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CBSE-XI-Physics

47: The Special Theory of Relativity

with Solutions - page 5
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  • #26
    Find the speed of an electron with kinetic energy (a) 1 eV, (b) 10 keV and (c) 10 MeV.
    Ans : If m0 is the rest mass of an electron and c is the speed of light, then kinetic energy of the electron = mc2 - m0c2 ...(1)
    If `` m=\frac{{m}_{0}{c}^{2}}{\sqrt{1-{v}^{2}/{c}^{2}}}``, then (a) Kinetic energy of electron = 1 eV = `` 1.6\times {10}^{-19}\,\mathrm{\,J\,}``
    From eq. (1), we get
    `` 1.6\times {10}^{-19}=\frac{{m}_{0}{c}^{2}}{\sqrt{1-{v}^{2}/{c}^{2}}}-{m}_{0}{c}^{2}``
    `` \Rightarrow \frac{1.6\times {10}^{-19}}{{m}_{0}{c}^{2}}=\left(\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1\right)``
    `` \Rightarrow \frac{1}{\sqrt{1-v/{c}^{2}}}-1=\frac{1.6\times {10}^{-19}}{9.1\times {10}^{-31}\times 9\times {10}^{16}}``
    `` \Rightarrow \frac{1}{\sqrt{1-v/{c}^{2}}}-1=0.019536\times {10}^{-4}``
    `` \Rightarrow \frac{1}{\sqrt{1-v/{c}^{2}}}=1+0.019536\times {10}^{-4}``
    `` \Rightarrow \frac{1}{\sqrt{1-v/{c}^{2}}}=\frac{1}{1.0000019536}``
    `` \Rightarrow 1-{v}^{2}/{c}^{2}=0.99999613``
    `` \Rightarrow {v}^{2}/{c}^{2}=0.00000387``
    `` \Rightarrow v/c=0.001967231=3\times 0.001967231\times {10}^{8}``
    `` =5.92\times {10}^{5}\,\mathrm{\,m\,}/\,\mathrm{\,s\,}`` (b) Kinetic energy of electron = 10 keV`` =1.6\times {10}^{-19}\times 10\times {10}^{3}\,\mathrm{\,J\,}``
    `` {m}_{0}{c}^{2}\left(\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1\right)=1.6\times {10}^{-15}``
    `` \Rightarrow 9.1\times {10}^{-31}\times 9\times {10}^{16}\left(\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1\right)=1.6\times {10}^{-15}``
    `` \Rightarrow \frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1=\frac{1.6\times {10}^{-15}}{9.1\times 9\times {10}^{-15}}``
    `` \Rightarrow \frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1=\frac{1.6}{9.1\times 9}``
    `` \Rightarrow \frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}=0.980838``
    `` \Rightarrow 1-{v}^{2}/{c}^{2}=0.962043182``
    `` \Rightarrow {v}^{2}/{c}^{2}=1-0.962043182``
    `` \Rightarrow {v}^{2}=0.341611359\times {10}^{18}``
    `` \Rightarrow v=0.584475285\times {10}^{8}``
    `` \Rightarrow v=5.85\times {10}^{7}\,\mathrm{\,m\,}/\,\mathrm{\,s\,}``
    `` `` (c) Kinetic energy of electron`` =10\,\mathrm{\,MeV\,}={10}^{7}\times 1.6\times {10}^{-19}\,\mathrm{\,J\,}``
    `` \Rightarrow \frac{{m}_{0}{c}^{2}}{2\sqrt{1-{v}^{2}-{c}^{2}}}-{m}_{0}{c}^{2}=1.6\times {10}^{-12}``
    `` \Rightarrow {m}_{0}{c}^{2}\left(\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1\right)=1.6\times {10}^{-12}``
    `` \Rightarrow \frac{1}{\sqrt{1-{v}^{2}\mathit{/}{c}^{\mathit{2}}}}-1=\frac{1.6\times {10}^{-12}}{9.1\times 9\times {10}^{-31}\times {10}^{16}}``
    `` \Rightarrow \frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1=1.019536\times {10}^{3}``
    `` \Rightarrow \frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1=1019.536+1``
