Earth's Magnetic Field

The Earth's magnetic field is a vital force that extends from the Earth's interior out into space, shielding our planet from harmful solar radiation and cosmic particles. Generated by the movement of molten iron in the Earth's outer core, this magnetic field acts like a giant invisible shield, playing a crucial role in maintaining life on Earth. Just like a compass needle aligns with the magnetic poles, countless real-life systems—such as navigation in ships, aeroplanes, and even smartphones—rely on this magnetic field for orientation. Beyond technology, many animal species, like birds and sea turtles, use the Earth's magnetic field to navigate during migration. In essence, this invisible force influences both nature and human activity in ways we often take for granted.

This Story also Contains

1. Earth's Magnetic Field
2. Solved Examples Based on Earth's Magnetic Field
3. Summary

Earth's Magnetic Field

The reason why, A bar magnet, when suspended freely, points in a north-south direction is due to the earth’s giant magnetic field.

The branch of Physics which deals with the study of Earth's magnetic field is called Terrestrial magnetism. It is also known as geomagnetism.

Various Terminologies about Earth’s Magnetism

Geographic axis- The Axis of rotation of Earth is called the Geographic axis.

Geographic Meridian: The vertical plane passing through the geographical axis is called Geographic Meridian.

Geographic Poles-The points where the Geographic axis cuts the surface of Earth are called Geographic poles (i.e Ng, Sg)

Magnetic axis- The axis of the huge magnet assumed to be lying inside the earth is called the magnetic axis.

The magnetic Equator- The circle on the earth's surface perpendicular to the magnetic axis is called the magnetic equator.

The angle between Magnetic and Geographical Axis- They make an angle of 11.5° with each other.

or we can say that the Earth’s magnetic field is similar to that of a bar magnet tilted 11 degrees from the spin axis of the Earth.

The Magnetic Elements

These define the Earth's magnetic field $\vec{B}$ at any point.
Following are the three magnetic elements of the earth:
1.Magnetic declination $(\theta)$
2. The angle of Dip or Magnetic Inclination ( $\delta$ )
3. The horizontal component of Earth's magnetic field $\left(B_H\right)$

Magnetic Declination $(\theta)$

Magnetic Declination is defined as the angle between geographic and magnetic meridian planes.

The Angle of Dip or Magnetic Inclination $(\delta)$

Magnetic dip or magnetic inclination at a place is defined as the angle at which the direction of the total strength of Earth’s magnetic field is made with a horizontal line in the magnetic meridian.

At the poles, the angle of dip = 90⁰ and at the equator, the angle of dip = 0⁰

Horizontal Component (H) of Earth’s Magnetic Field $\left(B_H\right)$

The intensity of the earth’s magnetic field can be resolved into two components

Horizontal Component $\left(B_H\right)$
Vertical Component $\left(B_V\right)$

So we can write $\tan \delta=\frac{\mathrm{B}_{\mathrm{V}}}{\mathrm{B}_{\mathrm{H}}}, \quad \sin \delta=\frac{\mathrm{B}_{\mathrm{V}}}{\mathrm{B}}, \cos \delta=\frac{\mathrm{B}_{\mathrm{H}}}{\mathrm{B}}$

Resultant Magnetic Field due to earth

$
\begin{aligned}
& B_H=B \cos \delta \\
& B_V=B \sin \delta \\
& B=\sqrt{B_H^2+B_V^2}
\end{aligned}
$
Earth's Magnetic field is horizontal only at the magnetic equator i.e when $\delta=0^0$ then $B_H=B$ and $B_V=0$ Earth Magnetic Field at the Pole- Since $\delta=90^{\circ}$ so $B_H=0$ and $B_V=B$

Important Points

• Isoclinic lines- The lines that pass through different places having the same angle of dip.
• An aclinic line- A line which passes through places having an angle of dip as 0⁰
• Isodynamic line-The lines drawn through places having the same B_H

Tangent law

When a small magnet is suspended in two uniform magnetic fields B and $B_H$ which are at right angles to each other.
The magnet comes to rest at an angle $\Theta$.
i.e for the below figure when

Magnet in Equilibrium
Then $M B_H \operatorname{Sin} \Theta=M B \operatorname{Sin}(90-\Theta) B \Rightarrow B_H \tan \Theta$ (tangent law) or $\tan \theta=\frac{B}{B_H}$

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Solved Examples Based on Earth's Magnetic Field

Example 1: Two small bar magnets are placed in a line with poles facing each other at a certain distance of d apart. If the length of each magnet is negligible as compared to d , the force between them will be inversely proportional to

1) $d$
2) $d^2$
3) $1 / d^2$
4) $d^4$

Solution:

Horizontal component of Earth's magnetic field

Earth's Magnetic field is horizontal only at the magnetic equator

it is denoted by $B_H$

$
F=\frac{\mu_0}{4 \pi}\left(\frac{6 M M^{\prime}}{d^4}\right)_{\text {in the end-on position }}
$

Hence, the answer is the option (4).

