At a point high in the Earth’s atmosphere, He²⁺ ions in a concentration of 2.8 x 10¹²/m³ are moving due north at a speed of 2.0 x 10⁶ m/s. Also, a 7.0 x 10¹¹ / m³ concentration of O⁻₂ ions is moving due south at a speed of 6.2 x 10⁶ m/s. Determine the magnitude and direction of the current density $\overrightarrow{j}$ at this point.

Verified step by step guidance

1. Step 1: Recall the formula for current density, which is given by: $\mathbf{j}=\sum nq\mathbf{v}$. Here, $n$ is the number density of the charged particles, $q$ is the charge of the particle, and $\mathbf{v}$ is the velocity vector of the particles.
2. Step 2: Calculate the contribution to the current density from the He²⁺ ions. The charge of a He²⁺ ion is $2e$, where $e$ is the elementary charge ($1.6\times 10⁻19$ C). Multiply the number density of He²⁺ ions ($2.8\times 10⁸$ m⁻³) by the charge ($2e$) and the velocity ($2.0\times 10⁶$ m/s, due north).
3. Step 3: Calculate the contribution to the current density from the O⁻₂ ions. The charge of an O⁻₂ ion is $-e$. Multiply the number density of O⁻₂ ions ($7.0\times 10⁷$ m⁻³) by the charge ($-e$) and the velocity ($6.2\times 10⁶$ m/s, due south).
4. Step 4: Add the contributions from the He²⁺ ions and the O⁻₂ ions to find the net current density. Be careful with the directions: north and south are opposite, so their contributions will have opposite signs. Use vector addition to combine the contributions.
5. Step 5: Determine the magnitude and direction of the resulting current density vector. The magnitude is the absolute value of the net current density, and the direction is determined by the sign of the resulting vector (north or south).

Key Concepts

Current Density

Current density, denoted as j, is a vector quantity that represents the amount of electric current flowing per unit area of a cross-section. It is calculated by multiplying the charge density by the velocity of the charge carriers. In this context, it helps quantify how the movement of charged ions contributes to the overall current in the atmosphere.

Charge Carriers

Charge carriers are particles that carry electric charge, such as electrons, ions, or holes. In this scenario, He²⁺ and O⁻₂ ions are the charge carriers, with their respective concentrations and velocities influencing the current density. Understanding the nature and behavior of these carriers is essential for analyzing the current flow.

Vector Addition

Vector addition is the mathematical process of combining two or more vectors to determine a resultant vector. In this problem, the current densities due to He²⁺ and O⁻₂ ions must be added as vectors, taking into account their magnitudes and directions. This concept is crucial for accurately determining the overall current density in the specified direction.