Sunday, March 23, 2008

Concept Review Ch.42 Photoelectric Effect and Wave-Particle Duality

There are some phenomena which can only be understood in terms of the particle theory of light.

Photoelectric effect can be understood only in terms of the particle nature of light.

Photons

The particles of light (according to the particle nature of light theory) are called photons.

Some important properties of photons are:

1. A photon always travels at a speed c = 299,792,458 m/s which is approximately equal to 3.0*10^8 m/s in vacuum.

2. We can say rest mass of photon is equal to zero.

3. Each photon has a definite energy and linear momentum.

4. Relation between properties of photon and properties of light waves.
E and p are energy and linear momentum of a photon of light.
ν and λ are the frequency and wavelength of the same light when it is considered (behaves) as a wave.

Then E = hν = hc/λ
p = h/λ = E/c ... (42.1)

wherein h is a universal constant known as the Planck constant and has a value 6.626*10^-34 J-s and is also equal to 4.136*10^-15 eV-s.

Therefore all photons of light of a particular wavelength λ have the same energy and the same momentum.

5. When a photon collides with a material particle, it may get absorbed and new photons may be emitted. Hence number of photons may decrease or increase due to collisions. But the total energy and the total momentum of colliding particles are conserved.

6. If the intensity of light of a particular wavelength is increased, there is an increase in the number of photons crossing a finite area in a given time. But the energy of each photons remains the same. The intensity is generated by increase in the number of photons.

(Note: The intensity of a wave is proportional to the square of the resultant electric field magnitude at that point. Note the equation of plane light wave travelling along the x direction (polarized-electric field may be along the y direction or z direction or in any other direction in the y-z plane) is written as

E = E0 Sin ω(t-x/v)

where E0 is the magnitude of the electric field at point x at time t. ω is the angular frequency and v is the speed of the wave.)

Photoelectric effect

When light of sufficiently small wavelength is incident on a metal surface, electrons ejected from the metal.

The electrons ejected from the metal are called photoelectrons.

A minimum energy, equal to the work function φ, must be given to an electron so as to bring it out of the metal.

Kinetic energy of photoelectrons

When light is incident on metal surfaces, photons collide with free electrons of the metal and may give all of their energy. When that energy is higher than φ, electrons try to come out of the metal but may lose some the energy in collisions with other particles in the metal. So the kinetic energy of the photoelectron may be anything between zero and (E-φ) where E is the energy supplied to the individual electrons

The maximum kinetic of the electron that comes out is

K max = hc/λ - φ = hv - φ

The above equation is called Einstein's photoelectric equation.

Threshold wavelength

If the incident wavelength is equal to λ0 = hc/φ the maximum kinetic energy is zero.

Photoelectric effect takes place only if λ≤λ0.

The wavelength λ0 is called the threshold wavelength for the metal. The corresponding frequency is called the threshold frequency (υ0).

Writing work function φ as hυ0 (h multiplied by frequency)

Kmax = h(υ - υ0)

Systematic Study of Photoelectric Effect – Experimental Arrangement

A cathode and anode in a sealed vacuum chamber with an ammeter, battery, rheostat and commutator connected in the external circuit is the experimental set up.

Saturation current: As the positive potential of the anode is increased, the current in the circuit keeps increasing and after certain potential remains constant. This constant current is called saturation current. At this stage all electrons that escaped from the cathode reach the anode.

Stopping potential: If the potential of the anode is made negative with reference to the cathode, the photoelectrons coming from the cathode are repelled and at certain value of negative potential the current completely stops. The smallest magnitude of the anode potential which just stops the photocurrent, is called the stopping potential.

Relation between Maximum kinetic energy of photoelectrons and stopping potential:

As a photoelectron travels from the cathode to the anode, the potential energy increases by eV0. This is equal to the decrease in the kinetic energy of the photoelectron.

The maximum kinetic energy a photoelectron will have is hc/λ - φ
Hence eV0. = hc/λ – φ

V0 = hc/e(1/ λ) – φ/e

Therefore the stopping potential V0 depends on th wavelength of the light falling on the cathode and the work function of the cathode metal. It does not depend on the intensity of light.

If an anode potential of -2.0 V stops the photocurrent from a metal when a 1 W source of light is used, the same potential of -2.0 V will stop the photocurrent when a 100 W source of light of the same wavelength is used.

But the saturation current increases as the intensity of light increases. This is because a larger number of photons now fall on the metal surface and hence a large number of photoelectrons are emitted. This will increase the saturation current.

The facts of photoelectric effect which wave theory is not able to explain

According to wave theory, it may take some time for the photoelectron to come out when low intensity light is used. But when the incident wavelength is less than the threshold wavelength, photoelectrons are coming out instantly.

According to wave theory, by using sufficiently intense light of any wavelength, an electron may be given the required amount of energy to come out. But experiments show the existence of threshold wavelength.

Matter Waves
Wavelength of electron

A large number of experiments are now available in which electrons interfere like waves and produce fringes. Electron microscope is built on the wave properties of electrons.

A relation for wavelength of electron was proposed by Louis Victor de Broglie.

The proposed expression for wavelength is

λ = h/p

Where p is the momentum of the electron and
h is the Planck constant.

This wavelength is called de Broglie wavelength.

This equation is applicable to a neutron or proton. It is applicable to light also.
When light shows its photon character, each photon has a momentum of p = h/λ.

We cannot use Newton’s laws to explain the movement of electron because classical mechanics fail for particles of very small size. For particle of linear size greater than 10^-4 cm, classical mechanic work.

For smaller particles, we should use quantum mechanics.

Work functions of some photosensitive metals

Cesium ...... 1.9 eV
Potassium.... 2.2 eV
Lithium...... 2.5 eV