Tag Archives: Teaching

Exam corrections

I’ve just finished correcting the last exam script of summer 2012.  No more corrections until August, yipee. That said, I don’t really mind correcting the semester exams, unlike most of my colleagues. One reason is that I see it as a form of feedback, if pretty shocking sometimes!

Oh joy

It’s probably true that correcting maths or physics exams is somewhat easier than fighting your way through hundreds of poorly-written essays. (I suspect it’s also less depressing – I often think the standard of literacy amongst our students is more worrying than their lack of mathematical ability). By the time I have corrected the first ten physics scripts of any course, I have usually committed every possible answer to memory, so the job goes quite quickly. Also,  I like a task that has a definite beginning, middle and end with room for targets and treats along the way…

In our college, exam scripts are corrected by name and the students sometimes campaign for anonymous marking. Little do they know that from a teacher’s perspective, it’s much harder to fail a person than a number, particularly if you know that student made a decent effort during the semester. Indeed, a great deal of correction time goes into trying to trying to find a few extra marks for the borderliners; if anything, I would expect pass marks to drop if anonymous marking was introduced.

The main downside of examinations is the administration. Combining exam results with attendance and continuous assessment marks, and getting the totals to the department in time for the course board meetings is no trivial task if one is teaching several different courses . Worse, there are always one or two students who seem to have appeared out of nowhere, with an ensuing search for their educational record and assessment results.

By the end of this week, the innumerable departmental meetings will be over, and we will be ready to meet the externs next week. After all that, the results become official and I will finally, finally get back to research…


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A letter to the Minister for Education

On Friday evening, I gave a public talk on the big bang at Blackrock Castle in Cork. I always enjoy giving public science talks, but this one was special (slides here). The venue was a beautiful castle overlooking the sea and I was enormously impressed with the science outreach work being done there by Dr Niall Smith, director of research at Cork Institute of Technology. I was equally impressed with the new observatory at the castle and the astronomy program of Niall and his postgraduate students. Superb work in a fantastic location, surely an inspiration for generations of young students.

Blackrock Castle in Cork: the white dome above the tower is the observatory

I left Cork early on Saturday morning in order to travel to Dublin to catch the High Flyers conference of the Institute of Physics (this is what physicists get up to on bank holiday weekends!). On my way to the meeting, I heard the Irish Minister for Education interviewed on RTE Radio One (Marian Finucane show, May 5th). The Minister had many interesting things to say on subjects such as RTE, the Catholic Church, a recent libel case in Ireland and the near-paralysis of political process in the United States (the latter is a most unusual topic for a politician over here). However, I was taken aback to hear him refer to “problems of productivity in the third level sector, particularly in the Institutes of Technology”, and disappointed that the interviewer didn’t seek some clarification on the comment.

I would very much like to know what the Minister meant by this comment. What do we understand by ‘productivity’ in the context of the third level education? How is it measured? Is it the number of students taught? Number of Noble prizes for research?  Perhaps some Soviet-style quota of engineers graduated? Like all Institute lecturers, I have a heavy teaching load; we produce legions of exactly the sort of science, computing and engineering graduates that Ireland so desperately needs. I must say I grow weary of generalizations like this about third level academics from journalists and politicians, and such a comment from the top man in education is pretty serious. Not a scintilla of evidence was offered by the Minister in support of his remark, just a casually delivered public insult to my colleagues and I.

Here’s the thing, Minister Quinn: like almost all lecturers in the Institutes of Technology (IoTs), I teach between four and five different courses per semester to degree level, a larger teaching load than any third level college in the world as far as I know; add research and outreach activity to this and it is no surprise I am in the office until 9 pm at least four days a week. In terms of prep, each semester typically presents at least one new module to teach, involving months of preparation over the summer, where I would hope to be concentrating on research, finishing my book and attending conferences. (I teach diverse courses in mathematics and physics to students in the departments of computing, engineering and science, not to mention more specialized modules in quantum physics, cosmology and particle physics – how many Harvard professors can boast such a wide teaching portfolio?).

‘Yes, but what about other IoT lecturers?’, the Minister will ask. I imagine I have a more accurate view of the work of my colleagues than the Minister’s advisors and I have no complaints. Indeed, the limited time I have for research arises because other lecturers take on the bulk of student administration (the large number of classes in the IoTs necessitates a great deal of admin; Year Tutors and Course Leaders spend a great deal of time keeping track of attendance, assessments, lab performance  and exam results). There are no easy lecturing jobs.

