Problem 76 - Fundamentals of Physics

Question

A current is established in a gas discharge tube when a sufficiently high potential difference is applied across the two electrodes in the tube. The gas ionizes; electrons move toward the positive terminal and singly charged positive ions toward the negative terminal. (a) What is the current in a hydrogen discharge tube in which \(3.1 \times 10^{18}\) electrons and \(1.1 \times 10^{18}\) protons move past a crosssectional area of the tube each second? (b) Is the direction of the current density \(\vec{J}\) toward or away from the negative terminal?

Step-by-Step Solution

Answer

(a) The current is -0.32 A. (b) The direction of \( \vec{J} \) is toward the negative terminal.

Step 1: Understanding current in terms of charge flow

Current (\( I \)) is defined as the rate of flow of charge. We calculate it using the formula:\[ I = \frac{Q}{t} \]where \( Q \) is the total charge and \( t \) is the time in seconds. In our case, we calculate the total charge by considering both the electrons and protons moving past a cross-sectional area of the tube each second.

Step 2: Calculate the charge of electrons and protons

The charge of a single electron (or proton) is \( e = 1.6 \times 10^{-19} \text{ C} \).Calculate the total charge for electrons:\[ Q_{e} = 3.1 \times 10^{18} \times (-1.6 \times 10^{-19} \text{ C}) = -4.96 \times 10^{-1} \text{ C} \]Calculate the total charge for protons:\[ Q_{p} = 1.1 \times 10^{18} \times (1.6 \times 10^{-19} \text{ C}) = 1.76 \times 10^{-1} \text{ C} \]

Step 3: Calculate the net charge flow per second

The net charge \( Q \) passing through the cross-section each second is the sum of the charges from electrons and protons:\[ Q = Q_{p} + Q_{e} \]\[ Q = 1.76 \times 10^{-1} \text{ C} - 4.96 \times 10^{-1} \text{ C} = -3.2 \times 10^{-1} \text{ C} \]

Step 4: Calculate the current

Using the formula for current:\[ I = \frac{Q}{t} = -3.2 \times 10^{-1} \text{ C/s} \]Thus, the current in the discharge tube is \( -0.32 \, \text{A} \). Negative sign indicates the direction of flow.

Step 5: Determine the direction of current density \(\vec{J}\)

The direction of current density \( \vec{J} \) is in the direction of positive charge flow. Since protons (positive charges) are moving toward the negative terminal, the direction of \( \vec{J} \) is towards the negative terminal.

Key Concepts

Gas Discharge Tube

A gas discharge tube is a fascinating device used to study electrical conduction through gases. It is made of a sealed glass tube containing a low-pressure gas like hydrogen. When a high potential difference is applied across its electrodes, the gas inside becomes conductive. This process is exciting because it allows us to see the effects of electric current in gases.

These tubes are integral to many applications, including lighting (like neon signs), and scientific measurements. They form a plasma, which emits light as it transitions between energy states. The study of gas discharge tubes helps us understand the fundamental properties of gases and how charged particles like electrons and ions behave under the influence of an electric field. This behavior is crucial for understanding current flow, especially in low-pressure environments.

Ionization

Ionization is the key mechanism that makes a gas discharge tube work. It's the process where atoms or molecules gain or lose electrons to become charged particles called ions. You can think of it as charging up a substance. In a gas discharge tube, this happens when the electric field is strong enough to pull apart electrons from the neutral gas atoms.

Here's how it occurs:

• An electron is stripped away from the atom, leaving a positively charged ion and free electron.
• The free electron can then collide with other neutral atoms, ionizing more of them.
• As more atoms become ionized, a cascade effect occurs, leading to a significant increase in charge carriers within the gas.

This ionization process is crucial for establishing an electric current in the discharge tube. It also helps us understand how matter responds to electrical fields, which is fundamental to many applications in physics and engineering.

Current Density

Current density, denoted as \( \vec{J} \), describes how electric current is distributed over a given area. Imagine how water flows through a pipe, spreading out over the pipe's cross-section. Current density focuses on the amount of current passing through a particular area, usually measured in amperes per square meter (A/m²).

Understanding current density allows us to visualize how evenly current is flowing. In the example of the gas discharge tube, electrons and protons move past a cross-sectional area each second. The direction and magnitude of \( \vec{J} \) result from how these charged particles travel within the tube.

• If current density is high, it indicates a strong flow of electric current.
• If it's low, the current is weaker.
• The direction of \( \vec{J} \) is based on the movement of positive charges.

In the discharge tube, because protons (positive charges) are moving toward the negative terminal, we determine the direction of \( \vec{J} \) is towards the negative terminal.

Charge Flow

Charge flow refers to the movement of electric charge through a conductor, such as the gas in a discharge tube. Current is essentially the rate of charge flow, and in our context, imagining it is like thinking about how many cars pass by a toll booth each second.

Let's break it down:

• Current is calculated using the formula \( I = \frac{Q}{t} \), where \( Q \) is total charge and \( t \) is time.
• In the hydrogen discharge tube, both electrons (negative charge) and protons (positive charge) contribute to this flow.
• Charges flow from regions of high potential energy to low potential energy, driven by voltage applied across the tube's electrodes.

Understanding charge flow is critical to mastering the basics of electricity. It tells us how efficiently a current can move through given circumstances, affecting everything from device performance to safety.

Hydrogen Ionization

Hydrogen ionization is an amazing process, particularly relevant in the context of gas discharge tubes filled with hydrogen gas. Here, when we apply a strong electric field, hydrogen atoms lose their electron, resultantly forming positive hydrogen ions or protons.

Why is this interesting?

• Hydrogen, the simplest element, gives a clear demonstration of ionization processes.
• Shows how electron removal and subsequent charge interactions are integral to electrical conduction.
• Ionized hydrogen acts similarly to more complex gases, offering a simpler model to study.

In our exercise, the number of electrons and protons moving signifies the ongoing ionization. A greater understanding of hydrogen ionization can help explain principles of chemistry, physics, and many technologies that rely on gas discharge tubes.