Problem 10 - Master Resource Book in JEE Main Physics

Question

The radius of the proton is about \(10^{-15} \mathrm{~m}\). The radius of the observable universe is \(10^{26} \mathrm{~m}\). Identify the distance which is half-way between, these two extremes on a logarithmic scale. (a) \(10^{21} \mathrm{~m}\) (b) \(10^{6} \mathrm{~m}\) (c) \(10^{-6} \mathrm{~m}\) (d) \(10^{0} \mathrm{~m}\)

Step-by-Step Solution

Answer

(b) \( 10^{6} \)

Step 1: Understand Logarithmic Scale

On a logarithmic scale, distance between two points is measured in powers of 10. The midpoint between two points on a log scale is the geometric mean of the two points.

Step 2: Calculate the Geometric Mean

To find the geometric mean of the proton's radius and the universe's radius, we use the formula for geometric mean: \( \sqrt{a \times b} \), where \( a = 10^{-15} \) and \( b = 10^{26} \).Substitute the values: \[ GM = \sqrt{10^{-15} \times 10^{26}} = \sqrt{10^{11}} \]

Step 3: Simplify the Geometric Mean

Simplify \( \sqrt{10^{11}} \) by using the property of exponents: \[ \sqrt{10^{11}} = 10^{11/2} = 10^{5.5} \]

Step 4: Compare Options

Convert \( 10^{5.5} \) into a more manageable form to compare with given options. Since \( 10^{5.5} \) is between \( 10^{5} \) and \( 10^{6} \), the closest whole number option is \( 10^{6} \).

Step 5: Determine the Correct Answer

The answer that best matches \( 10^{5.5} \) is option (b) \( 10^{6} \).

Key Concepts

Logarithmic Scale

Imagine a ruler that doesn't measure things in usual steps of 1, 2, 3, but in jumps of 10. This is what a logarithmic scale looks like. It's super helpful when we want to deal with very large or very small numbers like the size of a proton or the observable universe.
In the problem, instead of finding a middle ground by regular counting, we look for a balance in size or scale. This balance is known as the geometric mean. On a logarithmic scale, everything is seen in terms of powers of 10. So when we're halfway between two numbers, like the proton's size and the universe's size, we're actually finding the geometric mean of these numbers represented as powers of 10.
This method gives us a neat shortcut where the halfway point isn't just adding or subtracting, but finding an exponential "average". This helps us handle super different size scales in a manageable way.

Exponents

Have you wondered what those small numbers above 10 mean? Those are exponents. They tell us how many times 10 is multiplied by itself. For example, \(10^{3}\) means 10 times 10 times 10.
Exponents make it easy to write big numbers without all the zeros. Instead of writing out a million (\(1,000,000\)), we can say \(10^{6}\). It's not just a shortcut—it’s a way to instantly understand the scale of a number.
When calculating with exponents, we use simple rules. Multiplying numbers with the same base means we add their exponents, like \(10^2 \times 10^3 = 10^{2+3} = 10^5\). When dividing, we subtract exponents, such as \(10^5 / 10^2 = 10^{5-2} = 10^3\).
In our exercise, simplifying \( \sqrt{10^{11}} \) leads us to \(10^{5.5}\), showing how we handle complex calculations by adjusting exponents. It makes everything simpler and more concise.

Distance Measurement

Measuring vast distances, like from a tiny proton to the entire observable universe, requires some creative thinking. Traditional steps just don't cut it when you're dealing with such extremes.
While typical distance measurement is straightforward with units like meters or kilometers, these can get tricky at extremes of size. Here, a logarithmic approach becomes valuable as it evens out the playing field.
In this mathematical exercise, the distance halfway between two extremes, viewed through a logarithmic lens, gives us new insights by using the geometric mean. This point reflects a balanced scale between the two sizes, not by direct measurement, but by understanding the scale at a deeper level. This exciting way of measuring helps us not just see the numbers, but understand the relationship between incredibly large or small distances.