Final answer:
The range of the weak interaction mediated by the Z⁰ boson can be estimated using the Heisenberg uncertainty principle, leading to a range of approximately 10⁻¹⁸ meters.
Step-by-step explanation:
To estimate the range of the weak interaction mediated by the Z0 boson, we will use the Heisenberg uncertainty principle, ΔEΔt ≈ ℏ, where ℏ is the reduced Planck constant. The mass of the Z0 boson is given as 91 GeV/c2, which is equivalent to an energy (E = mc2) of 91 GeV. Using the conversion of GeV to joules (1 GeV = 1.602×10-10 J), we have the energy in joules as 91 x 1.602×10-10 J. Setting ΔE equal to this energy and solving for Δt will give us the lifetime of the boson. The reciprocal of this time (1/Δt) is related to the range of the force, Δx, through the uncertainty principle applied to position and momentum, ΔxΔp ≈ ℏ. With Δp as the momentum of a virtual particle (Δp ≈ E/c), we arrive at the desired range, Δx.
The calculation will not be exact due to the simplifications made, but it will give us an approximate value for the order of magnitude of the range of the weak nuclear force. The relationship between force range and particle mass implies that heavier exchange particles correspond to shorter force ranges. Hence, given the mass of the Z0 boson, the weak force's range is approximately 10-18 meters.