Problem 34 - Master Resource Book in JEE Main Physics

Question

How many wavelengths of \(\mathrm{Kr}^{86}\) are there in one metre? (a) \(1553164.13\) (b) \(1650763.73\) (c) \(652189.63\) (d) \(2348123.73\)

Step-by-Step Solution

Answer

The number of wavelengths of krypton-86 in one meter is 1650763.73, option (b).

Step 1: Understand the Problem

The problem is asking us to determine the number of wavelengths of krypton-\(\text{Kr}^{86}\) that fit into one meter. To do this, we first need to know the wavelength of krypton-\(\text{Kr}^{86}\) in meters.

Step 2: Recall the Wavelength of Krypton-86

Historically, the wavelength of krypton-\(\text{Kr}^{86}\) was used to define the meter until 1960. The wavelength is approximately \(\lambda = 606 \times 10^{-9}\) meters (606 nanometers).

Step 3: Calculate the Number of Wavelengths in One Meter

We calculate the number of wavelengths in one meter by dividing one meter by the wavelength of krypton-\(\text{Kr}^{86}\). This is given by \(\frac{1}{606 \times 10^{-9}}.\)

Step 4: Perform the Division

Perform the division calculation: \(\frac{1}{606 \times 10^{-9}} = \frac{1}{606 \times 10^{-9}} = 1650763.73.\)

Step 5: Select the Correct Answer

Among the given options, the calculated result corresponds to option (b) which is \(1650763.73\).

Key Concepts

Definition of Meter

The definition of a meter has evolved over time, adapting to our growing understanding of the universe and technology. Initially, it was based upon measurements that humans could easily relate to, such as the size of a pendulum or a fraction of the Earth's meridian. As precision and technology improved, so did our methods of defining a meter.

In the late 19th and early 20th century, a more scientifically rigorous definition was sought. This led to the adoption of using the wavelength of light to define the meter due to its consistency and reproducibility.

Ultimately, this paved the way for the 1983 definition, where the meter is defined as the distance light travels in vacuum in 1/299,792,458th of a second. This current definition relies on the constant speed of light, ensuring international uniformity and precision.

Historical Usage of Krypton-86

Krypton-86, a specific isotope of krypton, played a crucial role in defining the meter for several decades. In 1960, it was chosen because of its specific wavelength when used in a scientific process called a spectral line. The wavelength emitted by krypton-86 was found to be exceedingly consistent and reliable, making it ideal for precision measurement.

The wavelength of the krypton-86 spectral line is 606 nanometers. During its use, this consistent characteristic allowed for an exact length measurement system worldwide.

Even though krypton-86 is no longer used as the standard, its impact remains significant in the history of scientific measurement. It showcases how inventive methods are continually sought and adopted to hone the accuracy of fundamental units of measure.

Calculation of Wavelengths

Calculating the number of wavelengths of krypton-86 that fit into one meter is a delicate process, emphasizing precision in both understanding and calculation.

Firstly, we need the wavelength of krypton-86, which is 606 nanometers, or 606 x 10^-9 meters. To find out how many of these wavelengths fit into one meter, we use division:

• Divide 1 meter by the wavelength of krypton-86: \( \text{Number of Wavelengths} = \frac{1}{606 \times 10^{-9}} \).

When you perform this calculation, you arrive at approximately 1,650,763.73 wavelengths.

This exacting process helps us appreciate how specific units help map the reality in which they function. Division allows us to see how these tiny fragments of space add up to create something larger that is measurable and meaningful.