I love my job and stopped counting the overtime years ago. However, it is frustrating to hear the work of lecturers in the institutes and the universities denigrated by politicians who know nothing of what we do. The tragedy is, I suspect the binary system of universities and institutes has served Ireland very well, although few in charge of education seem to realize it. As they consider the future of the third level sector, I hope politicians and their advisors will make an effort to understand the current system, rather than indulge in unsupported generalizations.

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Last day of semester

Today was the last day of lectures in the first semester, hurrah. There’s something very satisfying about emptying out the teaching briefcase and filing the notes and overheads back on the bookcase until next year. (Yes, we have computers and data projectors in Waterford, but I still use overheads quite a bit). The students now have a study week followed by exams but for lecturers, it’s an ideal time to get back to research.

I’m frequently asked if WIT is a let down after Harvard, but I must say I enjoyed this semester no end. I taught maths (to 1st science), physics (to 1st engineering) and my ‘concepts in cosmology’ course to our physics students. I’m writing a book based on the latter so it was fun summarizing a chapter each week and presenting it in class as bullet points. After each lecture, I found myself rushing back to the office to rewrite a paragraph or re-jig an explanation – very satisfying!

Motivated students

Then there was the neutrino experiment; a superb opportunity for public lectures on relativity. Like almost all physicists, I expect this result is an anomaly because neutrinos are known to have a finite rest mass. I really enjoy explaining this in outreach lectures so long may the anomaly survive! The Trinity lecture was very satisfying, we got a great crowd including some very eminent physicists.

Now I have four weeks to work quietly on the book, uninterrupted by classes – what a job!

Update

Meanwhile, rumours continue to circulate in the media about a possible sighting of the Higgs boson. I haven’t heard anything in physics circles so I’m betting it’s a false alarm based on a misunderstanding of the purpose of next week’s roundup meeting at CERN (see here for more on the rumours). Still, I’ll be keeping an eye on the news on Tuesday!

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Back to school in Ireland

I finally left Harvard in the last week of August, having had a wonderful summer working quietly on The Book during the day and sailing on the Charles in the long summer evenings.

It’s nice to be back home too – no more going around in silly shorts, suncream and shades. Back at Waterford Institute of Technology in the southeast of Ireland, we are already in the second week of teaching term. The bad news is that thanks to the recession, teaching loads have been increased (increased productivity!) leaving almost no time at all for frivolous activities such as research. On the other hand, there is much discussion of the college being upgraded to full university status, mainly because the government thinks that an upgrade ay help attract industry to a region badly hit by the recession. So after all the valiant efforts of WIT researchers, it seems an upgrade may occur for political reasons…

How does the college seem after Harvard? Colleagues keep asking me this. Yes, I miss the beautiful Harvard campus, the incredible libraries and the superb seminars. However, the main day-to-day difference is one of organization. There seems to be a problem of chaotic timetabling in WIT for the first few weeks of every semester, at least in my department. It’s very stressful and leaves no time over for prep or research. I’ve never understood why this happens every year, as our staff and courses change relatively little. One reason might be that lecturers are left to decide who teaches what amongst themselves, pitting Alice against Bob. Give me a didactic Head of Department any day…

Waterford Institute of Technology

On the other hand, it’s great to be in a job with an influx of Hopeful Young People every year. I always think that academics are v lucky in this regard, it doesn’t really matter which college you are in. Another change is that I am moving to a smaller, quieter office yipee. There is a special place in hell reserved for managers who believe that academics work well in large open-plan offices. With students coming to the door and phones continually ringing, it’s impossible to get any work done between classes. Hopefully I’ll have some quiet evenings in my nice new office….

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Harvard vs Trinity College Dublin

I was recently invited to write a short piece on my impressions of Harvard for the Irish university blog University Diary. The piece is published today and can be read here

Update; here it is in full

[I’m fast approaching the end of my year as a research fellow at Harvard – what an experience! ‘So what was it like?’, a great many colleagues in Ireland have asked. Actually, Harvard reminded me very much of Trinity College Dublin, where I did my PhD – but on a larger-than-life scale.

First, the main Harvard campus is not unlike Trinity. Although the architecture dates from a different period, the campus consists of one large quadrangle, with other quadrangles branching off. All of these beautiful quads boast fine old buildings that serve as lecture halls, libraries, dining halls and student housing. This centralization gives Harvard a great ‘lived-in’ feeling; in this respect, it is resembles a large version of Trinity, in contrast with the dispersed, collegiate system of Oxford and Cambridge.

