Question

what do you mean by dual nature of matter? show that de-Broglie's wave length is inversly proportional to velocity of tha particle

Answer 1

Answer :

As we know that matter has dual nature that is particle nature as well as wave nature.

Particle nature means that the light always travel in a straight line. Particle nature is observed through the photoelectric effect.

Wave nature is illustrated through experiment on the diffraction and interference.

According to the De-Broglie's wave, every fast moving matter particle is associated with a wavelength known as De-Broglie's wavelength. He cosidered both the nature that is, particle nature and wave nature.

As per particle nature,

$E=mc^2$     .........(1)

As per wave nature,

$E=h\nu$      ..........(2)

From equation 1 and 2, we get

$mc^2=h\nu\\\\mc^2=h\frac{c}{\lambda}\\\\\lambda=\frac{h}{mc}$

But non of the particle has got velocity equal to speed of light.

Therefore, v < c

The formula will be,

$\lambda=\frac{h}{mv}$

From this we conclude that the wavelength is inversely proportional to the velocity of the light.

Answer 2

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According to the concept of De Broglie's concept of matter shows their two properties according to the conditions :

As Wave ( in state of moving ).

As Praticle ( in state of rest ) .

⠀⠀⠀⠀⠀━ So , that's why Matter has dual nature .

⠀⠀⠀⠀▪︎ According to the De Broglie's equation suggests that the wave length of any object is inversely proportional to its momentum ( or product of mass & Velocity )

Therefore,

Representation in Mathematical Eq. :

$\qquad \dag\:\:\bigg\lgroup \sf{ \lambda \:= \dfrac{h}{mv}}\bigg\rgroup$

⠀⠀⠀⠀Here , h is the Plank's Quantity , mv is the momentum ( product of mass(m) and Velocity(v) is known as momentum ) & $\bf \lambda$ is the wavelength.

Now , We can clearly see that ,

⠀⠀⠀⠀▪︎ Wave length is inversely proportional to the velocity of the particle.

$\qquad \dag\:\:\bigg\lgroup \sf{ \lambda \propto \dfrac{1}{v}}\bigg\rgroup$

⠀⠀⠀⠀Here , $\bf \lambda$ is the wavelength and v is the velocity.

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

⠀⠀⠀⠀⠀More To Know about Dual nature of Matter :

It was proposed by De Broglie .

It was proposed in 1923

It was verified by Davisson & Gerber by the Diffraction.

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