Step 1: Understanding the Concept:
Earth acts like a giant bar magnet with magnetic field lines emerging from the magnetic south pole and entering the magnetic north pole.
At any specific location on the Earth's surface, the total magnetic field vector (\( B \)) does not necessarily lie flat against the ground.
Instead, it usually points into or out of the Earth at an angle.
This vector is resolved into two mutually perpendicular components for analysis:
1. Horizontal Component (\( B_H \)): This is the part of the field that acts along the horizontal plane. This component is what influences the needle of a standard magnetic compass.
2. Vertical Component (\( B_V \)): This is the part of the field that acts straight down (or up).
The angle that the total magnetic field makes with the horizontal plane is called the Angle of Dip (\( \theta \)) or Magnetic Inclination.
Step 2: Key Formula or Approach:
Using simple vector resolution in a right-angled triangle where \( B \) is the hypotenuse and \( B_H \) is the adjacent side to the angle of dip \( \theta \):
\[ B_H = B \cdot \cos(\theta) \]
To find the total magnetic field \( B \) when the horizontal component and the dip angle are known, we rearrange the equation:
\[ B = \frac{B_H}{\cos(\theta)} \]
Step 3: Detailed Explanation:
From the problem, we are provided with the following data points:
- Horizontal component of Earth's field, \( B_H = 0.3 \) T.
- Angle of dip, \( \theta = 60^{\circ} \).
We need to find the total intensity \( B \).
The cosine of \( 60^{\circ} \) is a standard value used frequently in physics and mathematics:
\[ \cos(60^{\circ}) = 0.5 = \frac{1}{2} \]
Now, substitute the given values into the rearranged formula:
\[ B = \frac{0.3}{\cos(60^{\circ})} \]
\[ B = \frac{0.3}{0.5} \]
Mathematically, dividing by \( 0.5 \) is equivalent to multiplying by \( 2 \).
\[ B = 0.3 \times 2 \]
\[ B = 0.6 \text{ T} \]
This calculation tells us that at this specific location, the Earth's magnetic field is pointing quite steeply into the ground, such that only half of its total strength is projected onto the horizontal plane.
If we were at the magnetic equator, the dip angle would be \( 0^{\circ} \), and \( B \) would equal \( B_H \).
If we were at the magnetic poles, the dip angle would be \( 90^{\circ} \), and \( B_H \) would be zero because the field is entirely vertical.
Since our result is \( 0.6 \) T, this matches the value provided in option (B).
Step 4: Final Answer:
The total magnetic field of the Earth at the given place is \( 0.6 \) T.