Electromagnetism lab project
Contents 1. Overview of the course 2. How to analyse errors in measurements 3. How to make graphical representations (plots)
Overview Four lab experiments Biot Savart s law Helmholtz coil Coulomb s law Force on an electric charge You will do two experiments one on magnetostatics (Biot-Savart/Helmholtz) one on electrostatics (Coulomb/Force on a charge).
Overview Working in groups of 3-5 You have 3x3 hours of lab time Use that time to plan your experiment with assistants, ask questions about things you don t understand. Look for information from internet, textbooks etc.
What to do? Before setting up the experiment write a plan See the instructions for each experiment Once you are done show the plan to the assistant You can ask questions! Build your experiment according to your own plan Before you plug in the power and start measuring, ask one of the assistants to check your setup!
What to do? Write a report describing your results 5 pages including figures, concentrate on the results and error analysis See instructions in lab instructions and on mycourses page Deadline one week after the final lab session This will be marked, commented and returned You will have another week to return a second version, if you wish (only first hand-in is compulsory) Mark will be average of two reports in the case of resubmission
Errors in measurement Error in this context doesn t mean mistake or blunder In scientific measurements the term error refers to the limits of the precision of the measurement (also know as uncertainty) We state the measured quantity y with the error Δy in format x = y ± Δy For example, when measuring something with a simple ruler, we can only measure things with 0.5 mm accuracy So we state the length of the line as 4.9 ± 0.05 cm
Why should we care? Analysing errors gives us an idea of the limits of our accuracy Furthermore, it often helps us to point out the relationships between individual error sources which is useful for developing our measurements Accurate measurements are necessary for any industry, from IKEA cupboards to microelectronics.
Estimating error margins Let s say you measure a quantity y. What should be the error Δx? Error margins for any measured quantity depend on the equipment used Sometimes it is given by the equipment manufacturer Sometimes you have to use your own judgement Usually error margin is one or half of the smallest measurable decimal, for example, in the case of a normal ruler for example, it could be 1 or 0.5 mm. For repeated measurements, statistical methods can be used
Sources of errors Random errors are deviations from the real value of the measured quantity caused by unknown and unpredictable changes in the experiment. For example, when measuring the period of a pendulum with a stopwatch, random error is introduced by the experimenter Random error can be reduced by repeating the measurement Systematic errors are reproducible inaccuracies that are caused by incorrect use of the instruments For example, the instruments might be calibrated so that the readings are incorrect Systematic errors are often hard to detect and eliminate
Sources of errors (example) Let s say we try to measure acceleration due to gravity by dropping a ball from a window You measure the time to fall several times and get the following 1. 1.02 s The time is different each time, this is 2. 1.05 s caused by random error in the use of the 3. 0.94 s stopwatch. 4. 1.08 s d? d= 9 m Then you measure the distance from the window to the ground. But unfortunately your measuring tape is old and you measure the distance to be 1 meter shorter than it actually is. This is an example of systematic error
Propagation of errors Measuring the speed of a snail v x t 1 1 x t 0 0 0 h = 4.0 ± 0.5 cm 1 h = 15.0 ± 0.5 cm What about error? The maximum error we can get is the sum of the x1 x0 individual errors. t1 t2 The speed would be then 11 ± 1 cm/h Notice here how we round the uncertainty to one significant figure and round the answer to match. In general: for addition and substraction the total error is the sum of the errors: Δz x y
Propagation of errors Example: Measuring the density of metal cylinder m V m hr 3 2 7862 kg/m We get the following measurements m = 247 ± 1 g r = 10 ± 1 mm h = 100 ± 2 mm How to combine the errors?
Propagation of errors m hr 2 How much the result changes if m is changed by small amount? We can find it out by taking the derivative. 1 m hr 2 If we multiply this by Δm, we will get the error contribution from mass measurement 1 m 31.8 kg/m 2 hr 3
m = 247 ± 1 g r = 10 ± 1 mm h = 100 ± 2 mm Propagation of errors Now let s do this for all 3 variables and sum them together 1 m 31.8 kg/m 2 hr m m r r In the end we can state 3 m 2 r 1573 kg/m 3 hr h h m 1 h 157 kg/m 2 2 h r 3 3 m V 1761.8 kg/m 8000 2000 kg/m m hr 3 3 2
Total differential What we did for the cylinder was to calculate something called total differential Our final result is a function of a number of variables i.e. c f a, a,,...) ( 1 2 a3 The total error contribution from the individual error margins of the variables can be calculated by total differential c f a 1 a 1 f a 2 a 2...
Graphical representations After the measurements are done, you will need to find a way to represent the data. Most common way to do this is to plot the data in a graph
Fitting a line With graph representation, we can then extract more data from the results by fitting a line to them In this example, the slope of the line is the resistance according to Ohm s law V RI
Step-by-step 1. Choose axis so that we can present the model as a straight line Sometimes for this you will need to manipulate the data for this 2. Choose the axis range suitably Remember axis labels
Step-by-step 3. Straight line which gives the best fit to the measured data set is drawn to the graph. We ask you to use a least-squares method for this There are also many other methods to do this, including drawing by hand
Links Error analysis http://reference.wolfram.com/applications/eda/e xperimentalerrorsanderroranalysis.html (en) http://physics.aalto.fi/pub/kurssit/tfy- 3.15xx/Luentomat/eng_lab_instr.pdf (en) http://physics.aalto.fi/pub/kurssit/tfy- 3.15xx/Luentomat/Tulostenkasittely.pdf (fi)