An electron starts moving in a unifrom electric fied of strength `E = 10 kV//cm`. How soon after will the kinetic energy of the electron become equal to its rest energy ?

Explanation

From the law of relative conservation of energy
`(m_(0) c^(2))/(sqrt(1 - (v^(2)//c^(2))) -e Ex = m_(0) c^(2)`.
as the electron is at rest (`v = 0` for `x = 0`) initially.
Thus clearly `T = e Ex`.
On the other hand, `sqrt(1 - (v^(2)//c^(2))) = (m_(0) c^(2))/(m_(0) c^(2) + e Ex)`
or, `(v)/(c) = (sqrt((m_(0) c^(2) + e Ex)^(2) - m_(0)^(2) c^(4)))/(m_(0) c^(2) + e Ex)`
or, `ct = int c dt int ((m_(0) c^(2) + e Ex) dx)/(sqrt((m_(0) c^(2) + e Ex)^(2) - m_(0)^(2) c^(4)))`
`= (1)/(2 eE) int (dy)/(sqrt(y - m_(0)^(2) c^(4))) = (1)/(eE) sqrt((m_(0) c^(2) + e Ex)^(2) - m_(0)^(2) c^(4)) +` constant
The "constant" = 0, at `t = 0`, for `x = 0`,
So, `ct = (1)/(eE) sqrt((m_(0) c^(2) + e Ex)^(2) - m_(0)^(2) c^(4))`.
Finallay, using `T = e Ex`,
`c e E t_(0) = sqrt(T (T + 2 m_(0) c^(2)))` or, `t_(0) = (sqrt(T (T + 2 m_(0) c^(2))))/(e Ec)`