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A nucleus of mass M emits $γ -$ rays photon of frequency ‘v’. The loss of internal energy by the nucleus is : [Take ‘c’ as the speed of electromagnetic wave]

Option 1 -

$h v [1 + \frac{h v}{2 M c^{2}}]$

Option 2 -

Option 3 -

$h v [1 - \frac{h v}{2 M c^{2}}]$

Option 4 -

2 Views | Posted 2 months ago

Asked by Shiksha User

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Answered by

alok kumar singh | Contributor-Level 10

2 months ago

Correct Option - 1

Detailed Solution:

Using conservation linear momentum, we can write

$M v = \frac{h}{λ} = \frac{h ν}{c} \Rightarrow v = \frac{h ν}{M c}$

Loss of internal energy = Gain in Kinetic energy + energy of gamma ray

= $\frac{1}{2} M v^{2} + h ν = \frac{1}{2} M {(\frac{h ν}{M c})}^{2} + h ν = h ν [1 + \frac{h ν}{2 M c^{2}}]$

Using conservation linear momentum, we can write <math> <mrow> <mi>M</mi> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mi>h</mi> </mrow> <mrow> <mi>λ</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <mi>c</mi> </mrow> </mfrac> <mo>⇒</mo> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <mi>M</mi> <mi>c</mi> </mrow> </mfrac> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1816090878"> </o:OLEObject></xml><! [endif]->                                            Loss of internal energy = Gain in Kinetic energy + energy of gamma ray= <math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mi>M</mi> <msup> <mrow> <mi>v</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>h</mi> <mi>ν</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mi>M</mi> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <mi>M</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>h</mi> <mi>ν</mi> <mo>=</mo> <mi>h</mi> <mi>ν</mi> <mrow> <mo> [</mo> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <mn>2</mn> <mi>M</mi> <msup> <mrow> <mi>c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mrow> </math>  <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1816090879"> </o:OLEObject></xml><! [endif]->

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A nucleus of mass M emits $γ -$ rays photon of frequency ‘v’. The loss of internal energy by the nucleus is : [Take ‘c’ as the speed of electromagnetic wave]

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A nucleus of mass M emits $γ -$ rays photon of frequency ‘v’. The loss of internal energy by the nucleus is : [Take ‘c’ as the speed of electromagnetic wave]