Question

if two particles of equal masses have kinetic energy in the ratio 1:4 the ratio of their de broglie wavelength will be


dont copy paste the ans please
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Answer 1

Answer:

so answer should be 2:1 I hope my handwriting is legible

Answer 2

Given :

• Ratio of Kinetic Energies of two particles having same mass is 1 : 4

To Find :

• Ratio of their De-Broglie wavelength

Solution :

The relation between Kinetic energy and de-broglie wavelength is given by ,

${\boxed {\rm{ \lambda = \frac{h}{ \sqrt{2mKE} } }}}$

Where ,

• λ is wavelength
• h is planck's constant
• m is mass
• KE is Kinetic Energy

We are given that the particle's masses are equal so let the masses be 'm' and 'm'

Let ,

• Kinetic energy of the first particle be (KE)₁

• Kinetic energy of second particle be (KE)₂

• Debriglie wavelength of first particle be λ₁

• De-broglie wavelength of second particke be λ₂

We have ,

• (KE)₁ : (KE)₂ = 1 : 4

Let

• (KE)₁ = x , (KE)₂ = 4x

Now ,

$: \implies \sf \: \dfrac{ \lambda_1}{ \lambda_2} = \dfrac{ \frac{h}{ \sqrt{2m(KE)_1} } }{ \frac{h}{ \sqrt{2m(KE)_2} }} \\ \\ : \implies \sf \: \frac{\lambda_1}{\lambda_2} = \dfrac{ \frac{1}{ \sqrt{2m(x)} } }{ \frac{1}{ \sqrt{2m(4x)} } } \\ \\ : \implies \sf \: \dfrac{ \lambda_1}{\lambda_2} = \frac{ \sqrt{2m(4x)} }{ \sqrt{2m(x)} } \\ \\ : \implies \sf \: \frac{\lambda_1}{ \lambda_2} = \frac{ \sqrt{4x} }{ \sqrt{x} } \\ \\ : \implies \sf \: \frac{ \lambda_1}{ \lambda_2} = \frac{2 \sqrt{x} }{ \sqrt{x} } \\ \\ : \implies \sf \: \frac{ \lambda_1}{ \lambda_2} = \frac{2}{1} \\ \\ : \implies \sf {\boxed {\sf{ \blue {\lambda_1 : \lambda_2 = 2 : 1}}}}$

∴ The ratio of De-broglie wavelength of the given particles is 2 : 1