Physics

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Physics is not an easy subject to master. Talk to non-physicists and the most common view you’ll meet is that it is hard because it requires a great deal of intelligence, which is flattering but not necessarily true. What makes it so difficult is that it requires not only specialist knowledge in physics and maths but also a range of skills in order to put that knowledge to use. It takes time and effort to acquire these skills, but the effort must also be well directed.

• Learning physics
• Skills in physics
• Critical thinking in the laboratory
• Relating mathematics to physics
• Mathematical modelling

Learning physics

As a student you have to make the best of the teaching you receive. The difficulty is that whilst some lecturers and some courses will appeal to you others won’t. The simple truth is that it is not easy to identify the best way to teach physics, as Robert Ehrlich, of George Mason University, Virginia, USA, described recently. In an article entitled , ' How do we know if we are doing a good job in physics teaching? ', Dr Ehrlich wrote enthusiastically about the innovations he had made in teaching only to report that some students liked it and some didn’t. Virtually every lecturer you meet would most likely tell you the same thing. Even if you don’t like the way a subject is taught you have to make the most it. It will be largely up to you to acquire not only a detailed knowledge of specific areas of physics but also a variety of skills that you will need by the time you graduate.

Robert Ehrlich, 'How do we know if we are doing a good job in physics teaching' American Journal of Physics, Vol. 70(1) 2002 pp24 –29

As a physicist you will have to be able to;

• Understand mathematical arguments
• Relate mathematics to physics
• Design experiments
• Formulate theory
• Analyse experimental data
• Understand significance and errors
• Perform statistical analysis
• Evaluate the impact of new data on existing theories.
• Report the results of your work

How do you set about learning these skills? Knowledge and understanding are of course essential but you’ll also need to think critically. Critical thinking is central to the practise of physics. It serves two purposes: first it can help you to understand the physics, and second, by altering the way in which you approach problems, it can help you to achieve better results in the laboratory and in other assessments.

Critical thinking is central to both the process of designing an experiment and analysing the data.

In designing an experiment you need to;

• Be clear about what you are trying to measure
• Understand the physics that informs your choice of measurement; what are you going to choose as your dependent variable and why?
• Have a clear appreciation of the limitations of your instrumentation
• Understand the sources of errors in your measurements

In analysing the results of the experiment you need to:

• Manipulate and transform data
• Propagate the errors through to the final result
• Draw conclusions about the final result compared with your aims in performing the experiment
• Possibly re-design the experiment if you judge the outcome to be wanting

Finally, you also need to present the results of the experiment, either for assessment or for publication. Writing a report or making an oral presentation are skills in themselves that are an essential part of the physicist’s trade and which have to be acquired along with the more obvious physics skills. You will be judged as a physicist on the quality of what you write, so it is important to get it right. The readers of your report will want to know in essence that you have designed and executed the experiment properly; in short, that you have employed all these critical thinking skills.

For more advice, see critical and analytical thinking and the free audio download on critical thinking.

Relating mathematics to physics

Mathematics is a tool used by all physicists. Experimentalists often do not need as many mathematical techniques as a theoretician, but rare is the person who can get away without using any maths whatever. The ability to understand mathematical arguments and relate the mathematics to the physics is therefore essential.

Mathematics is a shorthand. Consider the familiar equation from radioactive decay

1.

Where N is the number of atoms and l is a constant. This is equivalent to saying, “the rate of decay in a radioactive sample is proportional to the number of atoms present and therefore decreases as the number of atoms decreases due to the decay”.

Written out, it appears much more long-winded than the mathematical form, but, more importantly, what does it mean? Does the sample somehow know how many atoms are present? Does one atom influence another? These are examples of critical thinking. By asking such questions we can get to an understanding of radioactivity that may not be immediately apparent from the mathematics. Re-writing the equation,

2.

Shows that the fractional change in the number of atoms present occurring in a fixed interval of time is constant. Therefore if the size of the sample is increased the number of radioactive events increases.

This can be seen very easily if we imagine a sample containing say N 0 atoms where the time taken for a small fraction, like 1 %, to decay is measured. If we had an identical sample we would expect that a similar measurement would yield a similar time. It won’t be exact because the process is essentially random and small differences are bound to occur. If we repeated this measurement hundreds of times we would expect hundreds of similar results from which we could take an average. However, the total change in a sample containing say 100N 0 atoms is the same as the total change in 100 samples containing N 0 atoms so if size does not matter then equation 2 holds.

Both questions have been answered, but is it physically sensible? Equations 1 and 2 are mathematical arguments based on the assumption that the decay of each atom is a random event uninfluenced by its neighbours. There are certain situations where this is clearly not the case. The fission of uranium or plutonium atoms can be triggered by the fission of a neighbouring atom, so this model would not apply here. That does not mean the maths is wrong, however. As a mathematical argument it is sound. The physics lies in identifying the assumptions about the physical world implied in the mathematics and deciding whether they apply. In the case of fission they do not, but in many other examples of radioactive decay the assumptions do apply. These equations therefore provide an adequate description of these processes.

For more advice, see studying mathematics.

Mathematical modelling

Model making in physics is the reverse of the above process. Here a mathematical argument has been examined to identify its implications along with the assumptions underlying it so that it can be better understood. This is a useful technique for the student of physics who wants to understand a mathematical argument but how do you go the other way and express a physical situation in mathematical form?

What makes physics seem hard is that it is difficult to understand how anybody might come to assert some of the common ideas as they are often expressed. Take equation (1) above, for example. It immediately begs a couple of awkward questions that were only answered by a long and detailed argument involving some critical thinking. If it takes so much work to understand such an equation how is it derived in the first place? It’s hard to imagine that you could come up with this equation from scratch without answering these questions beforehand, and therein lies the key. Most of the models you will encounter as an undergraduate are presented in their most useful form, not the form in which they were first derived.

If instead of assuming that the rate of decay is proportional to the number of atoms present, which is not intuitively obvious, you came up with the idea that the fractional change in the number of atoms is the same from one sample to the next under identical conditions, which is actually quite sensible, you would be far more likely to come up with equation (2) first. It is only a short mathematical step then to equation (1), which is not so physically obvious but mathematically is more useful because it is in a form that can be integrated.

Mathematical modelling therefore involves:

1. Identifying the key physical ideas that we want to express.
2. Often those ideas will be assumptions about the world that we don’t know are right. We make the assumption in its simplest form and express it mathematically.
3. The maths is then developed to find an equation or set of equations that are useful.

This then constitutes our mathematical description of the physical world, which we may have to alter by revising the assumptions if the theory and experiment don’t match. Mathematical arguments can of course be very complex and may require a very high level of mathematical skill. The essential idea is the same however. We must express a physical idea mathematically and then interpret the mathematical solution physically.

For more advice, see studying mathematics.

This content was written for skills4study by David Sands, author of Studying Physics.