Question

A proton in a cyclotron has a velocity of 2x108 m/s, which is nearly the speed of light. The uncertainty in the velocity is 1%. What is the minimum uncertainty in the position of the proton in meters

Answer 1

The minimum uncertainty in the position of the proton is 1.578 x 10⁻¹⁴ m

Step-by-step explanation:

Given;

velocity of the proton, v = 2 x 10⁸ m/s

uncertainty in the velocity, = 1 % = 0.01 v = 0.01 x 2 x 10⁸ m/s = 2 x 10⁶ m/s

The momentum of the proton is given by;

ΔP = mv = (1.67 x 10⁻²⁷)(2 x 10⁶ ) = 3.34 x 10⁻²¹ kgm/s

The minimum uncertainty in the position of the proton is given by;

`\delta x = (h)/(4\pi \delta P)\\\\ \delta x =(6.626*10^(-34))/(4\pi *3.34*10^(-21))\\\\ \delta x = 1.578*10^(-14) \ m`

Therefore, the minimum uncertainty in the position of the proton is 1.578 x 10⁻¹⁴ m