Problem 2162 - JEE Physics

Question

If the wavelength of light is \(4000^{\circ} \mathrm{A}\) then the number of waves in \(1 \mathrm{~mm}\) length will be (A) \(2.5\) (B) 2500 (C) 250 (D) 25000

Step-by-Step Solution

Answer

The number of waves in 1 mm length, given the wavelength of light is \(4000^{\circ} \mathrm{A}\), is 25000.

Step 1: Convert the given units into millimeters

First, we need to convert the given wavelength (in Angstroms) into millimeters. We know that \(1 \mathrm{~A} = 10^{-10} \mathrm{~m}\) and \(1 \mathrm{~mm} = 10^{-3} \mathrm{~m}\). Therefore, to convert Angstroms to millimeters, we can use the following formula: Wavelength (in mm) = Wavelength (in A) × \(10^{-10} \mathrm{~m}\) / \(10^{-3} \mathrm{~m}\) For the given wavelength of \(4000^{\circ} \mathrm{A}\): Wavelength (in mm) = \(4000 \times (10^{-10} / 10^{-3}) = 4000 \times 10^{-7}\) Now, we have the wavelength in mm.

Step 2: Calculate the number of waves

To find the number of waves in 1 mm length, we can use the formula: Number of waves (N) = Length / Wavelength In this case, the Length is 1 mm, and we have already calculated the Wavelength in step 1. Hence, N = 1 mm / (4000 x \(10^{-7}\) mm) N = \(1 / (4000 \times 10^{-7})\) N = \(10^{7} / 4000\) N = 25000

Step 3: Choose the correct answer

The number of waves in 1mm length is 25000. Therefore, the correct answer is: (D) 25000

Key Concepts

Wavelength Conversion

The concept of wavelength conversion is essential in wave optics. It involves changing the unit of the wavelength from one form to another to simplify calculations. In the exercise given, the wavelength of light is provided in angstroms. An angstrom is a very small unit of length commonly used to express the size of atoms and wavelengths. To convert wavelengths from angstroms to millimeters, you first need to understand the exact conversion factors:

• 1 angstrom ( ^{∘} A) is equal to 10^{-10} meters (m).
• 1 millimeter (mm) is equal to 10^{-3} meters (m).

To convert from angstroms to millimeters, you divide the value in angstroms by 10^7, because each angstrom is 10^{-10} m and each millimeter is 10^{-3} m, resulting in a multiplicative factor of 10^{-7} when converting directly to millimeters. This method simplifies the understanding of wavelengths in millimeters, making them easier to work with in further calculations.

Angstrom to Millimeter

In practical terms, converting angstroms to millimeters allows for easier computations in the context of everyday measurements and comparisons. Individuals typically think in terms of millimeters rather than the incredibly tiny angstroms. To illustrate:

• Given a wavelength of 4000 angstroms.
• Convert this into millimeters using the multiplication factor from the previous section.

This calculation is straightforward using multiplication:\[ \text{Wavelength in mm} = 4000 \times 10^{-7} \]By computing this, you get the wavelength in millimeters, which simplifies usage in further calculations, such as determining the number of waves within a certain distance like 1 millimeter, as shown in the exercise.

Number of Waves Calculation

Once you have the wavelength in the same unit as the length you're comparing it to, calculating the number of waves is simple. This exercise demonstrates this by determining how many wavelengths fit into a 1 mm length. You use the formula:\[ \text{Number of Waves} = \frac{\text{Length}}{\text{Wavelength}} \]For a wavelength of 4000 angstroms converted to millimeters (4000 x 10^{-7} mm), the calculation becomes:\[ \text{Number of Waves} = \frac{1 \, \text{mm}}{4000 \times 10^{-7} \, \text{mm}} = \frac{10^7}{4000} = 25000 \]This tells us there are 25,000 waves in 1 millimeter when the wavelength is 4000 angstroms. Understanding this computation is key to solving similar wave mechanics problems.

Wave Mechanics

Wave mechanics is a fundamental aspect of wave optics, explaining how waves propagate, interact, and can be quantified within different mediums. In our exercise, wave mechanics helps us understand how the wavelength relates to the number of waves present in a given space like 1 mm. This interaction and calculation rest on principles such as:

• Understanding wave units and easy conversion between them.
• Applying formulas to measure and predict the behavior of waves in various mediums.

Wave mechanics not only aids in calculating properties like wavelength and frequency, but also enables the practical use of these in technological applications such as fiber optics, laser machining, and various imaging technologies. Awareness of how waves operate at a microscopic level enriches comprehension in physics and engineering applications.