Question

Assume that the Earth's magnetic field is a dipole. At what distance above the Earth's surface is the magnitude of the field one-half of its value at the surface?

Short answer

The magnetic field is half its surface value at approximately 1656 km above the Earth's surface.

Step-by-step solution

Understand the Magnetic Field Model

The Earth's magnetic field is considered a dipole, meaning its field strength can be described using a dependency that decreases with distance. The magnetic field strength, \( B \), at a distance \( r \) from a dipole is inversely proportional to \( r^3 \). This can be represented by the equation \( B = \frac{B_0}{r^3} \), where \( B_0 \) is the field strength at the Earth's surface.

Set Up the Equation

To find the distance where the field is half of the surface value, express the condition \( B = \frac{1}{2} B_0 \). Here's the equation: \( \frac{1}{2} B_0 = \frac{B_0}{r^3} \). Since \( B_0 \) will cancel out from both sides, simplify to \( \frac{1}{2} = \frac{1}{r^3} \).

Solve for r

Solve the equation \( \frac{1}{2} = \frac{1}{r^3} \) for \( r \). This gives \( r^3 = 2 \). Solving for \( r \), take the cube root of both sides: \( r = \sqrt[3]{2} \approx 1.26 \).

Interpret r as a Distance Above Earth's Surface

The \( r \) calculated is the ratio of the distance from the Earth's center where the field is half the surface value. If the Earth's radius is \( R \), then \( r \) is the total distance from the Earth's center, so \( r = 1.26R \). The height above the surface is obtained by subtracting \( R \) from \( r \), thus the height \( h = 1.26R - R = 0.26R \). Using the Earth's average radius \( R \approx 6371 \text{ km} \), compute \( h \): \( h \approx 0.26 \times 6371 \approx 1656 \text{ km} \).

Key concepts

Magnetic Dipole

The Earth's magnetic field resembles a dipole, much like a giant bar magnet. This means it has two opposite poles, similar to north and south ends of a magnet. This dipole configuration is critical in understanding why the magnetic field weakens with increased distance from the Earth.
One key aspect of this model is that it simplifies the complexities of the Earth's magnetic field into a predictable pattern. The field lines emerge from the magnetic north and south poles and loop around the Earth, returning to the opposite pole.
This dipole nature is instrumental in explaining many phenomena we observe, like the auroras, which are significantly affected by the interaction between the Earth's magnetic field and solar particles. Understanding the dipole nature of Earth aids in predicting how its magnetic field behaves at different locations and heights.

Distance Calculation

When determining how the Earth's magnetic field changes with distance, we focus on a specific relationship. Magnetic field strength is inversely proportional to the cube of the distance from the dipole. This means as you move away from the dipole (the Earth, in this case), the field strength decreases rapidly—at a rate proportional to the distance raised to the third power.
The formula to calculate this is:

• \( B = \frac{B_0}{r^3} \)

Here, \( B_0 \) is the magnetic field strength at the Earth's surface, and \( r \) is the distance from the center of the Earth in terms of Earth radii. This rapid decrease explains why magnetic phenomena are most pronounced at the surface and become negligible at much greater distances.

Field Strength Dependency

The strength of the Earth's magnetic field is not constant but changes depending on your distance from the Earth. By nature of its dependency on distance, as you ascend into space, you'll find the magnetic field lessens.
If you need the field to be half its strength compared to the surface, you use the relationship:

• \( \frac{1}{2} = \frac{1}{r^3} \)

This expression shows that if you want the field strength halved, the distance from the Earth's center must increase such that the cube of this distance equals two. By solving for \( r \), where \( r^3 = 2 \), you find that \( r \approx 1.26 \), meaning you'll be at 1.26 times Earth's radius from its center.

Earth's Radius

The Earth's radius is a crucial value in calculations involving the planet's magnetic field. On average, the Earth's radius is about \( 6371 \text{ km} \). This parameter is pivotal because it provides a baseline for measuring distances related to Earth's properties.
In our exercise, knowing Earth's radius helps us transition from understanding theoretical distances (like \( r = 1.26R \)) to applicable distances in kilometers.

• For instance, the distance above Earth's surface where the magnetic field is half its surface value requires subtracting Earth's radius from this computed distance.
• This gives a height of \( h = 0.26 \times 6371 \approx 1656 \text{ km} \).

Thus, Earth's radius is not just a standalone measurement; it's an integral component to understanding and applying Earth-centric calculations.