Magnetic Field On The Axis Of A Loop

The magnetic field on the axis of a circular loop of radius $R$ at distance $x$ from the centre is $B=\frac{{\mu }_{0}I{R}^2}{2(R^2+x^2{)}^{3/2}}$. At the centre, $x=0$, this reduces to $\frac{{\mu }_{0}I}{2R}$, the maximum value. As $x$ grows, the denominator increases, so the field decreases monotonically along the axis. Far from the loop, where $x\gg R$, the expression approaches $B\approx \frac{{\mu }_{0}I{R}^2}{2x^3}$, which is the characteristic inverse-cube falloff of a magnetic dipole. The option that the field increases with distance contradicts this monotonic decrease. The option that it stays constant ignores the $x$-dependence entirely. The option of inverse-square falloff would describe a magnetic monopole, which does not exist; magnetic dipoles fall as inverse cube. NCERT derives this axial field and identifies the dipole limit. A quick check at $x=0$ recovers the familiar centre-of-loop result, confirming the formula's consistency. This inverse-cube behaviour at large distances is the unmistakable signature of a magnetic dipole and contrasts sharply with the inverse-square falloff of an electric point charge. Its physical root is that isolated magnetic poles do not exist; the loop necessarily has two effective poles whose fields nearly cancel at large distances, leaving only the slower-cancelling dipole term. Recognising this falloff helps students estimate fields without recomputing the full Biot-Savart integral.