Question

The earth's magnetic field at a certain place has a total strength of 0.5 Gauss and the horizontal component of 0.3 Gauss. Then the angle of dip at that place is

• A. $\tan^{-1}\frac{3}{4}$
• B. $\tan^{-1}\frac{4}{3}$
• C. $\sin^{-1}\frac{3}{4}$
• D. $\sin^{-1}\frac{3}{5}$ \

Hint:

Think of the classic 3-4-5 right triangle. Since the horizontal base is 3 and the hypotenuse is 5, the vertical side must be 4, making $\tan\delta = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}$.

Answer 1

The Correct Option is B

Step 1: Split the earth's field.
The total field $B$ has a horizontal part $B_{H} = B\cos\delta$ and a vertical part $B_{V} = B\sin\delta$. Here $\delta$ is the angle of dip.

Step 2: Find $\cos\delta$.
\[ \cos\delta = \frac{B_{H}}{B} = \frac{0.3}{0.5} = \frac{3}{5} \]
Step 3: Make a right triangle.
If the adjacent side is $3$ and the hypotenuse is $5$, the opposite side is $\sqrt{5^{2}-3^{2}} = 4$.

Step 4: Vertical component.
So $B_{V}$ matches the side $4$ in this triangle.

Step 5: Find $\tan\delta$.
\[ \tan\delta = \frac{B_{V}}{B_{H}} = \frac{4}{3} \]
Step 6: Write the dip angle.
So $\delta = \tan^{-1}\dfrac{4}{3}$, which is option 2.
\[ \boxed{\delta = \tan^{-1}\frac{4}{3}} \]