Question

The radius of the proton is about $10^{-15} m$. The radius of the observable universe is $10^{26}m$. Identify the distance which is half-way between these two extremes on a logarithmic scale.
a) $10^{21}m$
b) $10^{6}m$
c) $10^{-6}m$
d) $10^{8}m$

Answer 1

Hint : A logarithmic unit is a system that can be utilized to display a quantity on a logarithmic rule, that is, as being proportionate to the benefit of a logarithm function implemented to the proportion of the quantity and a reference quantity of the equal type.

Complete step-by-step solution:
Radius of the proton, $r = 10^{-15}m$
Radius of the observable universe, $R = 10^{26}m$
We need to find the distance which is half-way between these two extremes on a logarithmic scale.
Mid-point of the logarithmic scale is obtained by taking the sum of the log of radius divided by two.
Mid-point $= \dfrac{1}{2} ( log r + log R)$
$= \dfrac{1}{2} ( log 10^{-15} + log 10^{26})$
$= \dfrac{1}{2} (-15 + 26)$
$= \dfrac{11}{2} = 5.5$
Mid-point $= 10^{5.5} \simeq 10^{6}$
Distance which is half-way between these two extremes on a logarithmic scale is $10^{6} m$.
Option (b) is correct.

Note: A spectrum that contains overlapping of various colors is described as an impure Spectrum. In different words, in an impure spectrum, the colors are not separate and independent, but they flap. On the opposite hand, a spectrum with every color separate and independent is named a pure spectrum.