Example 2: The correct relation is where B_H = Horizontal component of earth's magnetic field; B_V = vertical component of earth's magnetic field and B = Total intensity of earth's magnetic field.

1) $B=\frac{B_V}{B_H}$
2) $B=B_V \times B_H$
3) $B=\sqrt{B_H^2+B_V^2}$
4) $B=B_H+B_V$

Solution:

Horizontal component of earth's magnetic field at any other Place

Total intensity can be resolved into horizontal component (B_H) and vertical component (B_v)

The angle between B_H & B_V is 90^o.

$B=\sqrt{B_H^2+B_V^2}$

Hence, the answer is the option(3).

Example 3: The earth's magnetic field at a certain place has a horizontal component of 0.3 gauss and a total strength of 0.5 gauss. The angle of the dip is
$
\begin{aligned}
& \text { 1) } \tan ^{-1} \frac{3}{4} \\
& \text { 2) } \sin ^{-1} \frac{3}{4} \\
& \text { 3) } \tan ^{-1} \frac{4}{3} \\
& \text { 4) } \sin ^{-1} \frac{3}{5}
\end{aligned}
$

Solution:

Resultant Magnetic Field due to earth

$\begin{aligned} & B_H=B \cos \phi \\ & B_V=B \sin \phi \\ & B=\sqrt{B_H^2+B_V^2} \\ & B^2=B_V^2+B_H^2 \Rightarrow B_V=\sqrt{B^2-B_H^2}=\sqrt{(0.5)^2-(0.3)^2}=0.4 \\ & \text { Now, Angle of dip }=\tan ^{-1}\left(\frac{B_V}{B_H}\right) \\ & \tan \phi=\frac{B_V}{B_H}=\frac{0.4}{0.3}=\frac{4}{3} \Rightarrow \phi=\tan ^{-1}\left(\frac{4}{3}\right)\end{aligned}$

Hence, the answer is the option(3).

Example 4: At some locations on Earth, the horizontal component of the E magnetic field is $18 \times 10^{-6} T$. At this location, a magnet of length 0.12 m and pole strength 1.8 Am is suspended from its mid-point using a thread, it makes $45^{\circ}$ horizontal in equilibrium. To keep this needle horizontal, the vertical force that should be applied at one of its ends is:

1) $1.3 \times 10^{-5} \mathrm{~N}$
2) $6.5 \times 10^{-5} \mathrm{~N}$
3) $3.6 \times 10^{-5} \mathrm{~N}$
4) $1.8 \times 10^{-5} \mathrm{~N}$

Solution:

$\begin{aligned} & A t 45^{\circ}, \\ & \mathrm{B}_{\mathrm{H}}=\mathrm{B}_{\mathrm{V}} \\ & \frac{\mathrm{F} l}{2}=M B_{\mathrm{V}}=\mathrm{m} \times 1 \times B_{\mathrm{V}} \\ & \mathrm{F}=\frac{2 \mathrm{mlB}}{1}=3.6 \times 18 \times 10^{-6} \\ & \mathrm{~F}=6.5 \times 10^{-5} \mathrm{~N}\end{aligned}$

Hence, the answer is the option (2).

Example 5: The vertical component of Earth's magnetic field is zero or the Earth's magnetic field always has a vertical component except at the

1) Magnetic Poles

2) Geographical Poles

3) Every Place

4) Magnetic equator

Solution:

Earth's Magnetic Field at the Equator
$
\begin{aligned}
& B_H=B \\
& B \cos \phi=B \\
& \cos \phi=1 \\
& \text { or } \phi=0^{\circ}
\end{aligned}
$
At the magnetic equator, the angle of dip is $\underline{0}^{\circ}$. Hence the vertical component $V=I \sin \phi=0$

Hence, the answer is the Option (4).

Summary

The Earth's magnetic field, generated by the movement of molten iron in its outer core, resembles a tilted bar magnet. It influences navigation systems and the migration of animals. Key concepts include the magnetic axis, magnetic equator, and the 11.5° tilt between the geographic and magnetic axes. The three magnetic elements—magnetic declination, dip or inclination, and horizontal component—define the field at any location. These principles explain phenomena like the Earth's field being horizontal at the equator and vertical at the poles.