Calculate the wavelength of an electron in a 10 MeV particle accelerator (1 MeV = 10⁶ eV).
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Given conditions ⇒
Energy of the Electron = 10 MeV
= 10 × 10⁶ eV.
= 10⁷ × 1.6 × 10⁻¹⁹ J.
= 16 × 10⁻¹³ J.
Now, We know the Mass of the Electron = 9.1 × 10⁻³¹ kg.
∵ E = 1/2 × mv²
where, E is the Energy, m is the mass of the electrons, and v is the velocity of the Electrons.
∴ 16 × 10⁻¹³ = 1/2 × 9.1 × 10⁻³¹ × v²
∴ v² = (32 × 10³¹ × 10⁻¹³)÷9.1
⇒ v² = 3.51 × 10¹⁸
⇒ v = √(3.51 × 10¹⁸)
⇒ v = 1.86 × 10⁹ m/s.
Momentum(p) = m × v
= 9.1 × 10⁻³¹ × 1.86 × 10⁹
= 16.926 × 10⁻²²
Now, Using the Formula,
λ = h/p
where h is the plank's constant = 6.63 × 10⁻³⁴ J-s.
⇒ λ = (6.63 × 10⁻³⁴) ÷ (16.926 × 10⁻²²)
⇒ λ = 0.39 × 10⁻¹² m.
∴ λ = 0.39 pm. [∵ 1 pm = 10⁻¹².]
Hence, the de-broglie wavelength of electron is 0.39 pm.
Hope it helps.
Energy of the Electron = 10 MeV
= 10 × 10⁶ eV.
= 10⁷ × 1.6 × 10⁻¹⁹ J.
= 16 × 10⁻¹³ J.
Now, We know the Mass of the Electron = 9.1 × 10⁻³¹ kg.
∵ E = 1/2 × mv²
where, E is the Energy, m is the mass of the electrons, and v is the velocity of the Electrons.
∴ 16 × 10⁻¹³ = 1/2 × 9.1 × 10⁻³¹ × v²
∴ v² = (32 × 10³¹ × 10⁻¹³)÷9.1
⇒ v² = 3.51 × 10¹⁸
⇒ v = √(3.51 × 10¹⁸)
⇒ v = 1.86 × 10⁹ m/s.
Momentum(p) = m × v
= 9.1 × 10⁻³¹ × 1.86 × 10⁹
= 16.926 × 10⁻²²
Now, Using the Formula,
λ = h/p
where h is the plank's constant = 6.63 × 10⁻³⁴ J-s.
⇒ λ = (6.63 × 10⁻³⁴) ÷ (16.926 × 10⁻²²)
⇒ λ = 0.39 × 10⁻¹² m.
∴ λ = 0.39 pm. [∵ 1 pm = 10⁻¹².]
Hence, the de-broglie wavelength of electron is 0.39 pm.
Hope it helps.
abhi178:
Bhiaya check first step , 10 MeV = 1.6 × 10^-12 J
Abhi, I have written, 16 x 10^-13 not 1.6.
But can you see how velocity of electron > velocity of light . It is violet Enstien theory, then how you can use de broglie concepts because debroglie is the application of quantum and photoelectric concept.
Speed of light = 3 × 10^8 but here you get speed of electron = 1.86 × 10^9 = 18.6× 10^8 > 3 × 10^8
Am I wrong ?
As we know that λ = h/mc or E = hc/λ is valid for only photons. When we talk about the Particles (Like Electrons), we have to take there speed only instead of the Speed of the Light. It was said by the Famous Chemist De-Broglie.
but how photons speed greater than speed of light. Is it possible ?
No, It can't be possible in the Reality. But as the question prescribed , Here whatever the value comes. We have to take it.
Ok , but it's really confusable . Thanks for helping
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Since rest mass energy of electron is 0.51 MeV so it has to be a reltivistic motion and we can't use the formulae P^2 /2m = E E^2 = p^2*c^2 + mc^2 will pe used to calculate p
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