Calculate the uncertainty in the position (in A) of an electron moving at a speed of (6.5 +/- 0.1)* 10^5m/s. The mass of an electron is 9.109x10^-31 kg.
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DxDp = h/4p
Where Dx is uncertainty of position, Dp is uncertainty in momentum, h is Planck's constant 5.26e-33 Joule seconds, and p is linear momentum, P = mv.
To calculate the uncertainty in the position of an electron, we can use the principle of uncertainty, also known as Heisenberg's uncertainty principle. According to this principle, the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) of a particle is always greater than or equal to the reduced Planck's constant, denoted as h-bar (ħ). The formula representing this relationship is:
Δx * Δp >= ħ/2
To find the uncertainty in position, we need to calculate the uncertainty in momentum first. The momentum (p) of an object can be calculated using the formula:
p = m * v
where m represents the mass and v represents the velocity.
Given:
Speed of the electron (v) = (6.5 ± 0.1) * 10^5 m/s
Mass of the electron (m) = 9.109 * 10^-31 kg
Calculating the momentum of the electron:
p = m * v
p = (9.109 * 10^-31 kg) * (6.5 * 10^5 m/s)
p = 5.92085 * 10^-25 kg.m/s
Now we can calculate the uncertainty in momentum (Δp). Since the uncertainty in velocity is given as ±0.1 * 10^5 m/s, we can use this value to represent the uncertainty in momentum as well.
Therefore, the uncertainty in momentum is:
Δp = ±0.1 * 10^5 kg.m/s
Finally, we can use Heisenberg's uncertainty principle to find the uncertainty in position (Δx). Rearranging the formula, we have:
Δx >= ħ/(2 * Δp)
Plugging in the values:
ħ = 1.0545718 × 10^-34 J.s (Planck's constant divided by 2π)
Δp = ±0.1 * 10^5 kg.m/s
Calculating the uncertainty in position:
Δx >= (1.0545718 × 10^-34 J.s) / (2 * (±0.1 * 10^5 kg.m/s))
Note: The uncertainty in position will be a positive value since we are dealing with magnitudes.
Therefore, the uncertainty in position (in meters) is given by:
Δx >= 5.272859 × 10^-40 m
Hence, the uncertainty in the position of the electron is approximately 5.27 × 10^-40 meters.