If the angle of dip at two places are `30^(@)` and `45^(@)` respectively, then the ratio of horizontal components of earth's magnetic field at the two places will be

Options

A. `sqrt(3):sqrt(2)`
B. `1:sqrt(2)`
C. `1:sqrt(3)`
D. `1:2`

Explanation

To find the ratio of the horizontal components of the Earth's magnetic field at two places where the angles of dip are \(30^\circ\) and \(45^\circ\), we can follow these steps:

### Step 1: Understand the relationship between the horizontal component and the angle of dip
The horizontal component of the Earth's magnetic field (\(B_H\)) is related to the total magnetic field (\(B\)) and the angle of dip (\(\delta\)) by the formula:
\[ B_H = B \cdot \cos(\delta) \]
where \(B\) is the total magnetic field and \(\delta\) is the angle of dip.

### Step 2: Write the expressions for the horizontal components at both locations
Let \(B_{H1}\) be the horizontal component at the first location (where \(\delta_1 = 30^\circ\)) and \(B_{H2}\) be the horizontal component at the second location (where \(\delta_2 = 45^\circ\)).
\[ B_{H1} = B \cdot \cos(30^\circ) \]
\[ B_{H2} = B \cdot \cos(45^\circ) \]

### Step 3: Substitute the values of \(\cos(30^\circ)\) and \(\cos(45^\circ)\)
Using known values:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
\[ \cos(45^\circ) = \frac{1}{\sqrt{2}} \]
Now we can substitute these values into the expressions for \(B_{H1}\) and \(B_{H2}\):
\[ B_{H1} = B \cdot \frac{\sqrt{3}}{2} \]
\[ B_{H2} = B \cdot \frac{1}{\sqrt{2}} \]

### Step 4: Find the ratio of the horizontal components
Now, we can find the ratio of the horizontal components:
\[ \frac{B_{H1}}{B_{H2}} = \frac{B \cdot \frac{\sqrt{3}}{2}}{B \cdot \frac{1}{\sqrt{2}}} \]
The \(B\) cancels out:
\[ \frac{B_{H1}}{B_{H2}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{\sqrt{2}}} \]

### Step 5: Simplify the ratio
To simplify the ratio:
\[ \frac{B_{H1}}{B_{H2}} = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{1} = \frac{\sqrt{3} \cdot \sqrt{2}}{2} = \frac{\sqrt{6}}{2} \]

### Final Answer
Thus, the ratio of the horizontal components of the Earth's magnetic field at the two places is:
\[ \frac{B_{H1}}{B_{H2}} = \frac{\sqrt{6}}{2} \]