The magnetic field of the earth at any point on its surface is determined by using three physical quantities which are called elements of earth’s magnetic field. They are:

• Declination: Declination at a point of earth’s surface can be defined as the
 acute angle between the geographic meridian and magnetic meridian of the earth. Geographic meridian is the imaginary vertical plane passing through the geographical axis of the earth and magnetic meridian is the imaginary vertical plane passing through the magnetic axis of the earth. The value of declination varies from place to place on the

earth’s surface. The maximum value of declination is 17o towards east or west. In general, its value at a place is constant but it changes slightly with time and the same value is obtained after 960 years.

• Inclination or Angle of Dip: The angle made by the direction of earth’s magnetic field or a freely suspended magnet with the horizontal direction is called inclination or angle of dip at that place. When a magnetic compass is pivoted or rotated about its centre of gravity, it rests pointing downwards in the northern hemisphere and pointing upwards in the southern hemisphere. The angle made by this needle with the horizontal gives the angle of dip. Its value varies from 0o to 90o. At equator, the angle of dip is 0o. The angle of dip is measured by using dip circle.
 A figure showing Angle of Dip with the help of Dip Circle Dip Circle
• Horizontal component of earth’s magnetic field: As the direction of earth’s magnetic field is inclined with the horizontal, it can be resolved into horizontal and vertical components.
The component of earth’s magnetic field along the horizontal direction at a place is called horizontal component of earth’s magnetic field at that place. Let ‘δ’ be the angle of dip. Then the horizontal and vertical components of the earth’s magnetic field ‘B’ are:
Horizontal component of earth’s magnetic field (BH) = Bcosδ …. (i)
Vertical component of earth’s magnetic field (Bv) = Bsinδ …. (ii)
Dividing equation (ii) by (i),
Bv/BH= Bsinδ/Bcosδ
or, Bv/BH= tanδ
∴ δ= tan-1(Bv/BH)
Squaring equations (i) and (ii) and adding both of them, we get,
BH2 + Bv2 = B2cos2δ + B2sin2δ
or, BH2 + Bv2 = B2(cos2δ + sin2δ) [∵ cos2δ + sin2δ = 1]
or, BH2 + Bv2 = B2
∴ B = √(BH2 + Bv2)

Ampere’s circuital law was discovered by André-Marie Ampere in the year 1826 A.D. This law states that, “The line integral of magnetic field B along any closed path in free space is equal to the vacuum permeability (μo) times the total current enclosed by the surface.” Mathematically,

Let us consider a straight conductor carrying current ‘I’. Then the magnetic field around the conductor are concentric circles.
Let ‘P’ be a point at a distance ‘r’ from the conductor. From point P, let us draw a circle of radius ‘r’. Then, the magnitude of magnetic field ‘B’ at any point on the circle is
B = μoI/2πr

And the direction of B is along the tangent to the circle at that point. If we consider an element of length ‘dl’ on the circle, then the angle between ‘dl’ and B is θ = 0o. Now, the line integral of magnetic field B along the closed path isHence, Ampere’s circuital law is proved.