Problem 58 - Fundamentals of Physics

Question

The magnetic dipole moment of Earth has magnitude \(8.00 \times 10^{22} \mathrm{~J} / \mathrm{T}\). Assume that this is produced by charges flowing in Earth's molten outer core. If the radius of their circular path is \(3500 \mathrm{~km},\) calculate the current they produce.

Step-by-Step Solution

Answer

The current is approximately \(2.08 \times 10^9\) Amperes.

Step 1: Understanding the Magnetic Dipole Moment

The magnetic dipole moment (\(m\)) is a measure of the strength of a magnetic source. It's given by the formula \(m = I \cdot A\), where \(I\) is the current and \(A\) is the area of the loop of current.

Step 2: Calculate the Area of the Circular Path

The area \(A\) for a circular path is given by the formula \(A = \pi r^2\), where \(r\) is the radius of the path. Here, \(r = 3500 \times 10^3\) meters. Hence, \(A = \pi \times (3500 \times 10^3)^2 \text{ m}^2\).

Step 3: Solve for Current (I)

Rearrange the formula \(m = I \cdot A\) to solve for current \(I\): \(I = \frac{m}{A}\). Substituting the given values, \(m = 8.00 \times 10^{22} \text{ J/T}\) and \(A = \pi \times (3500 \times 10^3)^2\), calculate \(I\).

Step 4: Perform the Calculation

Calculating the area: \[A = \pi \times (3500 \times 10^3)^2 \approx 3.85 \times 10^{13} \text{ m}^2\]. Then calculate the current: \[I = \frac{8.00 \times 10^{22}}{3.85 \times 10^{13}} \approx 2.08 \times 10^9 \text{ Amperes}\].

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment is a fundamental concept in the study of magnetism. It represents how strongly a system generates a magnetic field. The magnetic dipole moment, denoted as \( m \), is a vector quantity pointing from the south to the north pole of a magnetic source. It's crucial in determining how magnetic forces will interact with each other.
In practical terms, the magnetic dipole moment can be thought of as a measure of the internal strength of magnetic fields. It is calculated using the formula:

• \( m = I \cdot A \)

where:

• \( I \) is the electric current, measured in Amperes.
• \( A \) is the area through which the current flows, measured in square meters \( m^2 \).

Understanding this relationship allows us to see how Earth's massive magnetic field could be due to electric currents flowing in its outer core, as demonstrated by the given problem. The Earth's dipole moment of \(8.00 \times 10^{22} \mathrm{J/T}\) suggests an enormous flow of charge that produces Earth's magnetic shield.

Current Calculation

Current, denoted as \( I \), is essentially the flow of electric charge and is measured in units called Amperes. When dealing with problems involving the magnetic dipole moment, such as the Earth's magnetic dipole, calculating the current provides insight into the power of the magnetic field.
We start with the magnetic dipole formula:

• \( m = I \cdot A \)

By solving for \( I \), we rearrange the formula:

• \( I = \frac{m}{A} \)

With known values for \( m \) (\(8.00 \times 10^{22} \text{ J/T}\)) and \( A \) (as calculated in the next section), we can compute \( I \) to find that it is approximately \(2.08 \times 10^9\) Amperes. This high current is a theoretical approximation of the vast amount of electric charge circulating within the Earth, contributing to its magnetic characteristics.

Circular Path Area Calculation

The circular path area is essential when determining quantities like magnetic dipole moments related to current loops. For any circular loop, the area \( A \) can be determined with the formula:

• \( A = \pi r^2 \)

where \( r \) is the radius of the circle.
In our exercise, the circular path has a radius \( r = 3500 \text{ km} \), which is first converted to meters, resulting in \( 3500 \times 10^3 \text{ m} \). Plugging this into the formula yields:

• \( A = \pi \times (3500 \times 10^3)^2 \)

This calculation gives an area of approximately \( 3.85 \times 10^{13} \text{ m}^2 \). Understanding how to compute this circular area is crucial because it directly influences the magnitude of calculated quantities like the current in this context. This substantial area reflects the vast region over which Earth's electric currents might circulate, hence contributing to the generation of its magnetic field.