An accelerated electron has speed of \(5 \times 10^{6} \mathrm{~ms}^{-1}\) with an uncertainty of \(0.02 \%\). The uncertainty in finding its location while in motion is \(x \times 10^{-9} \mathrm{~m}\). The value of \(x\) is ____. (Nearest integer) [Use mass of electron \(=9.1 \times 10^{-31} \mathrm{~kg}, \mathrm{~h}=6.63 \times 10^{-34} \mathrm{Js}, \pi=3.14\) ]

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Atomic Structure

An accelerated electron has speed of

5 \times 1 0^{6} ms^{- 1}

with an uncertainty of

0.02%

. The uncertainty in finding its location while in motion is

x \times 1 0^{- 9} m

. The value of

x

is ____.
(Nearest integer)
[Use mass of electron

= 9.1 \times 1 0^{- 31} kg, h = 6.63 \times 1 0^{- 34} Js, π = 3.14

]

Answer: 58

The uncertainty in the speed is

0.02

per cent of the speed itself:

Δ v = \frac{0.02}{100} \times 5 \times 1 0^{6} = 1 \times 1 0^{3} m s^{- 1}

so the uncertainty in momentum is

Δ p = m Δ v = 9.1 \times 1 0^{- 31} \times 1 \times 1 0^{3} = 9.1 \times 1 0^{- 28} kg m s^{- 1}

. Taking the uncertainty principle at its limit,

Δ x = \frac{h}{4 π Δ p} = \frac{6.63 \times 1 0 ^{- 34}}{4 \times 3.14 \times 9.1 \times 1 0 ^{- 28}} = 5.8 \times 1 0^{- 8} m

Written in the form

x \times 1 0^{- 9}

m this is

58 \times 1 0^{- 9}

m, so

x = 58

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Integer answer