In end-on and broadside-on positions of a deflection magnetometer, if θ₁ and θ₂ are the deflections produced by short magnets at equal distances, then (tan θ₁) / (tan θ₂) is:
(A) 2:1
(B) 1:2
(C) 1:1
(D) None of these

In the case of a deflection magnetometer, when a short magnet is placed at equal distances from the magnetometer in two different positions—end-on and broadside-on—the deflections measured at each position depend on the magnetic field produced by the magnet.
Key Points:
• End-on position: In this position, the axis of the magnet is aligned along the line joining the magnet and the deflection magnetometer. The magnetic field strength produced by the magnet at the position of the magnetometer is strongest, and the deflection produced will be large.
• Broadside-on position: In this position, the axis of the magnet is perpendicular to the line joining the magnet and the deflection magnetometer. The magnetic field produced at the magnetometer is weaker compared to the end-on position, resulting in a smaller deflection.
The relationship between the deflections and the position of the magnet can be understood by the following principles:
• When the magnet is in the end-on position, the magnetic field strength BB at the location of the magnetometer is proportional to the distance from the magnet and the magnetic moment of the magnet.
• In the broadside-on position, the magnetic field strength BB at the location of the magnetometer is weaker than in the end-on position, and it also depends on the distance and the orientation of the magnetic moment.
Mathematically, the magnetic field at a distance rr from the center of a short magnet is given by:
• In the end-on position: Bend∝1r3B_{\text{end}} \propto \frac{1}{r^3}
• In the broadside-on position: Bside∝1r3B_{\text{side}} \propto \frac{1}{r^3}
Since the field strength is proportional to the inverse of the cube of the distance, but with different orientations for each position, we can say that the ratio of the deflections (θ1\theta_1 for end-on and θ2\theta_2 for broadside-on) will relate to the magnetic field strength inversely.
By the relation between deflection and magnetic field strength:
tan⁡θ1∝Bend\tan \theta_1 \propto B_{\text{end}} tan⁡θ2∝Bside\tan \theta_2 \propto B_{\text{side}}
Since the magnetic field is stronger in the end-on position, we find:
tan⁡θ1tan⁡θ2=2:1\frac{\tan \theta_1}{\tan \theta_2} = 2:1
Thus, the correct answer is: (A) 2:1