However, Harvard is situated in the quiet district of Cambridge, Boston, not Dublin city centre. As a result, it has been able to situate its growing graduate schools in the immediate area surrounding the main campus, unlike Trinity. Indeed, much of the area between the main campus and the Charles river is filled with Harvard buildings, from graduate schools in business, law and government to student housing; the whole area is now known as Harvard Square.

What about the academic side of things? Apart from a high number of staff who are stars in their field, it doesn’t feel all that different from other universities. What strikes one most is the sheer diversity of scholarship. Consider science; as well as world-renowned departments in mathematics and physics, Harvard also boasts a famous centre for astronomy and accompanying observatory. As well as prestigious departments in traditional disciplines such as chemistry, biology and the medical sciences, Harvard has a huge History of Science department and accompanying museum. Not many universities can boast these, or Harvard’s well-known programs in Science, Technology and Society.

Academic standards are sky high, as you might imagine. Although I have my doubts about some university ranking systems, there is no denying Harvard comes in at no.1 or 2 in almost every poll. So while TCD comes in at the top of the Irish rankings, Harvard comes in at the top of the world! For my money, this is not just a question of its ability to attract the very best because of its prestige and massive endowment (and yes, they do buy in top professors). It is also the close proximity of MIT and other Boston colleges that makes for a highly competitive, interactive academic environment, at least in the sciences. This is quite a unique situation; there is a daily level of intervarsity interaction that is far beyond that of Oxford and Cambridge, or Trinity and UCD say. Most physics seminars I attended had an even mix of MIT/Harvard personnel, irrespective of where the seminar took place. Indeed, regularly trotting off to MIT was a great treat; it’s a beautiful college where any scientist feels instantly at home, not to mention the awe-inspiring number of spin-off companies ringed around the college. Indeed, MIT’s success at innovation currently far surpasses that of Harvard. Of the ‘Nobel possibles’ I was made aware of (quite a few of those over here), at least as many were MIT. So there’s not much complacency amongst the Harvard scientists. Given the relatively small size of Dublin, it’s a pity this sort of daily interaction between the colleges doesn’t happen much.

What about undergraduate life at Harvard? Here, there is a huge difference with Trinity, and indeed between the American system and the situation in Ireland and Europe. Undergraduate fees at Harvard are in the region of 40-50 thousand dollars per annum, with few scholarships. This is true of a great many of the top colleges in the US and it has major implications for society. May we never go down this road, however bad the funding situation gets. You can also see how corporate jobs that cover kids’ health insurance and college fees have an urgent appeal.

As regards tuition, class sizes can be large (> 50), but there is a huge diversity of modules offered. Students typically have 2 plenary lectures per week, with smaller sectionals run by teaching assistants. There is great emphasis on continuous assessment, with corrections done by teaching assistants rather than the Prof (nice!). Sitting in on some classes, I couldn’t help noticing that a great many students spend precious class-time fooling around on the web, so I think I will ban internet connections in my lectures when I return home.

At postgraduate level, the financial situation is very different. While competition to get into the Harvard postgraduate program is intense, once accepted, the stipends for postgraduates are quite generous. I found the difference between the undergraduate and postgraduate populations quite noticeable; while the general student population is mainly made up of well-heeled young Americans, the postgraduate population seemed to be comprised mainly of Europeans and Asians. I had plenty of time to observe this in one my favourite venues, the Harvard Graduate School of Arts and Sciences. With its own building, dining hall, library and common room, this was a great place to meet scholars of all nationalities and a wide variety of disciplines. A great idea for any college! But isn’t it interesting that the research output of the great Ivy League colleges may rest on students who have in fact been trained in European and Asian universities? We should remember this before we adopt every fashionable trend in U.S. undergraduate education.

I’ve decided to stay in Boston for the summer, writing up my research before returning to WIT in September. I’ll certainly miss Harvard, MIT, and Cambridge, I don’t think I’ve ever been in such a vibrant academic environment. More generally, Cambridge Boston is a great place for a European; a liberal, highly-educated bastion of American society, blessedly free from the right wing ideology so increasing pervasive in the US. Back at home, it’s nice to think that the Irish IoTs may someday play MIT to our universities, but I think we have some way to go. More pragmatically, I find it a great drawback being too far from Dublin/Cork to interact with university colleagues on a daily basis…]

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Introductory physics: the photoelectric effect

One of the last lectures in an introductory physics course is usually a description of the photoelectric effect. This is because the effect is a beautiful manifestation of one of the astonishing discoveries of modern physics; that light, known to behave as an electromagnetic wave, can in some circumstances behave as a stream of discrete particles.

