# CCSU Photoelectric Effect Physics Study Material

**CCSU Photoelectric Effect Physics Study Material**

## CCSU Photoelectric Effect Study Material

**Photoelectric Effect :** When a beam of light of frequency in the blue or ultraviolet region falls on a metal plate, slow-moving electrons are emitted from the metal surface. This phenomenon is known as ‘photoelectric effect’. The electrons emitted by photoelectric effect are called ‘photo-electrons’ and the current produced is called ‘photoelectric current’. For photoelectric effect, light of high frequency is more effective than that of low frequency.

**Experimental Observation :** Lenard in 1900 studied the photoelectric effect experimentally. His apparatus (Fig.) consists of an evacuated tube having two plates, C (cathode) and A (anode). A varying potential difference can be applied between the plates C and A by means of a potential divider and a commutator K. The p.d. and the current in the circuit can be read by a voltmeter and a sensitive ammeter respectively.

**Dependence upon Intensity of Light :** Light of an appropriate fixed frequency is allowed to fall on the plate C. Photoelectrons are emitted from the surface of C. When the potential V of the plate A is made positive relative to the plate C, a steady saturation current flows in the circuit, whatever the (positive) value of V. This means that all the photoelectrons emitted from the plate C are reaching the plate A. If the intensity of the incident light is increased, the saturation current is found to increase in the same ratio (fig). This shows that **the photoelectric current, or the number of photoelectrons emitted per second is directly proportional to the intensity of the incident light.**

When the potential V of the plate A is made negative with respect to the plate C, the electrons are retarted. If the potential is small, only the slower electrons are pulled back and the current falls. As the negative potential is increased, more and more faster electrons are pulled back. Ultimately, at a certain negative potential V_{0 }, even the fastest electrons fail to reach A. The photoelectric current is then reduced to zero. An increase, however large, in the intensity of the incident light does not produce any current at the potential – V_{0} (fig.). **The (negative) potential V _{0 }at which the photoelectric current is just reduced to zero is called the ‘stopping potential’ or ‘cut-off potential’.** Since the energy of the fastest electrons is just annulled when they fall through the stopping potential, the stopping potential is a measure of the maximum kinetic energy of the photoelectrons. Thus, if K

_{max }be the maximum energy of the photoelectrons, then

**K _{max} = e V_{o},**

Where e is the electronic charge. Since at the stopping potential the photoelectric current cannot be obtained by increasing the intensity of the incident light, it is clear that **the stopping potential, or the maximum kinetic energy of the photoelectrons, does not depend upon the intensity of the incident light.** If the intensity of light be doubled, the photoelectric current will be doubled but the stopping potential will remain unchanged.

**Dependence Upon Frequency of Light :** If, after reaching at the stopping potential, the incident light is replaced by another light of higher frequency, the photoelectric current is re-established. This current again stops when the stopping potential is increased. Thus, higher the frequency of the incident light, higher will be the stopping potential (fig.), that is, higher will be the maximum kinetic energy of the photoelectrons.

If a graph is plotted between the frequency v of incident light and the maximum kinetic energy K_{max }(=eV_{0}) of the photoelectrons , a straight line is obtained (fig.) From this we conclude that** the maximum kinetic energy of the photoelectrons increases linearly as the frequency of the incident light increases.**

If the straight-line graph (between K_{max} and V) is produced back, it cuts the v-axis at a point corresponding to a particular frequency v_{0}. It means that the emission of photoelectrons cannot occur if the frequency of the incident light is below a certain value v_{o}, however strong the intensity of light. On the other hand, when the frequency of the incident light is above v_{0, }photoelectrons are emitted almost instantaneously with no time-lag, however poor the intensity of light.

**The lowest frequency (v _{0}) of light which can emit photoelectrons from a material is called ‘threshold frequency’ of that material. **It is different for different material surfaces.

**Failure of Wave Theory :** The above characteristics of the photoelectric effect cannot be explained on the basis of the wave theory of light. It is due to three main reasons:

- According to the wave theory, the energy carried by a light beam is distributed uniformly over a wavefront and
**is measured by the intensity of the beam.**Thus, when light falls on a metal surface, the energy of the wave should be transferred uniformly to the electrons in the surface before they are emitted out Obviously, the energy taken up by the electrons must increase as the intensity of light is increased. This is against the experimental observation that the maximum energy of the emitted electrons is independent of the light intensity. - Again, the wave theory suggests that light of any frequency, however, low it is, should be capable of ejecting electrons from a surface, provided only that the light is intense enough. Experiment, however, shows that light of frequency lower than a certain threshold value cannot eject photoelectrons, no matter how intense it is.
- Finally, the wave theory suggests that if the incident light is very feeble, the electron should take appreciable time before it acquires sufficient energy to come out from the surface. However, no detectable time-lag has ever been found between the falling of the light on the surface and the emission of the photoelectrons.

