Question

At a certain place, the angle of dip ${{30}^{o}}$ and the horizontal component of earth’s magnetic field is $0.50$ oersted. The earth’s total magnetic field (in oersted)
A. $\sqrt{3}$
B. $1$
C. $\dfrac{1}{\sqrt{3}}$
D. $\dfrac{1}{2}$

Answer 1

Earth is considered a large magnet having its own North Pole and South Pole. The component of the earth’s magnetic field is resolved into two components: horizontal and vertical. The magnitude of earth’s magnetic field varies from one place to another on the earth’s surface.

Complete step by step solution:
The earth’s magnetic field arises due to the Dynamo effect. According to this, earth is made up of three layers: core, mantle and crust. The outer core of earth consists of molten iron and nickel which generates convection currents resulting in the formation of streams of charged particles. The current produced by charged particles results in the formation of earth’s magnetic field. The earth’s magnetic field is very weak and therefore does not interfere with the experiments performed with strong magnets.
Now, imagine a magnetic dipole in the centre of the earth which has its own magnetic field lines. As the axis of rotation of the earth is tilted, there is the formation of geographic North and South Pole as well as magnetic North and South Pole.
The three elements of the earth’s magnetic field are:
Angle of declination: The angle between the geographic north and magnetic north is called angle of declination. The geographic north changes its position according to earth’s surface and time.
Angle of inclination or angle of dip: The angle made by the earth’s total magnetic field, $B$ with the surface of earth (i.e. its horizontal component, $H$ ) is called angle of dip. The angle of dip at magnetic equator is ${{0}^{o}}$ and at magnetic poles, the angle of dip is ${{90}^{o}}$ .
Horizontal component of the earth’s magnetic field: The intensity of the earth’s magnetic field can be resolved in two components: horizontal and vertical.
According to the question, the angle of dip,$\theta ={{30}^{o}}$ and the horizontal component,$H=0.50\,\,oersted$ . Therefore earth’s total magnetic field,$B$ is given by:
$\begin{aligned} & \cos \theta =\dfrac{H}{B} \\\ & \Rightarrow B=\dfrac{H}{\cos \theta } \\\ & \Rightarrow B=\dfrac{0.50}{\cos {{30}^{o}}} \\\ & \Rightarrow B=\dfrac{0.50}{\dfrac{\sqrt{3}}{2}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( \because \cos {{30}^{o}}=\dfrac{\sqrt{3}}{2} \right) \\\ & \Rightarrow B=\dfrac{1}{\sqrt{3}} \\\ \end{aligned}$
Therefore, option C is the correct answer.

Note:
In a bar magnet, the magnetic field lines are directed from the South Pole to the North pole inside the magnet and from the North Pole to the South Pole outside the magnet. This implies that magnetic field lines come out of the North Pole. However, the earth’s magnetic field is directed from the South Pole to the North Pole. This means that the field lines generate from the South Pole and go into the North Pole.