Question

The effective radius of a proton is about \(1 \times 10^{-15} \mathrm{~m} ;\) the radius of the observable universe (given by the distance to the farthest observable quasar) is \(2 \times 10^{26} \mathrm{~m}\) (see Table \(1-4\) ). Identify a physically meaningful distance that is approximately halfway between these two extremes on a logarithmic scale.

Short answer

The physically meaningful distance that is approximately halfway between a proton and the universe on a logarithmic scale is approximately \(4.47 \times 10^{5} \mathrm{~m}\) or 447 kilometers.

Step-by-step solution

Convert the radii to logarithmic scale

Taking base 10 log of both radii gives us for the proton \(log_{10}(1 \times 10^{-15}) = -15\) and for the universe \(log_{10}(2 \times 10^{26}) = 26 + log_{10}(2) \approx 26.3 \)

Calculate the average in logarithmic scale

Adding both values and dividing by 2 gives the average \( \frac{-15 + 26.3}{2} = 5.65 \)

Convert back to normal scale

This gives us \(10^{5.65} \approx 4.47 \times 10^{5} \mathrm{~m}\)

Key concepts

Proton Radius

The proton radius is a crucial measure in atomic physics. It represents the approximate size of a proton, one of the fundamental particles in all matter. Protons, along with neutrons and electrons, form the building blocks of atoms. Understanding their size aids in grasping the atomic structure of matter. The effective radius of a proton is tiny, estimated to be about \(1 \times 10^{-15} \text{ m}\). This is an incredibly small distance, a fraction of a micrometer. To better visualize this, consider that 1 millimeter is equivalent to \(10^6\) times the size of a proton. Even with such a tiny size, the proton radius has significant implications in fields like quantum mechanics and nuclear physics. It's essential for understanding scattering experiments and energy levels within an atom.

Universe Scale

The universe scale represents the largest distances known to humankind. This includes the radius of the observable universe, which stretches approximately \(2 \times 10^{26} \text{ m}\). These vast distances encompass all we can currently observe, from galaxies and stars to quasars.In astrophysics, the scale of the universe helps scientists comprehend the overall structure and age of the universe. It provides a framework for understanding cosmic phenomena like dark matter, cosmic microwave background radiation, and the expansion of the universe.The sheer size of the universe scale helps place cosmic distances into perspective, highlighting both the magnitude and mysteries of space exploration. It contrasts strikingly with the minuteness of atomic scales, emphasizing how much there is left to explore and understand.

Logarithmic Distance Calculation

A logarithmic scale is useful for comparing vastly different values, like the radius of a proton and the universe. This method simplifies understanding by compressing large and small scales into a manageable range.To find a meaningful distance halfway between the proton radius and universe radius, we use logarithmic calculations:

• Convert both distances to their base 10 logarithmic forms: \(-15\) for the proton and approximately \(26.3\) for the universe.
• Calculate the average on this scale by adding the two values and dividing by two, giving \(5.65\).
• Finally, convert this back to a linear scale, \(10^{5.65} \approx 4.47 \times 10^5 \text{ m}\).

This technique is beneficial because it helps us understand the relative positions on a scale where differences are not easily comparable. By using logarithms, extreme values become more accessible and analyzable, a valuable approach in both physics and other sciences.