Thus, wave theory fails in explaining the experimental observations regarding the photoelectric effect.

**Explanation from Quantum Theory :** According to the quantum theory, the energy carried by a light beam of frequency v is concentrated in indivisible packets called ‘photons’, each photon carrying an amount hv. The intensity of the beam depends merely upon the number of photons. Now, when light falls on a metal surface, a photon (energy hv) is completely absorbed by a single electron in the surface. Part of this energy is used in ejecting the electron against the attraction of the rest of the metal, and the rest is given to the electron as kinetic energy. If the intensity of light is doubled, the number of photons falling per second on the surface will be doubled, (the photon energy hv will remain the same) and hence the number of electrons emitted per second will also be doubled, their maximum energy remaining the same as before. Thus, the number of photoelectrons emitted per second varies directly as the light intensity, but their maximum energy is independent of the intensity. This is completely in agreement with experiment.

The maximum kinetic energy of the photoelectrons increases as the frequency of light, and hence the photon energy hv, increased as observed be experiment.

The existence of a threshold frequency also speaks in favour of the photon (quantum) theory. The electron cannot leave the metal until the energy given to it by a single photon hv) exceeds the energy required in liberating the electron against the attraction of the rest of the metal; no matter how many photons there are (that is, no matter how intense the incident light is).

Finally, the absence of a time-lag also follows from the photon theory. As soon as the first photon strikes the surface, it is immediately absorbed by some electron which escapes instantaneously, there being no time-lag.

The successful explanation of photoelectric effect in terms of quantum theory is one of the strongest evidence in favour of this theory.

## CCSU Laws of Photoelectric Emission Study Material

**Laws of Photoelectric Emission :** Lenard and Millikan gave the following laws derived from experiments on photoelectric effect :

- The number of photoelectrons emitted per second from the surface of a metal varies directly as the intensity of light incident on the surface.
- The maximum kinetic energy of the emitted photoelectrons is independent of the intensity of the incident light.
- The maximum kinetic energy of the photoelectrons increases linearly with increase in the frequency of the incident light.
- If the frequency of the incident light is below a certain lowest value, no photoelectrons are emitted from the metal, however strong the intensity of light. This lowest frequency (threshold frequency) is different for different metals.
- As soon as the light is incident on the surface of the metal, the photoelectrons are emitted instantly, that is, there is no time-lag between incidence of light and emission of photoelectrons.

## CCSU Einstein’s Photon Theory and The Photoelectric Equation Study Material

**Einstein’s Photon Theory and the Photoelectric Equation :** Einstein, in 1905, explained the photoelectric effect by introducing the photon theory of light. He assumed that light travels through space in indivisible packets of energy called ‘photons’ or ‘quanta’, the energy of a single photon being hv, where h is ‘Planck’s constant’ and v is frequency of the light.

When light falls on a metal surface, a photon (energh hv) is completely absorbed by a single electron in the surface. A part of this energy is used in ejecting the electron against the attraction of the metal, and the rest is given to the ejected electron as kinetic energy. All the electrons are not ejected electron as kinetic energy. All the electrons are not ejected from the ‘surface’ of the metal. The electrons which are ejected from within the metal, they expend some of their acquired energy in collisions with the atoms on their way to the surface. Thus, **electrons with different energies are emitted from the metal.** The electrons emitted from the surface of the metal have maximum kinetic energy because their energy is not lost by collisions.

Let us consider those photoelectrons which are emitted from very near the surface and do not suffer any collision. They emit with maximum kinetic energy K_{max}. Let W be the energy required to eject an electron against the attraction of the rest of the metal. W is called the ‘work function’ of the given metal. Thus, we can write

hv = W + K_{max }………………(i)

or K_{max} = hv – W

where hv is the energy of the photon absorbed by the electron in the metal.

If the energy of the photon absorbed by the electron is less than the work-function W of the metal, then the electron will not be emitted. Therefore, if for the given metal, the threshold frequency of light be v_{0, }then the threshold energy of the photon will be hv_{0}. Thus

W = hv_{0}

Substituting this value of W in eq. (i), we get

K_{max} = hv – hv_{0}

Or K_{max }= h(v – v_{0 })

If the maximum velocity of the emitted photoelectrons be v_{max }, then K_{max }= ½ m v_{max}^{2}. Hence, the above equation may also be written as

½ m v_{max}^{2 }= h(v – v_{0 })

The equation is called ‘Einstein’s photoelectric equation’. The laws of photoelectric effect can be explained by this equation.

## CCSU Explanation of Photoelectric Laws Study Material

**Explanation of Photoelectric Laws :** If the intensity of light of a given frequency v is increased, then the number of photons striking the surface per second will increase in the same ratio, but the energy hv of each photon will remain the same. Hence the number of photoelectrons emitted per second will correspondingly increase, but their maximum kinetic energy per second will correspondingly increase, but their maximum kinetic energy K_{max} will remain the same, as is clear from Einstein’s equation. Thus, laws (i) and (ii) of the photoelectric effect are explained.

