Orders of magnitude: the ratio of a typical atomic radius to a typical nuclear radius is approximately equal to which power of ten?

Introduction / Context:
Physics relies heavily on orders of magnitude to compare very different length scales. Understanding how atomic sizes compare to nuclear sizes helps explain why matter is mostly empty space and why nuclear forces act over extremely short ranges compared with electromagnetic interactions in atoms.

Given Data / Assumptions:

• Atomic radius scale: about 1 angstrom, which is 1 Å = 10^-10 m.
• Nuclear radius scale: about a few femtometres, e.g., 1 fm = 10^-15 m.
• We need the ratio r_atom / r_nucleus.

Concept / Approach:
Compute the ratio using canonical scales. If r_atom ≈ 10^-10 m and r_nucleus ≈ 10^-15 m, then r_atom / r_nucleus ≈ 10^-10 / 10^-15 = 10^5. While precise values vary by element and isotope (nuclear radius roughly scales as r0 * A^(1/3)), the 10^5 figure is the widely accepted order-of-magnitude comparison.

Step-by-Step Solution:
Set r_atom ≈ 10^-10 m (angstrom scale).Set r_nucleus ≈ 10^-15 m (femtometre scale).Form ratio: r_atom / r_nucleus = 10^-10 / 10^-15.Simplify powers: 10^(-10 − (−15)) = 10^5.

Verification / Alternative check:
Realistic ranges (e.g., 0.5–2.5 Å for atoms and 4–7 fm for many nuclei) still deliver a ratio near 10^5 within a factor of a few, confirming the order-of-magnitude choice.

Why Other Options Are Wrong:
10^8 or 10^12 or 10^15: Overstate the gap by several to many orders of magnitude.10^3: Understates the gap; nuclei are much smaller than atoms.

Common Pitfalls:
Confusing atom diameter with radius or using nanometres instead of angstroms without adjusting exponents, both of which can skew the ratio if not handled consistently.

Final Answer:
10^5