an electron inside spherical metallic particle of volume 10^ -12 metre cube has a kinetic energy about 10 eV. use uncertainty relations to find relative uncertainty in electron velocity
To find the relative uncertainty in the electron's velocity using the uncertainty principle, we need to determine the uncertainty in the position of the electron.
The uncertainty principle states that the product of the uncertainties in position (Δx) and momentum (Δp) of a particle is always greater than or equal to Planck's constant divided by 2π (h/2π).
Mathematically, the uncertainty principle can be expressed as:
Δx * Δp >= h/2π
Now, let's calculate the uncertainty in position (Δx). Since the problem states that the electron is inside a spherical metallic particle, we can assume that the position uncertainty is approximately equal to the diameter of the particle.
The volume of the particle is given as 10^-12 m^3. Therefore, the radius (r) of the particle can be calculated using the formula:
Volume = (4/3) * π * r^3
(r^3) = (3 * Volume) / (4 * π)
r = [(3 * 10^-12) / (4 * π)] ^ (1/3) ≈ 1.39 * 10^-4 m
The diameter (Δx) is twice the radius: Δx ≈ 2 * r ≈ 2 * 1.39 * 10^-4 m
Now that we have the uncertainty in position (Δx), we can determine the uncertainty in momentum (Δp) using the equation:
Δx * Δp >= h/2π
Solving for Δp:
Δp >= h/(2π * Δx)
Substituting the known values:
Δp >= (6.63 * 10^-34 Js) / (2 * π * 2 * 1.39 * 10^-4 m)
Calculating Δp:
Δp >= 2.41 * 10^-25 kg⋅m/s
Finally, to find the relative uncertainty in velocity (Δv/v), we divide Δp by the mass (m) and velocity (v) of the electron:
Δv/v = (Δp/mv)
The mass of an electron (m) is approximately 9.11 × 10^-31 kg. The velocity (v) can be calculated using the kinetic energy (K.E.) of the electron:
K.E. = 1/2 mv^2
v = sqrt(2K.E./m)
Substituting the known values:
v ≈ sqrt(2 * 10 eV * 1.6 * 10^-19 J/eV / 9.11 × 10^-31 kg)
Calculating v:
v ≈ 6.6 × 10^6 m/s
Now, we can calculate the relative uncertainty in velocity (Δv/v):
Δv/v = (Δp/mv) ≈ (2.41 * 10^-25 kg⋅m/s) / (9.11 × 10^-31 kg * 6.6 × 10^6 m/s)
Calculating Δv/v:
Δv/v ≈ 4.2 * 10^-12
Therefore, the relative uncertainty in the electron's velocity is approximately 4.2 * 10^-12.