Problem 79 - Fundamentals of Physics

Question

In a hypothetical fusion research lab, high temperature helium gas is completely ionized and each helium atom is separated into two free electrons and the remaining positively charged nucleus, which is called an alpha particle. An applied electric field causes the alpha particles to drift to the east at \(25.0 \mathrm{~m} / \mathrm{s}\) while the electrons drift to the west at \(88.0 \mathrm{~m} / \mathrm{s}\). The alpha particle density is \(2.80 \times 10^{15} \mathrm{~cm}^{-3} .\) What are (a) the net current density and (b) the current direction?

Step-by-Step Solution

Answer

(a) Net current density is \(8.96\, \text{A/m}^2\); (b) Current direction is west.

Step 1: Convert Units for Density

The alpha particle density needs to be converted from \( \text{cm}^{-3} \) to \( \text{m}^{-3} \). To do this, multiply the given density by \( 10^6 \) (since \(1 \text{ m} = 10^2 \text{ cm} \) and therefore \(1 \text{ m}^3 = 10^6 \text{ cm}^3\)). Thus, the particle density becomes \(2.80 \times 10^{21} \text{ m}^{-3}\).

Step 2: Calculate Charge Densities

He atoms are ionized into 2 electrons and 1 alpha particle (which has a charge of \(+2e\)). The electron charge density is \(-2 \times (2.80 \times 10^{21}) \times e\) and the alpha particle charge density is \(2.80 \times 10^{21} \times 2e\), where \(e = 1.6 \times 10^{-19} \text{ C}\).

Step 3: Determine Current Density for Electrons

The current density \( J_e \) for electrons is given by \( J_e = n_e \cdot q_e \cdot v_e \), where \( n_e = 2 \times 2.80 \times 10^{21} \text{ m}^{-3} \), \( q_e = -1.6 \times 10^{-19} \text{ C} \), and \( v_e = -88.0 \text{ m/s} \). Thus, \( J_e = 2 \times (2.80 \times 10^{21}) \times (-1.6 \times 10^{-19}) \times (-88.0) \). Calculate \( J_e \).

Step 4: Determine Current Density for Alpha Particles

The current density \( J_{\alpha} \) for alpha particles is \( J_{\alpha} = n_{\alpha} \cdot q_{\alpha} \cdot v_{\alpha} \), where \( n_{\alpha} = 2.80 \times 10^{21} \text{ m}^{-3} \), \( q_{\alpha} = 3.2 \times 10^{-19} \text{ C} \), and \( v_{\alpha} = 25.0 \text{ m/s} \). Thus, \( J_{\alpha} = (2.80 \times 10^{21}) \times (3.2 \times 10^{-19}) \times (25.0) \). Calculate \( J_{\alpha} \).

Step 5: Calculate Net Current Density

The net current density \( J \) is the sum of the current densities for electrons \( J_e \) and alpha particles \( J_{\alpha} \): \( J = J_e + J_{\alpha} \). Calculate \( J \).

Step 6: Determine Current Direction

If the net current density \( J > 0 \), the net current direction is towards the east (direction of the alpha particles). If \( J < 0 \), the direction is towards the west (direction of the electrons). Determine the direction based on the sign of \( J \).

Key Concepts

Ionization

Ionization is a process where an atom or a molecule loses or gains electrons, transforming it into an ion. In a fully ionized gas, like the helium gas in our fusion lab scenario, the helium atoms lose their electrons. This process results in the creation of two free electrons and a positively charged alpha particle for every helium atom. Ionization is crucial in many scientific applications, including plasma creation and fusion research.
Understanding ionization helps us explore how charges are separated within a material. This step is essential for analyzing the behavior of charged particles under the influence of forces like electric fields.

Alpha Particles

Alpha particles are positively charged ions that consist of two protons and two neutrons, making them identical to a helium nucleus. As a byproduct of the ionization process in our lab, these alpha particles carry a charge of "+2e", where "e" is the elementary charge. Their large mass compared to electrons makes their motion different and central to calculating the current density.
Being heavier, alpha particles move at slower speeds when subjected to electric fields. However, their positive charge means they drift towards the direction of the applied field. In our exercise, they drift eastward, which contributes positively to the net current density.

Electric Field

An electric field exerts a force on charges within it, causing them to move or accelerate. The field's direction and strength determine how particles like electrons and alpha particles drift. The force exerted by the electric field on a charged particle can be calculated using the equation \( F = qE \), where \( q \) is the charge of the particle and \( E \) is the electric field strength.
In the fusion lab scenario, the electric field causes alpha particles to drift to the east and electrons to drift to the west. The field is a crucial factor in determining how these particles contribute to current density and the direction of the net current.

Electron Drift

Electron drift refers to the movement of electrons in response to an electric field. Unlike alpha particles, electrons have a negative charge, meaning they move opposite to the direction of the electric field. In our scenario, electrons drift towards the west at a speed of \(-88.0 \text{ m/s}\).
This drift speed is usually much faster than that of heavier charged particles like alpha particles. Electron drift contributes a negative component to the overall net current density because of their opposite charge compared to alpha particles. Therefore, in calculating net current density, attention to their velocity and charge is essential.

Charge Density

Charge density expresses the quantity of electric charge per unit volume. It's significant when considering how charged particles are distributed throughout a space. Here, the alpha particle charge density is calculated using their density and respective charge. Similarly, the electron charge density involves their density and charge, considering they carry negative charges.
The entire current density of a system is linked to these charge densities through their motion under an electric field. The calculation of charge density for both electrons and alpha particles allows us to determine the individual contributions of each to the current density, which is essential for evaluating the net current density and its direction.