Question

The horizontal component of the earth’s magnetic field at a place is B and angle of dip is ${60^ \circ }$ . What is the value of the horizontal component of the earth magnetic field at equator?

Answer 1

We know that the earth’s magnetic field has two components which are horizontal component and vertical component. In the question, we are provided with the horizontal component which is given by ${B_H} = K\cos \delta$ . Now, we are given an angle of dip and the magnetic field component. So, we can put the values and we can obtain the value of constant K. By using the value of constant K, we can calculate the horizontal component of the magnetic field at the equator as we know that the angle of dip at the equator is zero.

Formula used:
${B_H} = K\cos \delta$
Where,
${B_H} =$ Horizontal component of the earth’s magnetic field
$K =$ Resultant of the earth’s magnetic field
$\delta =$ Angle of dip

Complete step by step solution:
According to the question, the horizontal component of the earth’s magnetic field at a place is B.
So, putting it in the formula, we get
${B_H} = K\cos \delta = B$
Also, we are given the angle of dip which is ${60^ \circ }$.
So, putting the value of angle of dip in the above equation, we get

$$B = K\cos \delta \\\ B = K\cos {60^ \circ } \\\$$

We know that, $\cos {60^ \circ } = \dfrac{1}{2}$ , so the equation becomes:

$$B = K\cos {60^ \circ } \\\ B = K \times \dfrac{1}{2} \\\ B = \dfrac{K}{2} \\\ K = 2B \\\$$

Now, we need to find the horizontal component of the earth’s magnetic field at the equator.
At the equator, the angle of dip is zero.
So, using the formula we get

$${B_H} = K\cos \delta \\\ {B_H} = K\cos {0^ \circ } \\\$$

We know that, $\cos {0^ \circ } = 1$
Also, the value of K is calculated to be $2B$ .
Therefore, the equation becomes

$${B_H} = K\cos {0^ \circ } \\\ {B_H} = 2B \times 1 \\\ {B_H} = 2B \\\$$

Hence, the horizontal component of the earth’s magnetic field at equator is $2B$.

Note: We know that the horizontal component of the earth’s magnetic field at the equator can be calculated by putting the angle of dip to be zero. We need to be careful while calculating the horizontal component as it differs from the vertical component and the resultant magnetic field. The value of angle of dip should be kept in mind while calculating the magnetic field components at different places.