Through what potential difference should an electron be accelerated to give it a speed of (a) 0.6c, (b) 0.9c), and (0.99c)?

Explanation

`eV - m_0 C^(2) = m_0 c^(2) / 2 (sqrt 1- v^(2) / c^(2) ) `
` = 9.1 xx 9 xx 10^(-15) / 1.6`
` = 9.1 xx 9 / 1.6 xx 10^(-15)`
` rArr eV = (9 .1 xx 9 / 1.6 + 81.9) xx 410^(-15)`
` rArr eV = (81.9 / 1.6 + 81.9 )xx 10^(15))` xx 10^(-15)`
` rArr eV = 133. 0875 xx 10^(-15)
`:. V = 133.0875 xx 10^(-15) / 1.6 xx 10^(-19)`
` = 83.179 xx 10^(4) = 831 KV`
` (b) `eV - m_0 C^(2) = m_0 c^(2) / 2 (sqrt 1 -v^(2) / c^(2)) `
` rArr eV - 9.1 xx 10^(-31) xx 9 xx 10^(16)`
` = 9.1 xx 9 xx 10^(-15) / 2(sqrt 1 - 0.81c^(2) / c^(2))`
` rArr eV - 81.9 xx 10^(-15) `
` = 9.1 xx 9 xx 10^(-15) / 2 xx 0.435 `
` eV = (81.9 / 0.87 - 81.9 )xx 10^(-15)`
` = 12.237 xx 10^(-15)`
`rArr V = 12.237 xx 10^(-15) / 1.6 xx 10^(-19) `
`= 76 / 48 KV`
when ` v = 0.99 c `
` eV - m_0c^(2) = m_0 c^(2) / 2(sqrt 1 - v^(2) / c^(2) + m_0c^(2) `
` rArr eV = 9.1 xx 10^(31) xx 9 xx 10^(16) / 2 (sqrt 1 -(0.99)^(2))`
+ 9.1 xx 10^(-21) xx 9 xx 10^(6)`
` rArr eV = 372. 18 xx 10^(15) `
` rArr V = 372.18 xx 10^(-15) / 1.6 xx 10^(-19) `
` V = 272.6 xx 10^(4)`
` V = 2.726 xx 10^(6) = 2.7 MeV`