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Wave Optics

Question

Determine the speed of electron in n = 3 orbit of He⁺. Is the non-relativistic approximation valid?

Solution

The speed of electron in nth orbit is given by

v = \frac{2 π {KZe}^{2}}{nh}

For, He, Z = 2 and n = 3

Therefore,

v = \frac{2 π K 2 e^{2}}{3 h}

\Rightarrow

= \frac{4 \times 3.14 \times 9 \times 10^{9} (1.6 \times 10^{- 19})^{2}}{3 \times 6.6 \times 10^{- 34}}

v = 1.46 \times 10^{6} m / s

Now, the ratio of the speed of electron to the speed of light would tell us if relativistic or non-relativistic approximation is true.

So,

\frac{v}{c} = \frac{1.46 \times 10^{6}}{3 \times 10^{8}} = 0.048

<< 1

Hence non-relativistic approximation is valid.