a magnetic needle free to rotate in a vertical plane, orients itself vertically at a certain place on earth. what are the values of horizontal component of earth's mag field and angle of dip at that place?

At the two magnetic poles, the magnetic needle rests vertically.
Hence, at poles, $\delta = {90}^0$
As, Horizontal component, H= R cos $\delta = 0^0$
And at a place on the magnetic poles, earth's magnetic field is perpendicular to the surface of earth, i.e., vertical.
Therefore, angle of dip ($\delta$) = 90⁰
The angle that a magnetic needle makes with the horizontal plane at any specific location is dip angle.