05 g steel ball bearings as weights and determine the spring constant of the spring from a graph of force against extension. Be careful with the spring: it is easily damaged. Remove the ball bearings when you have finished this section. Determine the magnetic moment of the dipole (μa) in your system via the following procedure: Connect the coil system in the reversed Helmholtz configuration. Turn the current up to about 3 A (the maximum) and back down. You should see the magnet turn in its holder to align with the field generated by the Helmholtz coils. It is experiencing the torque (turning force) mention earlier. • Now increase the current from zero to about 3 A, and for at least eight different currents, measure the elongation of the spring. Note your results in the table. . Plot a graph to determine the magnetic moment μɑ of the dipole: o You can work out the force, F, from the spring constant that you calculated earlier and the field gradient, dB/dz, from equation 2. o Is the graph linear as you might expect from equations 1 and 2? If not, can you think why? What assumption have you made in using equation 2? o Check your value of dipole moment against that from the manufacturers of the experiment... https://www.teachspin.com/magnetic-force (Important Note: If you have done any reading around this topic you may have come across the electric dipole... two equal and opposite charges separated by a short distance. This has some very important differences from the magnetic dipole. The two electric charges can exist separately but magnetic poles can't. There's no such thing as 'magnetic charge' or a magnetic monopole (single pole). In fact, the magnetic field of a bar magnet is generated by tiny loops of electric current associated with electrons orbiting around their host nuclei. This is just the same as the Helmholtz coil itself works, but on a much smaller, atomic scale. While magnetic monopoles don't exist we can treat a bar magnet mathematically as though they do.) Current, I/A dB/dz Spring Extension, Force, F/N z/m