The first hint of this dual nature of light arose from Planck’s study of blackbody radiation in 1900 (see post on radiation). Planck found that he could only predict the observed spectrum of radiation from a hot body if it was assumed that the radiation was transferred between the body and the walls of a container in tiny, discrete packets or quanta of energy, each quantum having an amount of energy given by E = hf ; here f is the frequency of the radiation and h is a fundamental constant of nature (extremely small) that became known as Planck’s constant.

This assumption was regarded as something of a puzzling mathematical trick until a young Einstein suggested in a famous paper that it was the light itself (as opposed to some transfer process) that was quantized i.e. the blackbody spectrum could be described by assuming that light was behaving like a stream of extremely small, discrete bundles of energy, each of energy E = hf. This was a bold assumption as the wave properties of light were well established, but Einstein backed up the idea by showing it explained several other puzzling phenomena, not least the photoelectric effect.

The photoelectric effect was a well-known phenomenon whereby light incident on a metal could cause electrons to be released by the metal (measurable as an electric current). A great puzzle was that the effect ocurred only for light above a certain frequency, characteristic of the metal under investigation; this result was completely inexplicable in terms of the familiar wave theory of light.

Light of a certain frequency incident on a metal causes a current to flow

Einstein showed that the photoelectric effect could be easily explained if the incoming light was behaving as a stream of discrete packets (or photons) of energy. Invoking the conservation of energy, he predicted that the maximum kinetic energy (K.E.) of electrons liberated from the metal would be given by

K.E.e =  hf  –  W0

where each incoming photon of light has an energy of hf and W0 is the binding energy (or work function) of the metal. Clearly, electrons could be released from the metal only if the incoming light was of a frequency such that hf  >  W0 , irrespective of the intensity of the radiation! Could it be that simple? The experimentalist Phillipe Lenard disliked Einstein’s idea intensely and set about disproving it in a series of experiments; years of careful experimentation showed that Einstein’s theory was exactly right in its predictions (see here for more details).

Experimental measurement of the photoelectric effect: no electrons are emitted below the cut-off frequency

The explanation of the photoelectric effect was a significant breakthrough in physics as it represented the first unequivocal evidence of duality; the phenomenon whereby light can behave as a wave in some situations and as a stream of particles (or quanta of energy) in others. This duality formed a cornerstone of the new quantum theory and was later found to be a universal truth of the microworld –  entities known as  ‘particles’ such as electrons and even atoms were in turn found to exhibit wave behaviour.  Indeed, the quantum equation E = hf is as important in modern physics as E = mc2 and it was for his explanation for the photoelectric effect (not for special or general relativity) that Einstein was awarded the 1921 Nobel Prize in physics.

Historical note

Philosophers and journalists often claim that ‘Einstein disliked quantum theory’. It should be clear from the above that Einstein was one of the major pioneers of quantum physics; his view of quanta of light was far ahead of its time and was at first strongly resisted by the scientific establishment (including Planck). What Einstein disliked was a later interpretation of quantum theory known as the Copenhagen interpretation, a view of the quantum world that is still debated today.

Excercise

If light of wavelength 780 nm is incident on sodium metal (work function of 3.6 x 10-19 J), calculate the maximum kinetic energy of emerging electrons. (Hint: recall that wavelength and frequency are related by c = fλ and note that h = 6.6 x 10-34 Js )

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Introductory physics: the lens

A spectacular application of the phenomenon of refraction (see previous post) is the lens. Just as a focusing mirror is used to obtain an image of a distant object (see post on mirrors), a lens is used to focus light by refraction. The difference is that the light is transmitted through a lens – it is refracted once entering the lens and again as it passes out again. Lenses are cut from parabolic surfaces in such a way that distant rays are brought to a focus at the focal point.

As with mirrors, there are two types of lenses, depending on the curvature of cut: a convex lens causes parallel rays of light to converge to a real focus, while a concave lens cause the light to appear to diverge from a virtual focus.

As with mirrors, the position of an image will depend on the distance of the object from the lens (but the image of a distant object will of course be at the focal point of the lens). Amazingly, the same equation applies: for an object a distance u from a lens of focal length f, the location v of the image can be found from the relation

1/u1/v =   1/f

(Note that for a distant object u = and hence v = f ). The magnification m of the image can be calcuated from the equation m =  –v/u, as before.