It is also seen from Einstein’s equation that the maximum kinetic energy K_{max }of the photoelectrons will increase almost linearly with increase in the frequency v lof the incident light. This is law (iii) of photoelectric effect.

According to Einstein’s equation, if v < v_{o }, then the kinetic energy of the photoelectrons would be negative which is impossible. This means that if the frequency v of the incident light is less than the threshold frequency v_{o}, photoelectrons cannot be emitted whatever be the intensity of light. This is law (iv).

Finally, as soon as the first photon falls on the metal it is immediately absorbed by some electron which escapes instantaneously. Thus, there is no time lag between incidence of light (photon) on the metal and emission of electron. This is law (v).

According to the above Einstein’s explanation, one photon of the incident light would emit one electron from the metal. This, however, does not mean that as many photons falling on the metal are involved in many other processes besides emission lf electrons. Hence the ratio of the number of electrons emitted to the number of photons incident on the surface quite less than 1. This ratio can be increased by special treatment of the surface.

## CCSU Millikan’s Experimental Verification Study Material

**Millikan’s Verification :** Einstein’s equation was tested by Millikan who measured the maximum energies of the photoelectrons emitted by a number of alkali metals over a wide range of light frequencies. His apparatus is shown in Fig.

Three different alkali metal surfaces are mounted on a drum D placed inside an evacuated chamber. The drum can be rotated from outside so that any surface can be scraped clean be a sharp knife K and then brought before a window W*. A beam of monochromatic light passing through the window falls on the surface C. The emitted photoelectrons are collected by an electrode A by means of a potential difference applied between C and A through a potential-divider. The galvanometer H measures this photoelectric current.

The electrode A is given a small negative potential with respect to C, and the photoelectric current is read on the galvanometer. As the negative potential is increased, the current falls. Ultimately, at a certain potential V_{0} at which the current is just reduced to zero is called the ‘stopping potential’.

Millikan plotted a graph between the stopping potential and the frequency of light over a wide range of frequencies, and obtained a straight line (fig.) Parallel lines were obtained for other metallic surfaces.

Now, according to Einstein’s equation, we have

½ mv_{max}^{2 }= h(v-v_{0}), …………….(i)

Where v_{o} is threshold frequency for a surface. We know that the stopping potential v_{o} is a measure of the maximum kinetic energy of the photoelectrons, that is,

eV_{0 }= ½ mv_{max}^{2 } ………………….(ii)

Eq. (i) and (ii) give

eV_{0 }= h(v-v_{0})

or V_{0} = (h/e)v – (h/e)v_{0 } ….(iii)

since h and e are constants and v_{0 }is constant for a given surface, eq. (iii)

indicates that the graph between the stopping potential V_{0 }and the frequency of light v must be a straight line. This is actually the case, as found by Millikan. Hence Einstein’s equation is of the correct form.

**Determination of Planck’s Constant (h) :** Eq. (iii) shows that the slope of the straight-line graph is h/e, and its intercept on the potential-axis is –h/e v_{0}. Hence by measuring the slope of the straight-line obtained and, using the known value of e, the value of h can be obtained.

**Determination of Threshold Frequency and Work Function :** Eq. (iii) shows that when V_{0 }= 0; v = v_{0}. That is, the intercept of the straight line on the v-axis gives the threshold frequency v_{0 }for the particular metal surface. This, when multiplied by h, gives the work function h v_{0}.

### CCSU A Graph between the Frequency of Light falling on a Metallic Surface and the Kinetic Energy of the Photoelectrons Emitted Study Material

The Einstein’s photoelectric equation is

K_{max }(=½ mv_{max}^{2}) = hv –hv_{0}

Where K_{max }is the maximum kinetic energy of the emitted photoelectrons, v is the frequency of the incident for the emitting surface.

If W be the work function of the emitting surface, then

W = hv_{0}

And so K_{max }= hv – W

A graph between K_{max }and v would be a straight line, cutting the v-axis at v_{0 }(Fig.). The slope of this line is h.

**Effect of Change in Light Intensity :**A change in light intensity means a change in the number of photons falling per second on the metal. This will cause a corresponding change in the number of photoelectrons emitted. Thus, if the light intensity is increased, the number of photoelectrons will increase, but their energy will remain the same. Hence the K_{max }– v graph will remain unchanged.**Effect of Change of Target Material :**The work function is the work that must be done to take an electron through the emitting surface from just beneath it against the attraction of the rest of the metal. Obviously, a change in metal would change the work function W (and hence the threshold frequency v_{0}) which will change the energy of the photoelectrons. The new graph will be another straight line parallel to the previous line; parallel because the slope h is fixed.

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