Problem 2116 - JEE Physics

Question

If \(\lambda_{\gamma} \lambda_{\mathrm{x}}\) and \(\lambda_{\mathrm{m}}\) are the wave lengths of the \(\gamma\) -rays, \(\mathrm{x}\) rays and micro waves respectively in space then (A) \(\lambda_{\gamma}>\lambda_{\mathrm{x}}>\lambda_{\mathrm{m}}\) (B) \(\lambda_{\gamma}<\lambda_{\mathrm{x}}<\lambda_{\mathrm{m}}\) (C) \(\lambda_{r}=\lambda_{x}=\lambda_{m}\) (D) \(\lambda_{\gamma}<\lambda_{\mathrm{m}}<\lambda_{\mathrm{x}}\)

Step-by-Step Solution

Answer

The correct answer is (B) \(\lambda_{\gamma}<\lambda_{\mathrm{x}}<\lambda_{\mathrm{m}}\), as gamma-rays have the shortest wavelengths, x-rays have the next shortest wavelengths, and microwaves have the longest wavelengths.

Step 1: Identify the wavelengths of gamma-rays, x-rays, and microwaves

Gamma-rays have the shortest wavelengths among these three types of radiation, which is typically less than 0.1 nm. X-rays have the next shortest wavelengths typically between 0.1 nm and 100 nm. Microwaves have the longest wavelengths typically between 1 mm and 100 cm.

Step 2: Compare the wavelengths

Based on the identified wavelengths: - Gamma-rays (\(\lambda_{\gamma}\)) have the shortest wavelengths. - X-rays (\(\lambda_{\mathrm{x}}\)) have the next shortest wavelengths. - Microwaves (\(\lambda_{\mathrm{m}}\)) have the longest wavelengths. Thus, we can state that \(\lambda_{\gamma}<\lambda_{\mathrm{x}}<\lambda_{\mathrm{m}}\).

Step 3: Choose the correct answer

Comparing the given options with our conclusion from Step 2, we find that option (B) matches our result: (B) \(\lambda_{\gamma}<\lambda_{\mathrm{x}}<\lambda_{\mathrm{m}}\) So, the correct answer is (B).

Key Concepts

Gamma Rays

Gamma rays are a type of electromagnetic radiation with extremely short wavelengths. Typically measuring less than 0.1 nanometers (nm), these wavelengths are shorter than those of both X-rays and microwaves. Gamma rays are generated by nuclear reactions and radioactive decay.
They possess high energy, which allows them to penetrate many materials, making them useful in medical imaging and treatments, such as cancer radiotherapy. However, their high energy also means they can be harmful to living tissues, requiring careful handling and shielding.

X-Rays

X-rays are another form of electromagnetic radiation with wavelengths ranging from 0.1 nm to 100 nm. These fall between gamma rays and ultraviolet light in the electromagnetic spectrum.
X-rays are commonly used in medical diagnostics to view the inside of the body. Due to their ability to pass through soft tissues while being absorbed by denser materials like bones, they create images that highlight contrasts within the body.

• Soft X-rays have a range closer to the ultraviolet spectrum.
• Hard X-rays have higher energy levels, putting them closer to gamma rays in terms of penetrating power.

Microwaves

Microwaves oscillate at longer wavelengths compared to gamma rays and X-rays, usually from 1 millimeter to 100 centimeters. This makes them part of the electromagnetic spectrum just above radio waves.
Microwaves are well-known for their use in microwave ovens, where they revolve at specific frequencies to heat food by exciting water molecules.
They also play a vital role in communication technologies, such as in transmitting signals for mobile phones, Wi-Fi, and satellite communications. Their longer wavelengths and lower energy mean they interact less destructively with biological tissues compared to gamma and X-rays.

Wavelength Comparison

Understanding the comparison of wavelengths in the electromagnetic spectrum is key to grasping how different types of radiation behave and are used.

• Gamma Rays: Less than 0.1 nm, have the highest energy and shortest wavelengths.
• X-Rays: Range from 0.1 nm to 100 nm, have moderate energy.
• Microwaves: Span from 1 mm to 100 cm, are lower in energy compared to the others.

This hierarchy is crucial because it affects how different types of radiation penetrate materials and interact with matter. As shown in the original exercise, the ordering by wavelength is: \( \lambda_{\gamma} < \lambda_{\mathrm{x}} < \lambda_{\mathrm{m}} \). This highlights how gamma rays penetrate materials more deeply than X-rays, which in turn penetrate more than microwaves.