An electron has a speed of 500 m/Sec with uncertainty of 0.002% what is the uncertainty in position?
To determine the uncertainty in position of an electron, we can make use of the Heisenberg uncertainty principle, which states that it is impossible to simultaneously determine the exact position and momentum (which is related to speed) of a particle.
The uncertainty principle is expressed as:
Δx * Δp ≥ h/(4π)
Where Δx represents the uncertainty in position, Δp is the uncertainty in momentum, and h is Planck's constant (h ≈ 6.626 x 10^(-34) Js).
Given that the uncertainty in momentum is related to uncertainty in speed by:
Δp = m * Δv
Where m is the mass of the electron and Δv is the uncertainty in speed.
Since the mass of an electron is approximately 9.109 x 10^(-31) kg, and the speed uncertainty is given as 0.002% of the speed, we can calculate the uncertainty in momentum.
Δv = 0.002% * 500 m/s
= 0.002/100 * 500 m/s
= 0.01 * 500 m/s
= 5 m/s
Δp = (9.109 x 10^(-31) kg) * (5 m/s)
= 4.5545 x 10^(-30) kg m/s
Now, we can rearrange the uncertainty principle equation to solve for Δx:
Δx ≥ h/(4π * Δp)
Δx ≥ (6.626 x 10^(-34) Js)/ (4π * 4.5545 x 10^(-30) kg m/s)
Δx ≥ 0.091 x 10^(-4) m
Therefore, the uncertainty in position of the electron is approximately 0.091 x 10^(-4) meters.
To calculate the uncertainty in position, we can use the Heisenberg uncertainty principle, which states that the uncertainty in position multiplied by the uncertainty in momentum must be equal to or greater than h/(4π), where h is the reduced Planck's constant (equal to 6.626 x 10^-34 J·s).
The momentum of an electron can be calculated using the formula:
p = m*v
where p is the momentum, m is the mass, and v is the velocity.
Given that the speed of the electron is 500 m/s and we want to calculate the uncertainty in position, we need to convert the speed to a velocity by multiplying it by the mass of the electron, which is approximately 9.10938356 × 10^-31 kg.
So, the momentum is:
p = (9.10938356 × 10^-31 kg) * (500 m/s)
p ≈ 4.55469178 × 10^-28 kg·m/s
Now we can calculate the uncertainty in momentum:
∆p = (0.002/100) * p
∆p = (0.002/100) * 4.55469178 × 10^-28 kg·m/s
∆p ≈ 9.10938356 × 10^-31 kg·m/s
Using the Heisenberg uncertainty principle, we can find the uncertainty in position (∆x):
∆x * ∆p ≥ h/(4π)
∆x * 9.10938356 × 10^-31 kg·m/s ≥ (6.626 x 10^-34 J·s) / (4π)
Solving for ∆x:
∆x ≥ (6.626 x 10^-34 J·s) / [(4π) * 9.10938356 × 10^-31 kg·m/s]
∆x ≥ 7.26263 × 10^-4 m
Therefore, the uncertainty in position (∆x) is approximately 7.26263 × 10^-4 meters.