If $\Theta_1$ and $\Theta_2$ be the apparent angles of dip observed in two vertical planes at right angles to each other, then the true angle of $\operatorname{dip} \Theta$ is given by:-

$\tan ^2 \Theta=\tan ^2 \Theta_1+\tan ^2 \Theta_2$

$\cot ^2 \Theta=\cot ^2 \Theta_1-\cot ^2 \Theta_2$

$\tan ^2 \Theta=\tan ^2 \Theta_1-\tan ^2 \Theta_2$

$\cot ^2 \Theta=\cot ^2 \Theta_1+\cot ^2 \Theta_2$

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Concepts Covered - 1

Earth's magnetism

The reason why, A bar magnet, when suspended freely, points in 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 called Terrestrial magnetism. It is also known as geomagnetism.

Various Terminologies about Earth’s Magnetism-

Fig 14

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

Geographic Meridian-A vertical plane passing through the geographical axis is called Geographic meridian.

Geographic Poles-The points where 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 ( $B_H$ )

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 which the direction of the total strength of Earth’s magnetic field makes with a horizontal line in the magnetic meridian.

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

Horizontal component (H) of Earth’s magnetic field ( $B_H$ )-

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

Horizontal Component( $B_H$
Vertical Component( $B_V$ )

So we can write $\begin{array}{c}{\tan \delta=\frac{\mathrm{B_V}}{\mathrm{B_H}}, \quad \sin \delta=\frac{\mathrm{B_V}}{\mathrm{B}}} , {\cos \delta=\frac{\mathrm{B_H}}{\mathrm{B}}}\end{array}$

Resultant Magnetic Field due to earth

$B_{H}=B\cos \delta$

$B_{V}=B\sin \delta$

$B=\sqrt{B_{H}^{2}+B_{V}^{2}}$

Earth 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 $\delta=90^0$ 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.
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 of B_H

Tangent law-

When a small magnet is suspended in two uniform magnetic field B and $B_{H}$ which are at right angles to each other.