From Maxwell's theory we learnt that light is an electromagnetic wave consisting of coupled electric and magnetic oscillations that move with the speed of light and exhibit typical wave behaviour. Let us try to explain the experimental observations of photoelectric effect using wave picture of light.
(i) When light is incident on the target, there is a continuous supply of energy to the electrons in the metal surface.
(ii) According to wave theory, light of greater intensity should impart greater kinetic energy to the liberated electrons (Here, intensity of light is the energy delivered per unit area per unit time). But this does not happen. The experiments show that maximum kinetic energy of the photoelectrons does not depend on the intensity of the incident light.
(iii) According to wave theory, if a sufficiently intense beam of light is incident on the surface, electrons will be liberated from the surface of the target, however low the frequency of the radiation is. From the experiments, we know that photoelectric emission is not possible below a certain minimum frequency. Therefore, the wave theory fails to explain the existence of threshold frequency.
(iv) Since the energy of light is spread across the wavefront, the electrons which receive energy from it are large in number. Each electron needs considerable amount time (a few hours) to get energy sufficient to overcome the work function and to get liberated from the surface. But experiments show that photoelectric emission is almost instantaneous process (the time lag is less than 10^-9 s after the surface is illuminated) which could not be explained by wave theory.
Thus, the experimental observations of photoelectric emission could not be explained on the basis of the wave theory of light.

Principle:
(i) The wave nature of the electron is used in the construction of microscope called electron microscope.
(ii)The resolving power of a microscope is inversely proportional to the wavelength of the radiation used for illuminating the object under study.
(iii) Higher magnification as well as higher resolving power can be obtained by employing the waves of shorter wavelengths.
(iv) De Broglie's wavelength of electron is very much less than (a few thousand less) that of the visible light being used in optical microscopes.
(v) As a result, the microscopes employing de Broglie waves of electrons have very much higher resolving power than optical microscope.
(vi)  Electron microscopes giving magnification more than 2,00,000 times are common in research laboratories.

Working:
(i) The construction and working of an electron microscope is similar to that of an optical microscope except that in electron microscope focussing of electron beam is done by the electrostatic or magnetic lenses.
(ii) The electron beam passing across a suitably arranged either electric or magnetic fields undergoes divergence or convergence thereby focussing of the beam is done.
(iii) The electrons emitted from the source are accelerated by high potentials.
(iv) The beam is made parallel by magnetic condenser lens; When the beam passes through the sample whose magnified image is needed, the beam carries the image of the sample.
(v) With the help of magnetic objective lens and magnetic projector lens system, the magnified image is obtained on the screen. These electron microscopes are being used in almost all brands of science.

Construction:
(i) It consists of an evacuated glass or quartz bulb in which two metallic electrodes - that is, a cathode and an anode are fixed.
(ii) The cathode C is semi-cylindrical in shape and is coated with a photo sensitive material.
(iii) The anode A is a thin rod or wire kept along the axis of the semi-cylindrical cathode.
(iv) A potential difference is applied between the anode and the cathode through a galvanometer G.

Working:
When cathode is illuminated, electrons are emitted from it. These electrons are attracted by anode and hence a current is produced which is measured by the galvanometer. For a given cathode, the magnitude of the current depends on
(i) The intensity to incident radiation and
(ii) The potential difference between anode and cathode.

(i) When a photon of energy hv is incident on a metal surface, it is completely absorbed by a single electron and the electron is ejected.
(ii) In this process, a part of the photon energy is used for the ejection of the electrons from the metal surface (photoelectric work function Φ₀) and the remaining energy as the kinetic energy of the ejected electron. From the law of conservation of energy,
\(\\ \\ \\ hv=\phi { _{ 0 }+\cfrac { 1 }{ 2 } { mv }^{ 2 } }\)   ......(1) 
(iii) where m is the mass of the electron and v its velocity.

(iv) If  we reduce the frequency of the incident light is reduced, the speed or kinetic energy of photo electrons is also reduced. At some frequency v₀ of incident radiation, the photo electrons are ejected with almost zero kinetic energy.
Then the equation becomes.
\({ hv }_{ 0 }=\phi _{ 0 }\)    ......(2) 
(v) Where v₀ is the threshold frequency. B rewriting the equation, we get 
\(hv={ hv }_{ o }+\cfrac { 1 }{ 2 } { { mv }^{ 2 } }\)   ......(3) 
The equation is known as einstein's photoelectric equation.
(vi) If the electron does not lose energy by internal collisions, then it is emitted with maximum kinetic energy K_max. Then
\({ K }_{ max }=\cfrac { 1 }{ 2 } { mv }^{ 2 }_{ max }\)    ......(4) 
(vii) where v_maxis the maximum velocity of max the electron ejected. The equation (1) is rearranged as follows:
\({ K }_{ max }=hv-{ \phi }_{ 0 }\) 

A graph between maximum kinetic energy K_max of the photoelectron and frequency v of the incident light is a straight line.

i) According to wave theory, the energy in a light wave is spread out uniformly and continuously over the wavefront.
The energy absorbed by each electron in time t is given by
E = IAt
With this energy absorbed, the most energetic electron is released with K_max by overcoming the surface energy barrier or work function ϕ₀ and this is expressed as
K_max = IAt - ϕ₀    (1)
Thus, wave theory predicts that for a unit time, at low light intensities when IA < ϕ₀ no electrons are emitted. At higher intensities, when IA ≥ ϕ₀, electrons are emitted. This implies that the higher the light intensity, the greater will be K_max.
K_max is dependent only on the intensity under given conditions - that is, by suitably increasing the intensity, one can produce a photoelectric effect even if the frequency is less than the threshold frequency. So the concept of threshold frequency does not even exist in wave theory.
ii) According to wave theory, the intensity of a light wave is proportional to the square of the amplitude of the electric field \(({ E }_{ 0 }^{ 2 })\). The amplitude of this electric field increases with increasing intensity and imparts an increasing acceleration and kinetic energy to an electron.
Now I is replaced with a quantity proportional to \(({ E }_{ 0 }^{ 2 })\) in equation (1). This means that K_max should not depend at all on the frequency of the classical light wave which again contradicts the experimental results.
(iii) If an electron accumulates light energy just enough to overcome the work function, then it is ejected out of the atom with zero kinetic energy. Therefore, from equation (1),
0 = IAt - ϕ₀
t = \(\frac { { \phi }_{ 0 } }{ IA } =\frac { \phi _{ 0 } }{ I(\pi r^{ 2 }) } \)
By taking the atomic radius r = 1.0 x 10^-10 m and substituting the given values of I and ϕ₀, we can estimate the time interval as
t = \(\frac { 2.14\times 1.6\times 10^{ -19 } }{ 1.60\times 10^{ -6 }\times 3.14\times (1\times 10^{ -10 })^{ 2 } } \)
= 0.68 x 10⁷ s ≈ 79 days.
Thus, wave theory predicts that there is a large time gap between the incidence of light and the ejection of photoelectrons but the experiments show that photoemission is an instantaneous process.