A moving proton and electron have the same de-Broglie wavelength. If K and P denote the K.E. and momentum respectively. Then choose the correct option:

Option 1 -

$K_{p} = K_{e} a n d P_{p} = P_{e}$

Option 2 -

$K_{p} < K_{e} a n d P_{p} < P_{e}$

Option 3 -

$K_{p} < K_{e} a n d P_{p} = P_{e}$

Option 4 -

$K_{p} > K_{e} a n d P_{p} = P_{e}$

3 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

Raj Pandey | Contributor-Level 9

2 months ago

Correct Option - 3

Detailed Solution:

Proton electron

Charge : +e -e

Wavelength : $λ$ $λ$

KE : K_P &

...more

Proton electron

Charge : +e -e

Wavelength : $λ$ $λ$

KE : K_P K_e

Momentum : P_P P_e

As we know KE = $\frac{P^{2}}{2 m}$

Also, de-broglie wavelength, $?$ is given by

$λ = \frac{h}{ρ} o r λ = \frac{h}{\sqrt[]{2 m K}}$

Again $λ = \frac{h}{P}$

$?$ P → same for both

$∴ P_{P} = P_{e}$

less

Proton                electronCharge :              +e                        -eWavelength :     <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1815921221"> </o:OLEObject></xml><! [endif]->        <math> <mrow> <mi>λ</mi> </mrow> </math>                 <math> <mrow> <mi>λ</mi> </mrow> </math> <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1815921222"> </o:OLEObject></xml><! [endif]->KE :                      KP                         KeMomentum :     PP                         PeAs we know KE = <math> <mrow> <mfrac> <mrow> <msup> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> </math> <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1815921223"> </o:OLEObject></xml><! [endif]->Also, de-broglie wavelength, <math> <mrow> <mi>? </mi> </mrow> </math> <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1815921281"> </o:OLEObject></xml><! [endif]->is given by <math> <mrow> <mi>λ</mi> <mo>=</mo> <mfrac> <mrow> <mi>h</mi> </mrow> <mrow> <mi>ρ</mi> </mrow> </mfrac> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>o</mi> <mi>r</mi> <mtext> </mtext> <mtext> </mtext> <mi>λ</mi> <mo>=</mo> <mfrac> <mrow> <mi>h</mi> </mrow> <mrow> <mroot> <mrow> <mn>2</mn> <mi>m</mi> <mi>K</mi> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> </mrow> </math> Again <math> <mrow> <mi>λ</mi> <mo>=</mo> <mfrac> <mrow> <mi>h</mi> </mrow> <mrow> <mi>P</mi> </mrow> </mfrac> </mrow> </math> <math> <mrow> <mo>? </mo> </mrow> </math>  P →  same for both <math> <mrow> <mo>∴</mo> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mi>P</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> </mrow> </math>

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