Category Archives: Teaching

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/u +  1/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?

Electrical devices (TVs, stereos etc.) are connected to a voltage supply by an electrical circuit. The only difficult thing about circuits is that devices can be connected either in series or in parallel.

If connected in series, the same current runs through each device since there is no alternative path. However, the voltage across each device is different: from V = IR, the largest voltage drop will be across the largest resistance (just as the largest energy drop occurs across the largest waterfall in a river). As you might expect, the total resistance (or load) of the circuit is the sum of the individual resistances.

On the other hand, electrical devices can also be connected in parallel. In this case, each device is connected directly to the terminals of the voltage source and hence experiences the same voltage. Here, there will be a different current through each device since I = V/R. A counter-intuitive aspect of parallel circuits is that the total resistance of the circuit is lowered as you add in more devices (the physical reason is that you are increasing the number of alternate paths the current can take).

Parallel circuit: each device is connected directly to the battery terminals

Which is more useful? Household electrical devices are connected in parallel because it is easier (for the manufacturer) if every device sees the same voltage and it also turns out to be more efficient from the point of view of power consumption.

A more complicated type of circuit is the combination circuit: here some resistors are connected in series, others in parallel. In order to calculate the current through a given device, the trick is to replace any resistors in parallel with the equivalent resistance in series and analyse the resulting series circuit.

Combination circuit

Problem

Assuming a resistance of 100 Ohms for each of the resistors in the combination circuit above, calculate the current through each if a voltage of 12 V is applied.

We saw in the last post that energy that can be transferred by conduction and convection, two very different molecular processes. But it is the third mechanism of heat transfer that is the most surprising.

In radiation, the transfer of energy is not a molecular process at all. Instead, the energy is carried as an electromagnetic wave, that is a wave consisting of oscillating electric and magnetic fields. The fields are self-perpetuating (and mutually perpendicular) because the changing electric field induces a magnetic field and the changing magnetic field induces an electric field.

The discovery of electromagnetic radiation emerged late in the 19th century. From Maxwell’s theory of electromagnetism, it was realised that light itself consists of an electromagnetic wave: however, it took Einstein to realise in 1905 that electromagnetic waves travel from the sun to earth through a vacuum i.e. do not need a medium in which to travel (unlike conduction or convection).

The rate of radiation from the sun (or any body hotter than its surroundings) is proportional to the fourth power of its temperature i.e. is extremely sensitive to temperature. Radiation also depends on a property of the body known as emissivity. Emissivity is a measure of how well a material emits radiation and is determined by atomic processes within the body. For this reason, a good emitter is also a good absorber, if it is placed in an environment where it is cooler than its surroundings (a perfect absorber is called a blackbody, because it will absorb all light incident on it). The opposite of a good absorber or emitter is a reflector, an object which can neither absorb nor emit heat. Polished metals and bright materials tend to be goodish reflectors: for this reason white clothes are worn when playing cricket and tennis in hot countries (they reflect both heat and light, keeping the player cool and easy to see).

A hot body does not radiate energy at a particular frequency, but at all frequencies – from waves of high energy and frequency (gamma rays) to low-energy ones (radiowaves). The low energy waves have low frequencies but long wavelengths since the wavelength of a wave is inversely proportional to its frequency. The full range of frequency (or wavelength) of radiation is called the electromagnetic spectrum. One of the great unifying moments in physics occured when it was realised that radiowaves, microwaves, infra-red heat, visible light, ultra-violet light, X-rays and gamma rays are all versions of the same thing – they are simply electromagnetic waves of different frequencies (and wavelengths).

Even a blackbody body dies not radiate equally at all frequencies. The distribution of radiation vs frequency (i.e. the spectrum of radiation) depends on the temperature. A body at extremely high temperatures will radiate predominantly at high frequencies, while a body at very low temperatures will radiate predominantly at much lower frequencies. Below is a picture of the emisson spectrum of a blackbody, measured at several different temperatures.

This spectrum is of great interest in fundamental physics, because it turns out that it cannot be predicted using the laws of classical physics. In the early years of the 20th century, Planck and Einstein showed that the blackbody spectrum could only be explained if it was assumed that light behaves as a stream of discrete particles in some circumstances. This duality i.e. light behaving as wave in some circumstances and as a stream of particles in others forms the basis of the famous quantum theory (and was later found to be true of matter as well as of radiation i.e. the tiniest ‘particles’ of matter such as electrons can exhibit wave behaviour!)

Question

In cosmology, the cosmic background radiation is a faint background radiation that permeates the entire universe.  It is radiation that is almost as old as the universe itself, dating back to the time after the Big Bang when the universe had expanded and cooled just enough for the first atoms to form, allowing radiation to travel freely (up to this point in time radiation was scattered by the different particles) . Do you think the cosmic background radiation will be hot or cold? At what frequency do you think it is observed? What kind of spectrum might be expected?

One the great surprises about heat energy is that the transfer of heat can occur by any or all of three very different mechanisms.

In conduction, heat transfer occurs by a process of molecular collision. If you heat up one end of a bar of iron, the energy is transferred from the hot end to the cold by atoms or molecules bumping into one another i.e. while there is a net drift of energy from hot to cold, the molecules do not change their respective positions. This is the primary method of heat transfer in solids and it works best of all in metals (because loosely bound electrons play a role).  It is also an efficient method of conduction in liquids, but occurs hardly at all in gases. In gases, a low density of atoms or molecules inhibits conduction very effectively – hence air is an excellent insulator.

This fact is used to good effect in double glazing; a layer of air between two panes of glass allows one to have good light and views in a house without too much heat loss. Similarily, modern mountaineers keep warm by wearing many thin layers as the air trapped between each set of layers acts as an effective insulator.

Conduction in a solid: the molecules do not change position much

Heat transfer can also occur by the process of convection. In this case, heat energy is transferred by a movement of molecules. The classic example is hot air rising: on a hot day, air close to ground absorbs heat from the earth’s  surface, expands, and rises because it has become less dense than other air. Cooler and denser air then rushes down from above to fill its place, only to be heated in turn and a cycle is set up. As you might expect, this an important method of heat transfer in gases, and convection currents are responsible for everything from sea breezes at shore to major wind patterns around the globe.

Sea breeze close to shore on a hot day

Convection also occurs in liquids: indeed, convection currents are of great importance in the oceans of the world. For example, the seas around Ireland are warmer than might be expected for our latitude. This warming is a result of the famous North Atlantic Drift, a huge ocean current that is part of a giant conveyor belt that delivers heat from the seas off South America all the way up to the seas near Greenland. One of the concerns of global warming is that as the ice caps melt, this current may weaken or even shut down: in which case Ireland and Britain could become very cold indeed!

The North Atlanic Drift keeps Ireland’s climate mild

What is the third process of heat transfer? Well, that is a different story altogether…

The relation between heat and temperature (last post) is not always straightforward: in some cases, large quantities of heat can be supplied to a substance without any observable change in its temperature at all!

When does this happen? It happens when matter changes state i.e. when a solid melts to liquid, or a liquid changes to gas. In fact, it takes a lot of energy to convert a solid to a liquid even if the solid is at the melting point, and it takes even more energy to convert a liquid to a gas even at the boiling point. Neither of these ‘phase changes’ can happen unless energy is continually supplied and there is no rise in temperature rise during the phase change; for this reason the energy consumed is known as latent heat (from the Latin for hidden). It’s interesting physics and quite fundamental at the same time; for example the liquid/gas phase change requires much more energy than the solid/liquid because the intramolecular forces must be completely broken down (as opposed to being weakened in the solid/liquid case). It’s also fascinating to observe that as a solid gradually melts into liquid, the resulting liquid stays stubbornly at the melting point temperature until all of the solid has undergone the change of state (ditto for gas).

Heating curve for ice: the temperature stops rising at 0 and 100 degrees Celcius during the phase changes

A good example of an application of the physics of phase change can be found in an electric kettle: a sensor detects when the top layer of water begins to bubble and quickly switches off the heating element – as opposed to boiling water in an open saucepan, a wasteful process where a lot of water is converted into useless steam, meanwhile consuming a large amount of energy.

Electric kettle : clever device

A change of state can also happen in the reverse direction: when a gas changes to liquid, or liquid to solid, energy is released. This process is exploited in the refridgerator, for example.

In a fridge, a gas-to-liquid phase change extracts heat, keeping the fridge cold inside, while a liquid-to-gas change outside the fridge dumps heat

The whole business of state change raises interesting questions about the nature of matter. Why do some substances exist as solids at room temperature and pressure, others as liquid or gas? To answer this requires a discussion of molecular bonding. Another common question is this: if supplying enough heat to a solid gives you a liquid, and eventually a gas, what happens if you keep supplying heat to the gas?

The answer is that if you supply enough energy, you eventually get another phase change as the atoms of the gas become ionized i.e. electrons are stripped off the atoms of the gas. In this case, the gas becomes a plasma, the fourth state of matter. Plasmas are plentiful in nature: a star exists as a plasma, as does lightning and even fire. Plasmas can also be produced in the lab under extreme conditions, for example by laser bombardment or by particle collisions in accelerators.

A supernova is not a liquid or a gas but a plasma

Update

Here is a super little YouTube video on plasmas by rock band They Might Be Giants, many thanks MattW