Axial And Equatorial Fields Of A Bar Magnet

A short bar magnet behaves as a magnetic dipole of moment $m$. At a distance $r$ on the axis (the end-on position), the field is $B_{\text{axial}}=\frac{{\mu }_0}{4\pi }\frac{2m}{r^3}$. At the same distance on the equatorial line (the broadside position), the field is $B_{\text{equatorial}}=\frac{{\mu }_0}{4\pi }\frac{m}{r^3}$. Taking the ratio, the common factors $\frac{{\mu }_0}{4\pi r^3}$ and $m$ cancel, leaving $\frac{B_{\text{axial}}}{B_{\text{equatorial}}}=\frac{2}{1}$. So the axial field is exactly twice the equatorial field at equal distances. The option $1:2$ inverts the ratio. The option $1:1$ ignores the factor of two that distinguishes the two positions. The option $4:1$ overstates the difference, perhaps confusing the dipole result with an inverse-square scaling. This relationship is directly stated in the NCERT magnetism chapter and parallels the electric dipole. A direction check also confirms the axial field points along the dipole moment while the equatorial field points opposite to it, but their magnitude ratio remains $2:1$. This factor-of-two asymmetry mirrors exactly the electric dipole result and reflects the underlying dipole structure shared by bar magnets and small current loops alike. In the laboratory, a deflection magnetometer exploits these end-on and broadside positions, since placing a magnet in the so-called tan-A and tan-B positions relative to a compass needle lets one compare magnetic moments and even measure the horizontal component of the Earth's field.