What is de Broglie wavelength associated with electron moving under a potential difference of 10000v?

1. Dual Nature of Radiation (Wave-Particle Duality):

Electromagnetic radiation is an emission with a dual nature, i.e., it has both wave and particle aspects. In particular, the energy conveyed by an electromagnetic wave is always carried in packets whose magnitude is proportional to the frequency of the wave. These packets of energy are called photons. The energy of a photon is E=hν, where h is Planck's constant and ν is the frequency of the wave.

Few properties of light like the photoelectric effect could not be explained using wave theory of light (electromagnetic waves) and are rather explained by particle nature, so to explain behaviour of electromagnetic waves in different cases we assume different nature of it.

Sometimes, it behaves like a particle (called a photon) which explains how light travels in straight lines.

Sometimes, it behaves like a wave which explains how light bends (or diffracts) around an object.

This is called the dual nature of light (electromagnetic waves).

2. Photon Theory:

According to Planck's quantum theory, light consists of photons as energy packets having the following properties:

(i) Each photon is of energy E=hν=hcλ, where h is Planck's constant with value, h=6.63×10-34 J s=4.14×10-15 eV s, c is the speed of light in vacuum, c=3×108 m s-1 and λ is the wavelength of the wave.

(ii) All photons travel in a straight line with the speed of light in vacuum.

(iii) Photons are electrically neutral.

(iv) Photons have zero rest mass.

(v) Photons are not deflected by electric and magnetic fields.

(vi) The equivalent mass of a photon while moving is given by m=Ec2=hνc2=hcc2λ=hcλ.

(vii) Momentum of the photon, p=Ec=hνc=hλ.

(viii) Number of photons of the wavelength λ emitted in t second from a lamp of power P is n=Ptλhc.

3. Radiation Pressure P:

The electromagnetic wave transports not only energy but also momentum and hence can exert radiation pressure on a surface due to the absorption and reflection of the momentum.

Consider I be the intensity of the radiation, A be the projected area of the body, F be the radiation force and E be the energy of a photon.

Case I: Perfect absorption (black body):

If photons are being absorbed perfectly, the final momentum of photons is zero. The force exerted, works in the direction of incident beams,

F=Ec=IAc

Radiation pressure, P=Ic

Case II: Perfect reflection:

F=2IAc,

Radiation pressure, P=2Ic

Case III: Partial reflection:

Let a fraction of photons reflected, i.e., reflection coefficient be ρ. Then the fraction of photons absorbed is (1-ρ).

Force, F=2ρIAc+1-ρIAc

Radiation pressure, P=2ρIc+1-ρIc

Case IV: Suppose, beams are incident at an angle θ. Let the surface area be A. The point to remember is that surface area A and beam cross-sectional area Acosθ are two different terms.

Considering to be perfectly reflecting,

Force, F=2IAcosθccosθ

Radiation pressure, P=FA=2Iccos2θ

4. Photoelectric Effect:

The photoelectric effect is a process where electrons are ejected from a surface by the action of light (electromagnetic radiation). The process was discovered by Heinrich Hertz in 1887 . Attempts to explain the effect by classical electromagnetic theory but failed. In 1905, Albert Einstein presented an explanation based on the quantum concept of Max Planck.

(i) Observation of the experiments on the photo-electric effect:

(a) The emission of photoelectrons is instantaneous.

(b) The number of photoelectrons emitted per second is proportional to the intensity of the incident light.

(c) The maximum velocity with which electrons emerge is dependent only on the frequency and not on the intensity of the incident light.

(d) There is always a lower limit of frequency called threshold frequency below which no emission takes place, however high the intensity of the incident radiation may be.

(ii) Explanation: Einstein suggested that when a light beam is an incident on a metal surface, the free electrons of the metal absorb the entire energy of an incident photon during its collision with it. If this electron gets sufficient energy in this manner to do work against the surface adhesion of the given metallic surface and escape, then it leaves the metal and a photoelectron is found.

(iii) Conclusion:

(a) For an electron to escape from a metallic surface by doing work against its attractive force and get out of the force field of the metallic surface, a minimum amount of energy is required to supplied to electrons. This minimum energy required for an electron to escape from a metallic surface is called the work function of the given metal which is characteristic of the material and hence, different for different metals. Work function of a given metal is generally represented by the symbol ϕ.

(b) Einstein's photoelectric equation:

Photon energy =KEmax of electron + work function

⇒hν=KEmax+ϕ

ϕ= Work function = energy needed by the electron in freeing itself from the atoms of the metal ϕ=hν0.

(c) The minimum frequency of light corresponding to which the energy of a photon is equal to the work function of a given metal is called threshold frequency of that metal and the corresponding wavelength is called threshold wavelength,

hν0=ϕ or ν0=ϕh,

where vo is called threshold frequency.

So, hcλ0=ϕ or λ0=hcϕ, where λ0 is called threshold wavelength.

(d) Clearly, when a light beam of frequency less than ν0 or wavelength greater than λ0 is incident, then no photoelectrons can be emitted, no matter how high is the intensity of the incident beam.

(e) Suppose a photon transfers energy more than the work function of the given metal, then the photoelectron may be ejected with kinetic energy Kmax=hν-ϕ or less than that, because a part or all of the extra energy may be lost during several collisions that the electron makes before emission.

(f) Other forms of the equation:

Kmax=hν-ν0

Kmax=hc1λ0-1λ0

(g) Stopping potential: If the polarity of the battery is reversed and the applied potential is gradually increased, then the photo-current starts decreasing. This is because the electrons are retarded and most of the electrons are unable to reach the opposite electrode. It is observed that when the applied retarding potential is increased, the photocurrent eventually becomes zero. This potential is known as the stopping potential and depends only on the material of the photocathode and the frequency of light.

If Vs be the stopping potential, then, eVs=hν-ϕ

The stopping potential Vs depends only on the metal and does not depend on the intensity of incident light.

(h) A graph between the intensity of light and photoelectric current is found to be a straight line as shown in the figure. Photoelectric current is directly proportional to the intensity of incident radiation. In this experiment, the frequency and retarding potential are kept constant.

(i) A graph between photoelectric current and the potential difference between cathode and anode is found as shown in the figure.

(j) Graphs between maximum kinetic energy of electrons ejected from different metals and frequency of light used are found to be straight lines of same slope as shown in the figure.

Here m1, m2 and m3 are for three different metals.

5. Matter Waves (de-Broglie Idea):

As wave behaves like material particles, similarly matter also behaves like waves. According to de-Broglie, a wavelength of the matter-wave associated with a particle is given by λ=hp=hmv, where p=mv is the momentum, m is the mass and v is the velocity of the particle.

In other words, beams of electrons and other forms of matter exhibit wave properties including interference and diffraction with a de-Broglie's wavelength given by λ=hp (wavelength of a particle).

If a particle of mass m moving with velocity v, then

(i) kinetic energy of the particle, K=12mv2=p22m.

(ii) momentum of the particle, p=mv=2mK.

(iii) the de-Broglie's wavelength associated with the particles is λ=hp=hmv=h2mK.

(iv) de-Broglie's wavelength associated with the charged particles:

(a) For an electron, λe=12.27×10-10V m=12.27V A∘. So, λ∝1V.

(b) For proton, λp=0.286×10-10V m=0.286V A∘

(d) For α-particles, λα=0.101V A∘

Note: If an electron mass me is accelerated through a potential difference of V volt, then 12mev2=eV or V=2eVme,

∴ λ=hmev=h2eVme