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 possible assignments – Two in electrostatics – Two in magnetostatics Working in groups of 4 or 5 Equipment for experiments is available here – 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, the book etc.

What to do? Before the experiment – write a plan – You can start writing the plan today. You can take a look at the equipment, read the book, ask questions from assistants – The plan needs to be confirmed by an assistant before starting the experiment Make 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! Write a report describing your results – 7 pages including the essay and figures, concentrate on the results and error analysis – Deadline one week after the final lab session – The report gives 50 % of the grade for this part – Re-submission is possible!

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 Δy? 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 rulerexample, 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 s s s s The time is different each time, this is caused by random error in the use of the stopwatch. d? Then you measure the distance from the window to the ground. d= 9 m 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 What about error? The maximum error we can get is the sum of the individual errors. 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: 0 h = 4.0 ± 0.5 cm1 h = 15.0 ± 0.5 cm

Propagation of errors Example: Measuring the density of metal cylinder 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 How much the result changes if m is changed by small amount? We can find it out by taking the derivative. If we multiply this by Δm, we will get the error contribution from mass measurement

Propagation of errors Now let’s do this for all 3 variables and sum them together In the end we can state m = 247 ± 1 g r = 10 ± 1 mm h = 100 ± 2 mm

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. The total error contribution from the individual error margins of the variables can be calculated by total differential

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

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 – xperimentalErrorsAndErrorAnalysis.html (en) xperimentalErrorsAndErrorAnalysis.html – xx/Luentomat/eng_lab_instr.pdf (en) xx/Luentomat/eng_lab_instr.pdf – xx/Luentomat/Tulostenkasittely.pdf (fi) xx/Luentomat/Tulostenkasittely.pdf