Problem 19 - Master Resource Book in JEE Main Physics

Question

The mean free path of the electrons in a discharge tube is \(20 \mathrm{~cm}\). The length of the tube is \(15 \mathrm{~cm}\) only. Then length of Crooke's dark space is (a) \(5 \mathrm{~cm}\) (b) \(20 \mathrm{~cm}\) (c) \(15 \mathrm{~cm}\) (d) \(25 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The length of Crooke's dark space is 15 cm.

Step 1: Understanding the Problem

We need to determine the length of Crooke's dark space in a discharge tube. We're given the mean free path of the electrons, which is the average distance traveled by the electrons between collisions, is 20 cm. The length of the tube itself is 15 cm.

Step 2: Identifying Relevant Information

The Crooke's dark space is part of a discharge tube where electron collisions with gas molecules have not yet occurred prominently. It begins from the cathode and its length is less than or equal to the mean free path if the tube is shorter than the mean free path.

Step 3: Comparing Tube Length and Mean Free Path

Since the entire length of the tube is 15 cm and the mean free path is 20 cm, the electrons can travel the entire tube length without frequent collisions. Therefore, the Crooke's dark space would extend over the entire length of the tube.

Step 4: Conclusion

Crooke's dark space is based on the shortest of either the tube's length or the mean free path. As the tube length is 15 cm which is less than the mean free path, the dark space will cover the tube's entire 15 cm length.

Key Concepts

Discharge Tube

The discharge tube is an essential device in the study of cathode rays and the behavior of electrons in vacuum or gas-filled environments. It is a sealed glass tube containing gas at low pressure through which an electric current is passed. This tube is notable for facilitating the phenomenon of gas discharge, where electricity is transmitted through the gas, often leading to glowing displays.

In operation, a discharge tube usually consists of two electrodes: a cathode and an anode, each connected to a high voltage source. When the electric field is applied, it accelerates electrons from the cathode towards the anode. The movement of electrons and interactions with the gas molecules produce light and other phenomena, making discharge tubes useful in various applications like neon signs and fluorescent lamps.

• It acts as a vessel to observe electrical discharges.
• Often used to study the properties of gases and electron behavior.
• Key in historical experiments leading to the discovery of the electron.

Electron Collisions

Electron collisions in a discharge tube are a critical part of understanding gas discharge processes. As electrons move from the cathode to the anode, they collide with gas molecules in their path, leading to various outcomes.

These collisions can be elastic or inelastic. Elastic collisions result in minimal energy loss, while inelastic collisions can lead to changes in electron energy, often causing the gas molecules to become excited and emit light—a characteristic glow associated with discharge tubes.

The concept of mean free path is crucial here. It refers to the average distance an electron travels between collisions. A longer mean free path means fewer collisions over a given distance, which is pivotal in characterizing phenomena like Crooke's dark space. Understanding these collisions helps explain the behavior of electrons under different conditions and is essential for applications like particle accelerators and plasma televisions.

• Involves interaction between electrons and gas molecules.
• Results in light emission or excitation of molecules.
• Influences the characteristics of discharge, like color and intensity.

Crooke's Dark Space

Crooke's dark space is a fascinating part of the discharge phenomenon occurring within a discharge tube. Named after the scientist William Crookes, it is observed as a dark region near the cathode in the tube. It arises because electrons close to the cathode have not yet gained sufficient energy to cause significant gas excitation.

This zone is characterized by having a lower density of luminescent activity. The length of Crooke's dark space can be influenced by the mean free path of electrons and the tube's length. If a tube is shorter than the electron's mean free path, as seen in the problem, the dark space extends over the full tube length. Understanding Crooke’s dark space is crucial for explaining initial electron acceleration and the resulting light phenomena.

• Occurs near the cathode within a discharge tube.
• Indicates a low collision region where electron activity starts.
• Helps in understanding electron acceleration phases.

Cathode Rays

Cathode rays are streams of electrons observed in discharge tubes when a high voltage is applied across the electrodes. Originating from the cathode, these rays can travel through the discharge tube, interacting with gas molecules and resulting in phenomena like fluorescence or glow discharge.

Historically, the study of cathode rays led to the discovery of the electron by J.J. Thomson. Cathode rays paved the way for the development of early television tubes and oscilloscopes. Within the discharge tube, cathode rays are depicted as straight-line paths unless influenced by external fields, demonstrating the fundamental properties of electron movement.

Their study revealed:

• The nature of electrons as negatively charged particles.
• Fundamental insights into atomic structure.
• Applications in technology like cathode ray tubes and electron microscopes.