    `` \Rightarrow \sqrt{1-{v}^{2}/{c}^{2}}=0.000979877``
    `` \Rightarrow 1-{v}^{2}/{c}^{2}=0.99\times {10}^{-6}``
    `` \Rightarrow v=2.999999039\times {10}^{8}\,\mathrm{\,m\,}/\,\mathrm{\,s\,}``
    Page No 458:
  • #26-a
    1 eV,
    Ans : Kinetic energy of electron = 1 eV = `` 1.6\times {10}^{-19}\,\mathrm{\,J\,}``
    From eq. (1), we get
    `` 1.6\times {10}^{-19}=\frac{{m}_{0}{c}^{2}}{\sqrt{1-{v}^{2}/{c}^{2}}}-{m}_{0}{c}^{2}``
    `` \Rightarrow \frac{1.6\times {10}^{-19}}{{m}_{0}{c}^{2}}=\left(\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1\right)``
    `` \Rightarrow \frac{1}{\sqrt{1-v/{c}^{2}}}-1=\frac{1.6\times {10}^{-19}}{9.1\times {10}^{-31}\times 9\times {10}^{16}}``
    `` \Rightarrow \frac{1}{\sqrt{1-v/{c}^{2}}}-1=0.019536\times {10}^{-4}``
    `` \Rightarrow \frac{1}{\sqrt{1-v/{c}^{2}}}=1+0.019536\times {10}^{-4}``
    `` \Rightarrow \frac{1}{\sqrt{1-v/{c}^{2}}}=\frac{1}{1.0000019536}``
    `` \Rightarrow 1-{v}^{2}/{c}^{2}=0.99999613``
    `` \Rightarrow {v}^{2}/{c}^{2}=0.00000387``
    `` \Rightarrow v/c=0.001967231=3\times 0.001967231\times {10}^{8}``
    `` =5.92\times {10}^{5}\,\mathrm{\,m\,}/\,\mathrm{\,s\,}``
  • #26-b
    10 keV and
    Ans : Kinetic energy of electron = 10 keV`` =1.6\times {10}^{-19}\times 10\times {10}^{3}\,\mathrm{\,J\,}``
    `` {m}_{0}{c}^{2}\left(\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1\right)=1.6\times {10}^{-15}``
    `` \Rightarrow 9.1\times {10}^{-31}\times 9\times {10}^{16}\left(\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1\right)=1.6\times {10}^{-15}``
    `` \Rightarrow \frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1=\frac{1.6\times {10}^{-15}}{9.1\times 9\times {10}^{-15}}``
    `` \Rightarrow \frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1=\frac{1.6}{9.1\times 9}``
    `` \Rightarrow \frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}=0.980838``
    `` \Rightarrow 1-{v}^{2}/{c}^{2}=0.962043182``
    `` \Rightarrow {v}^{2}/{c}^{2}=1-0.962043182``
    `` \Rightarrow {v}^{2}=0.341611359\times {10}^{18}``
    `` \Rightarrow v=0.584475285\times {10}^{8}``
    `` \Rightarrow v=5.85\times {10}^{7}\,\mathrm{\,m\,}/\,\mathrm{\,s\,}``
    `` ``
  • #26-c
    10 MeV.
    Ans : Kinetic energy of electron`` =10\,\mathrm{\,MeV\,}={10}^{7}\times 1.6\times {10}^{-19}\,\mathrm{\,J\,}``
    `` \Rightarrow \frac{{m}_{0}{c}^{2}}{2\sqrt{1-{v}^{2}-{c}^{2}}}-{m}_{0}{c}^{2}=1.6\times {10}^{-12}``
    `` \Rightarrow {m}_{0}{c}^{2}\left(\frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1\right)=1.6\times {10}^{-12}``
    `` \Rightarrow \frac{1}{\sqrt{1-{v}^{2}\mathit{/}{c}^{\mathit{2}}}}-1=\frac{1.6\times {10}^{-12}}{9.1\times 9\times {10}^{-31}\times {10}^{16}}``
    `` \Rightarrow \frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1=1.019536\times {10}^{3}``
    `` \Rightarrow \frac{1}{\sqrt{1-{v}^{2}/{c}^{2}}}-1=1019.536+1``
    `` \Rightarrow \sqrt{1-{v}^{2}/{c}^{2}}=0.000979877``
    `` \Rightarrow 1-{v}^{2}/{c}^{2}=0.99\times {10}^{-6}``
    `` \Rightarrow v=2.999999039\times {10}^{8}\,\mathrm{\,m\,}/\,\mathrm{\,s\,}``
    Page No 458: