A magnetic needle free to rotate about the vertical direction (compass ) point `3.5` west of the geographic north . Another magnetic needle free to rotate in a vertical plane parallel to the magnetic meridian has its north tip pointing down at `18^(@)` with the horizontal . The magnitude of the horizontal component of the earth ' magnetic field at the place is known to be `0.40 G` . what is the direction and magnitude of the earth's magnetic field at the place ?

Explanation

To solve the problem, we need to find the direction and magnitude of the Earth's magnetic field at the given location. We are provided with the following information:

1. The magnetic declination is \(3.5^\circ\) west of geographic north.
2. The angle of dip (or inclination) is \(18^\circ\) downward from the horizontal.
3. The horizontal component of the Earth's magnetic field, \(B_H\), is \(0.40 \, \text{G}\).

### Step-by-Step Solution:

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