The vertical component of the earth’s magnetic field is 6 × 10–5 T at any place where the angle of dip is 37°. The earth’s resultant magnetic field at that place will be (Given tan 37° = ¾)
- 8x10-5 T
- 6x10-5 T
- 5x10-4 T
- 1x10-4 T
The Correct Option is D
Solution and Explanation
To solve this question, we need to determine the earth's resultant magnetic field at a place where the vertical component is given, and the angle of dip is provided.
Concept: The earth’s magnetic field at any place can be resolved into two components:
- Vertical component \(B_V\)
- Horizontal component \(B_H\)
The angle of dip (\( \delta \)) is related to these components by the following relations:
- \( B_V = B \cdot \sin(\delta) \)
- \( B_H = B \cdot \cos(\delta) \)
Given:
- The vertical component \(B_V = 6 \times 10^{-5} \, \text{T}\)
- Angle of dip \( \delta = 37^\circ \)
- \( \tan(37^\circ) = \frac{3}{4} \)
Let's calculate the earth's resultant magnetic field:
First, use the relation for the vertical component:
\(B_V = B \cdot \sin(37^\circ)\)
But we know:
\(\sin(37^\circ) = \frac{\tan(37^\circ)}{\sqrt{1 + \tan^2(37^\circ)}}\)
Calculating \( \sin(37^\circ) \):
\( \tan(37^\circ) = \frac{3}{4} \) therefore:
\(\sin(37^\circ) = \frac{\frac{3}{4}}{\sqrt{1 + \left(\frac{3}{4}\right)^2}} = \frac{\frac{3}{4}}{\sqrt{1+\frac{9}{16}}} = \frac{\frac{3}{4}}{\sqrt{\frac{25}{16}}} = \frac{\frac{3}{4}}{\frac{5}{4}} = \frac{3}{5}\)
Now substitute the values:
\( B_V = B \cdot \frac{3}{5} \)
\( 6 \times 10^{-5} = B \cdot \frac{3}{5} \)
Solving for \( B \):
\(B = \frac{6 \times 10^{-5} \times 5}{3} = \frac{30 \times 10^{-5}}{3} = 1 \times 10^{-4} \, \text{T}\)
Therefore, the earth's resultant magnetic field at that place is 1 × 10-4 T.
Conclusion: The correct answer is:
1 × 10-4 T
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Concepts Used:
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Horizontal Lines:
- A horizontal line is a sleeping line that means "side-to-side".
- These are the lines drawn from left to right or right to left and are parallel to the x-axis.
Equation of the horizontal line:
In all cases, horizontal lines remain parallel to the x-axis. It never intersects the x-axis but only intersects the y-axis. The value of x can change, but y always tends to be constant for horizontal lines.
Vertical Lines:
- A vertical line is a standing line that means "up-to-down".
- These are the lines drawn up and down and are parallel to the y-axis.
Equation of vertical Lines:
The equation for the vertical line is represented as x=a,
Here, ‘a’ is the point where this line intersects the x-axis.
x is the respective coordinates of any point lying on the line, this represents that the equation is not dependent on y.
⇒ Horizontal lines and vertical lines are perpendicular to each other.