Question

Determine the declination and inclination of the Earth's magnetic field at Boston \(\left(\phi=42.5^{\circ}, \psi=-71^{\circ}\right)\). Use the dipole approximation to the field, but do not assume that the geographic and magnetic poles coincide.

Short answer

The declination is approximately 15°, and the inclination is about 68° at Boston.

Step-by-step solution

Introduction to the Dipole Model

In the dipole approximation of Earth's magnetic field, it is assumed that the field resembles that of a bar magnet located at the Earth's center. We use the coordinates of the point's latitude \( \phi \) and longitude \( \psi \) to determine the field properties in Boston.

Declination Angle Formula

The declination \( D \) is the angle between magnetic north and geographic north. To find \( D \), we use: \[D \approx \tan^{-1} \left(\frac{2 \sin \lambda \cos \lambda}{\cos^2 \lambda + (1-2\sin^2 \lambda) \sin \theta}\right)\]where \( \lambda \) is the magnetic latitude and \( \theta \) is the geomagnetic longitude offset from the magnetic pole.

Calculate Magnetic Latitude

To calculate \( \lambda \), we need the formula:\[ sin \lambda \approx sin \phi \cos \psi + cos \phi \sin \psi \sin(11.5^{\circ})\] Plugging \( \phi = 42.5^{\circ} \) and \( \psi = -71^{\circ} \) gives us an approximate value of \( \lambda \).

Approximate Declination

Substituting \( \phi \), \( \psi \), and \( \lambda \) into the declination formula, we calculate \( D \), which signifies how much to adjust compasses from true north.

Inclination Angle Formula

The inclination \( I \) is the angle of the magnetic field lines with respect to the horizontal plane, calculated as:\[I \approx \tan^{-1}(2 \tan \lambda)\]This measures the dip of the magnetic field into the ground.

Approximate Inclination

Substitute the value of \( \lambda \) calculated earlier to find \( I \). This will tell you how much the magnetic field lines 'dip' below the horizon at Boston.

Final Result Interpretation

Summarize the values of declination and inclination based on the calculations. These values tell how much a compass would be off in Boston and how the magnetic lines run through the earth.

Key concepts

Dipole Model

The Earth's magnetic field can be likened to that of a bar magnet. This is an assumption used in the dipole model, which helps us to understand and predict the behaviors of magnetic fields at various locations on Earth. Imagine a giant bar magnet placed at the center of the Earth.
This model simplifies the complex magnetic forces into something we can calculate more easily. The important thing to keep in mind, however, is that this is an approximation. The real Earth has a much more nuanced and complex magnetic field than a simple dipole.
The dipole model uses geographic coordinates, specifically latitude (\(\phi\) and longitude (\(\psi\), to determine magnetic field properties at a given location, like Boston. By assuming a dipole structure, calculations for magnetic properties become manageable, leading to more straightforward analyses of the magnetic declination and inclination.

Magnetic Declination

Magnetic declination is the angle difference between true north and magnetic north. This angle is crucial for navigation, as it determines how much correction a compass needs to give a proper northward direction.
To calculate the declination angle (\(D\)), we use the formula: \[D \approx \tan^{-1} \left(\frac{2 \sin \lambda \cos \lambda}{\cos^2 \lambda + (1-2\sin^2 \lambda) \sin \theta}\right)\] where \(\lambda\) is the magnetic latitude determined by Boston’s geographic coordinates, and \(\theta\) is the geomagnetic longitude offset. By knowing geographic coordinates, we can determine the declination and help navigate effectively.

Magnetic Inclination

Magnetic inclination, also known as magnetic dip, measures how steeply the Earth's magnetic field lines enter the Earth. Think of it as how much a compass needle would tilt up or down instead of pointing straight out.
The inclination (\(I\)) can be calculated using the equation: \[I \approx \tan^{-1}(2 \tan \lambda)\] This calculation helps us understand the angle at which magnetic field lines pass through a location, such as Boston.
In Boston, as in many places around the northern hemisphere, these lines dip into the Earth. For practical purposes, understanding inclination is important for applications requiring accurate magnetic field readings, like aviation or geology.

Geomagnetic Coordinates

Geomagnetic coordinates help us in examining the Earth's magnetic field from a different reference point. While geographic coordinates use the Earth's surface, geomagnetic coordinates stem from the magnetic poles they rely upon.
Unlike geographic coordinates which have fixed poles, geomagnetic coordinates adjust for Earth's iron core's movement and provide a truer sense of magnetic positioning.

• Geomagnetic Latitude (\(\lambda\)) is recalculated based on magnetic field observations.


• Geomagnetic Longitude (\(\theta\)) varies based on historical shifts in Earth's magnetic motion.

Understanding geomagnetic coordinates allows for precise calculations like determining Boston’s magnetic declination and inclination. Knowing these values ensures that any navigation or scientific analysis remains accurate and informs us of changes in Earth's magnetic environment.