Applications

Lenses are used extensively in everyday life. The most common example is of course spectacles. No one knows when spectacles were first invented (12th century?), but they have been used throughout the ages to improve defective human eyesight.

Typically, spectacle lenses are concave (diverging) lenses are made from glass or plastic. This is because the most common eyesight defect is myopia (shortsightedness), a condition where the natural lens of the eye focuses too strongly i.e. an image is formed short of the retina. A diverging lens of the right strength placed in front of the eye will cause the image to be projected back on the retina as normal.

Concave (diverging) lens used to correct myopia

In the case of hyperopia (the longsightedness that occurs commonly in older people), the eye muscles are weakened and an image is formed beyond the retina; this is corrected by placing a convex (converging) lens in front of the eye in order to strengthen it i.e. shorten the focal length of the eye’s natural lens.

Converging lens used to correct longsightedness

A modern application is the contact lens: this operates on the same principle as above, but the lens is made of a soft fabric that can be worn directly on the pupil. A third option nowadays is laser surgery; in this case the focal length of the eye’s natural lens is adjusted directly (and permanently) by laser treatment.

Lenses and science

Lenses played a pivotal role in the development of science. In the 17th century, advances in lens technology led directly to the invention of the microscope, a device that revolutionized our view of the world of the very small: and to the development of the telescope, an invention that revolutionized our view of the solar system and ultimately the entire universe.

Exercises

1. If an object 5 cm high is placed 30 cm in front of a convex (converging) lens of focal length 20 cm, calculate the position and height of the image.  Is the image real or virtual?

2. As a shortsighted person ages, can the onset of longsightedness cancel myopia?

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Introductory physics: refraction

Light can be refracted as well as reflected: in refraction, light is transmitted, but changes direction as it passes from one medium to another. This familiar phenomenon occurs because light has different velocities in different media. The behaviour can be explained using wave theory but luckily the main results can be described using simple geometrical optics.

The tube appears bent because of the refraction of light

It can be shown that when a ray passes from one medium (1) to another (2) that is denser, the ray of light is bent towards the normal to the interface – and when a ray passes from a medium to a less optically dense one the ray is bent away from the normal. More quantitatively, the angle of incidence i (the angle between an incident ray and the normal) and the angle of refraction r (the angle between the refracted ray and the normal) are related by the simple equation

sin i/sin r = n2/n1

where n is a property of a medium known as its refractive index (related to the velocity of light in the medium)

Sell’s law: sin i/sinr = n2/n1

Note the reversibility of light: if a ray bends closer to the normal upon entering water, it bends further from the normal upon leaving it i.e. all the diagrams work in either direction.

Some typical values for the index of refraction are:

Substance n
Air 1.00
Water 1.33
Glass 1.50±
Plastic 1.40±
Diamond 2.42

Apparent depth

One consequence of refraction is the phenomenon of apparent depth; essentially, this means that a pool of water is deeper than it appears. From the simple diagram below, you can see why (remember the observer sees the image as the intersection of the two diverging beams).

The apparent depth is always shallower than the real thing, so perhaps it should be renamed apparent shallowness! It is just as well nature works this way round – if water was shallower than it appears, children would crack their heads every time they dived into a swimming pool.

Total Internal Reflection

A curious phenomenon can occur when light travels from a dense medium to a less dense one. Since a ray of light is bent away from the normal as it enters a less dense medium, it follows that at some critical angle of incidence, the refracted ray can be 90 degrees to the normal, i.e. travel along the boundary between the media. Further, rays at angles of incidence larger than the critical angle will not transmitted at all, but reflected back into the first medium. This phenomen is known as total internal reflection; the phenomenon is exploited heavily in telecommunications, where waves are transmitted undiminished over large distances in optical fibres.

TIR: at large angles of incidence, the light is simply reflected back into the medium

PROBLEMS

1. If a ray of light enters water from air at an angle of incidence of 60o, calculate the angle of refraction from the table above.

2. If a person looking down vertically into a pond sees a fish apparently 18 cm below the surface, calculate the actual depth of the fish in the pond.

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Introductory physics: focusing mirrors

Focusing mirrors are mirrors cut from a parabola of reflecting material; the parabola is fabricated in such a way that distant rays will be bent through a single point i.e. the focus of the mirror. In fact, there are two types of curved mirrors; converging mirrors made from parabolas that are concave in shape , and diverging mirrors that are made from parabolas that are convex. In either case, the focal length of the mirror is half the radius of the sphere from which it is cut.

