Step 1: Defining the Positions.
The axial position (End-on position) refers to a point located on the line passing through the two poles of the magnet. The equatorial position (Broadside-on position) refers to a point located on the perpendicular bisector of the magnetic axis.
Step 2: Analyzing Magnetic Field Direction.
At the axial position, the magnetic field direction is along the direction of the magnetic dipole moment (from South to North pole). At the equatorial position, the magnetic field direction is opposite to the direction of the magnetic dipole moment (from North to South pole).
Step 3: Comparing Field Magnitude.
For a short bar magnet at the same distance ($r$), the magnetic field strength at the axial position ($B_{axial}$) is exactly twice the strength of the field at the equatorial position ($B_{equatorial}$). Mathematically:
$$B_{axial} = 2 \times B_{equatorial}$$