A compass needle which is allowed to move in a horizontal plane is taken to a geomagnetic pole. It:

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1. **Earth's Magnetic Field at Poles:** At the geomagnetic poles, the magnetic field lines of the Earth are directed vertically (perpendicular to the surface). This means the angle of dip (or inclination, $I$ or $\delta$) is $90^\circ$ .
2. **Horizontal Component:** The horizontal component of the Earth's magnetic field ($H_E$) is related to the total magnetic field ($B$) and the angle of dip by the equation: 
   $$ H_E = B \cos \delta $$
   At the poles, since $\delta = 90^\circ$, $H_E = B \cos 90^\circ = 0$. Thus, the horizontal component of the Earth's magnetic field is zero.
3. **Effect on Compass:** A compass needle is designed to move in a horizontal plane and aligns itself with the horizontal component of the magnetic field. The torque ($	au$) exerted on the needle is given by $	au = m H_E \sin 	heta$. Since $H_E = 0$, the torque is zero regardless of the orientation ($	heta$) of the needle.
4. **Conclusion:** With no restoring torque to align it to a specific direction, the compass needle is in neutral equilibrium and will remain in any position in which it is placed.

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