From the diagram below, you can see that in the case of a converging (concave) mirror, parallel rays are focused down to an image at the focal point (this is the point of such a mirror). In this type of mirror the rays reflected by the mirror actually pass through F and it is therefore a real focus.

Converging (concave) mirror

By contrast, parallel rays appear to come from a focal point behind the mirror in the case of a diverging mirror. i.e. the focus is virtual.

Diverging (convex) mirror

There is a simple set of rules to follow when finding the position of an image in curved mirrors:

1. Rays parallel to principal axis are reflected through the principal focus

2. Rays through the principal focus are reflected parallel to the principal axis

3. Rays passing through the centre of curvature are reflected back along their own path

These rules are not mysterious but smply a result of how the mirrors are fabricated.

In the diagram above, note that the object is close to the converging mirror, but outside of the focal length. Using the first 2 rules above, the intersection of the reflected rays gives the position of the image. You can see the image is inverted and diminished.

Image tracing in a diverging mirror

More quantiatively, for any object a distance u from the mirror of focal length f, the location v of the image can be found from the ‘mirror’ equation

1/u +   1/v =   1/f

Note that there are only 2 variables in this equation since f is fixed for a given mirror. Typically, one uses the formula to find the location of the image of an object a given distance from the lens. One can also calculate the height of the image; this is because the magnification m of the mirror is given by the equation

m =   –v/u

Note: in using both the above formulae, we use the convention that any distance that is real object is taken as a positive.

Correction

Actually, focusing mirrors are cut from parabolic surfaces, not spherical ones – I forgot this. See comment below by Norman.

Problems

1. An astronomer is observing a distant star with a reflecting telescope: use the mirror formula above to calculate where the photographic plate should be positioned. What kind of magnification can one expect?

2. If an object 5 cm high is placed 40 cm in front of a converging mirror of focal length 20 cm, calculate the position and height of the image.  Is the image real or virtual?

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Introductory physics: reflection

As we saw in a previous post, visible light is simply one portion of the electromagnetic spectrum i.e. visible light consists of electromagnetic waves of a certain frequency travelling at a speed of 3 x 108 m/s (recall also that light can exhibit properties of both waves and particles, a property referred to as quantum wave–particle duality.)

The macroscopic properties of light had been studied for many years before its quantum properties were known. Such properties include transmission, reflection and refraction; the study of these phenomena is known as geometrical optics.

For example, it was realised centuries ago that light travels in straight lines (unlike sound): this can be demonstrated by placing a few pieces of cardboard with pinholes in their centres in a line. On placing a light source in front of A, the light will only be transmitted if the three pinholes are in a straight line.

The light can be seen by the observer if and only if the holes are in a straight line

Using one pinhole, one can form an image of a distant object as shown below: this is the basis of the famous camera obscura.

Rays of light can be convergent, divergent, or parallel. Rays emerging from a source diverge (think of a child’s drawing of the sun); on the other hand, rays arriving at an observer from a distance arrive parallel. Most useful of all, it was soon realised that a good image of an object could be got by causing incoming rays to converge using optical instruments – more on this later.

Reflection

When light falls on a smooth highly polished surface it is reflected i.e. turned back on its path.   A piece of polished metal, or indeed any shiny object makes a good reflector. [One reflecting material that is very much in the news at the moment is ice. The arctic is currently experiencing a global warming more pronounced than anywhere else in the world; this is thought to be caused by the fact that, as the polar ice cap gradually melts to water, it causes a reduction in the reflection of sunlight (water does not relect heat and light very well). This in turn causes further warming, an effect known as a positive feedback loop].

In reflection, a ray of light emerges at the same angle it went in (technically we say the angle of incidence equals the angle of reflection, where both angles are measured relative to the normal to the surface at the point of contact); this makes reflection images rather easy to draw (see below).

Plane Mirror

Glass mirrors have a thin layer of silvering deposited on the back of the glass which is protected.   An IMAGE is produced in the mirror.  The location of the image is got by simply the intersection of the reflected rays. A few trials soon show that the image in a plane mirror is always

– the same size as the object and the same way up

– as far behind the mirror as the object is in front

– laterally inverted

– virtual

Virtual images are images which are formed in locations where light does not actually reach. Light does not actually pass through the location on the other side of the mirror; it only appears to an observer as though the light is coming from this location. (The opposite is a real image; a real image can be focused on a screen, whereas a virtual image can not). In the case of the plane mirror the image is virtual because the rays APPEAR to be diverging from a point behind the mirror.

The reflected rays form a diverging beam which APPEAR to come from A’

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