Esercizi su Massa, Energia, Q. di m. Relativistiche N° 5

v = \frac{q R B}{m} = \frac{q R B}{\frac{m_{0}}{\sqrt{1 - β^{2}}}} \Rightarrow \frac{v}{c} = \frac{q R B}{m_{0} c} \cdot \sqrt{1 - β^{2}} \Rightarrow β^{2} = {(\frac{q R B}{m_{0} c})}^{2} \cdot (1 - β^{2}) \Rightarrow β^{2} [1 + {(\frac{q R B}{m_{0} c})}^{2}] = {(\frac{q R B}{m_{0} c})}^{2} \Rightarrow β = \frac{\frac{q R B}{m_{0} c}}{\sqrt{1 + {(\frac{q R B}{m_{0} c})}^{2}}} = \frac{1}{\sqrt{\frac{1}{{(\frac{q R B}{m_{0} c})}^{2}} + 1}} = \frac{1}{\sqrt{{(\frac{m_{0} c}{q R B})}^{2} + 1}}

Sostituendo i valori numerici:

β = \frac{1}{\sqrt{{(\frac{9.11 \cdot 1 0^{- 31} 3 \cdot 1 0^{8}}{1.6 \cdot 1 0^{- 19} \cdot 0.2 \cdot 0.03})}^{2} + 1}} = \frac{1}{\sqrt{{(0.2646..)}^{2} + 1}} = \frac{1}{\sqrt{1.081..}} = 0.962

L'energia cinetica si ricava facilmente, nota β:

K = \frac{m_{0} c^{2}}{\sqrt{1 - β^{2}}} - m_{0} c^{2} = m_{0} c^{2} (\frac{1}{\sqrt{1 - β^{2}}} - 1) = 9.11 \cdot 1 0^{- 31} 9 \cdot 1 0^{16} (\frac{1}{\sqrt{1 - {0.962}^{2}}} - 1) ≃ 2.17 \cdot 1 0^{- 13} J = \frac{2.17 \cdot 1 0^{- 13}}{1.6 \cdot 1 0^{- 13}} M e V ≃ 1.36 M e V