Category Archives: Introductory physics

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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Introductory physics: resistivity

We have seen that if a voltage V is applied to a device, the current I that flows is limited by the resistance R of the device according to I = V/R. Hence a material with high resistance will pass little current (insulator), while a material with low resistance will pass a large current (conductor).

In order to make a meaningful comparison of the resistances of different materials, we need to allow for the fact that resistance depends on how much of the material is present. Hence, we define the resitivity ρ of a material as its resistance per unit length L and cross-sectional area A e.g.

ρ = RA/L

Note that resistivity is a fundamental property of a material, like density. The room-temperature resistivites of some common conductors and insulators are listed below (just click on the table to see it properly)

What is most noticeable is that the resisitivities shown vary over a huge range, from 10+17 Ωm for quartz  to 10-8 Ωm for silver. Amongst solids, metals like silver have by far the lowest resistivities i.e. are the best electrical conductors – this is because the atoms of a metal have many electrons that are somewhat shielded from the nucleus and relatively free to move around. Hence, if a voltage is applied to a metal you have a steady supply of extremely light, charged particles to carry the current from one end to the other. Quartz, on the other hand, is an extremely good insulator because the electrons are tightly bound to individual atoms and there are almost no free charge carriers available for the conduction of electricity.

In between the conductors and insulators on the table lies a very interesting type of material called a semiconductor: these are materials that are normally insulators, but whose resistivity can be dramatically altered by the addition of impurities (doping). Semiconducting materials are extremely important in the manufacture of electronic devices and circuits and lie at the heart of the microelectronic revolution.

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How is resisitivity measured in the lab? First, you measure the resistance of a material by monitoring the current through it as a function of applied voltage (see previous post). Then you measure the length and cross-sectional area of the material and calculate its resistivity from the formula above.

The slope of the graph V/I gives the resistance and a measurement of length and cross-sectional area is then used to calculate the resistivity

Note

The inverse of resistivity is conductivity, measured in (Ωm)-1. Many tables list the conductivity of materials rather than the resistivity.

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Introductory physics: circuits

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.

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Introductory physics: the relation between voltage and current

We have established that voltage is simply energy per unit charge (see last post). What then is current and how does it relate to voltage?

Electric current is a flow of charge, just as a river current is a flow of water. By definition, an electric current I is the amount of charge q flowing per second, hence I = q/t . Current is measured in Colombs per second (also called Amperes, see below). However, we noted last day that the charge on the electron is only a tiny fraction of a Coulomb – hence a current of 1 Coulomb per second corresponds to an awful lot of electrons running around. (How many?)

The lamp lights because the current goes through it to complete the circuit

Since charge will only flow if there is a voltage difference between the terminals of a circuit (last day), you might expect that there is a simple relation between voltage and current. In fact, the German scientist Georg Ohm was the first to discover that there is a linear relationship between the two in many materials. Ohm’s law states that the current I passing through a material connected to an energy source V is given by the equation I = V/R. Here, R is the constant of proportionality and is called electrical resistance and you can see why from the equation: a material with a very large value of R will pass almost no current (electrical insulator), while another material with very small R will yield a large current for the same voltage (good electrical conductor).

Many materials have a linear relation between voltage and current – the slope of the graph is the material’s resistance

Notes

1. Ohm’s law is a bit of a misnomer – it is not a universal law of physics but simply a property of some materials (many materials have a nonlinear response to voltage, including your cat)

2. Current can be considered a fundamental physical quantity in its own right and indeed the ampere is defined as a fundmental unit (see here). However, it’s much better to define it in terms of electric charge, since this is more fundamental.

3. Some unfortunate people quote Ohm’s law as V = IR and play silly games with triangles. In my opinion, I = V/R conveys the physics of the situation much more clearly.

4. It seems from Ohm’s law that a material with zero resistance could pass infinite current! No such materials are known, but some materials have extremely low resistance at very low temperatures – known as superconductors. A good application of superconductivity can be found at the Large Hadron Collider, where protons are guided around the ring by magnets made of superconducting material: this reduces power consumption enormously but the snag is that the experiments have to be done at at extremely low temperatures.

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Introductory physics: voltage

What exactly is voltage? If you ask an engineer, she will probably tell you that voltage drives electric current. And so it does – but what is it? What is its nature? ‘Some sort of energy‘, you might expect. And so it is, although the technical answer is that voltage is electric potential energy per unit charge.

In physics, energy is simply the capacity to do work. Potential energy is the expression we use to convey the fact that an object can have energy simply due to its position or configuration;  a stretched rubber band will do work if released (snap back), as will a compressed spring (spring out), or a brick held aloft (fall on someone’s toe). Indeed, students usually encounter potential energy first in the latter context; any object lifted to a height in the earth’s gravitational field acquires potential energy equal to the amount of work done to get it to that point.  Plus, if you remove the restraint holding it in place, the object will fall and do precisely this amount of work on the ground as it lands (all of its original potential energy is converted to kinetic energy). So you can think of potential energy as work waiting to happen.

A lifted object has potential energy because work was done to get it there; this energy is converted back to work if it is released

Last week, we saw that any electric charge sets up an electric field which will repel like charges and attract unlike ones. Hence it takes work to bring a test charge into the field of a like charge so if we do this we give it electric potential energy ( if you remove the restraint, the charge will rush away). The amount of work done and hence the potential energy acquired will depend on the size of the charge you bring up, so we define instead the electric potential energy per unit charge, also known as the potential. To be strictly correct, potential should be measured relative to something, so physicists talk of potential difference, defined as the difference in potential between the point in question and zero field. Since energy is measured in joules, potential is measured in joules per coulomb or volts and hence potential also became known as voltage. So voltage, potential and potential difference are all the same thing.

In a battery, a potential difference is maintained between the terminals. Charge cannot flow from one terminal to the other because they are not connected. However, if a conducting path between the terminals is provided (by connecting them by wire), a current will flow in the circuit.

A battery and circuit (tnote that the direction of current is defined as the direction +ve charge would move for historical reasons)

More

Since voltage is defined as energy per unit charge, it should be obvious that the product of voltage and charge is energy (or work)  i.e.  W = qV. Thus if a charge of 1 Coulomb is moved through a potential difference of 1volt, 1 joule of work is done.

However, the charge on a single electron is not 1 Coulomb, but a minute 1.6E-16 Coulombs. Hence in the world of particle physics, one typically deals in tiny, tiny amounts of energy. For convenience, we define the unit electron-volt (eV) as the work that is done when a single electron moves through a potential difference of 1 volt.

Question

How many eVs  there are in 1 Joule of energy? The maximum energy achievable at the Large Hadron Collider (LHC) in Switzerland is 14 TeV – show that this corresponds to only 2.2 microjoules of energy. (Note that although this is a small amount of energy, the energy density is enormous because the cross-sectional area of the colliding particle beams is extremely small).

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Introductory physics: the concept of field

In a first course in physics, it is usually in electrostatics that one first encounters the concept of a field.

Everybody knows that like charges repel, while unlike charges attract. The quantitative version of this rule is Colomb’s law, which is the observation that the force between two electric charges A and B is given by F  =  k.q(A).q(B)/r2 where q represents electric charge and r is the separation of the charges (k is is a constant determined by the medium in which the charges are situated). Note that if the charges are like, the force comes out positive, so a repulsive force is positive in sign, which is what you might expect since work must be done to bring the charges together. On the other hand, if the charges are unlike, the force comes out negative (which also makes sense as the charges want to be together anyway).

However, the concept of force isn’t all that useful if one wants to know the effect of a given electric charge (A) on the world. It is clear from Coulomb’s law above that the force experienced by any charge B due to A will also depend on the magnitude of B i.e. each charge you bring up to A will experience a different force! Physicists get around this problem by defining the field due to charge A as the force a test charge brought up to A will experience divided by the magnitude of that test charge. Hence, while  every charge brought up to A will experience a different force, they will all experience the same field (F  =  k.q(A)/r2) – clever huh?

We can even draw pictures of the field due to A, simply by drawing lines representing the direction in which an electric charge will move. Unfortunately, the convention is that we draw the direction a positive charge will move (unfortunate because we now know that it is the negatively charged electron that’s doing the moving: the convention is historical)

Electric field between two unlike charges (red is +ve)

The concept of a field is not limited to electricity; it is used throughout physics. For example, you and I experience slightly different forces due to the earth’s gravity. This is because the gravitational force depends on the product of both the earth’s mass M and personal mass m (F = GMm/r2 where G is a constant). However we all experience exactly the same field: dividing by personal mass, the gravitational field due to the earth is given by GM/r2.


The earth’s gravitational field is directed towards its centre

Note

You may have noticed that the equation for gravitational force above looks very like that for the electric force, with charge replaced by mass (both forces decrease with the square of increasing distance). However, gravity is a much, much weaker force; the gravitation constant G is orders and orders of magnitude smaller than the electric constant k and you only notice a body’s gravitational field if it has the mass of a planet!

That said, it is now believed that there is a deep connection between electricity and gravity; indeed, particle physicists and cosmologists believe that all four of the fundamental forces of nature (gravity, electromagnetism and two nuclear forces) originally formed one superforce, which gradually split off into four separate forces as the uiverse expanded and cooled. We already have strong theoretical and experimental verification that two of the fundamental forces originally comprised one force, and it is one of the great ambitions of theoretical physics to describe all four forces in a single mathematical framework (unified field theory). Within that program, one of the great puzzles is why the gravitational force is so much weaker than all the others.

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