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Circular current loops produce some of the most frequently tested field configurations in JEE. At the center of a single loop of radius $R$: $B = \mu_0 I/(2R)$. For $N$ turns: $B = \mu_0 NI/(2R)$. For an arc subtending angle $\theta$ (radians): $B = \mu_0 I\theta/(4\pi R)$ — this is the fraction $\theta/(2\pi)$ of the full loop field.

On the axis at distance $x$: $B = \mu_0 IR^2/(2(R^2+x^2)^{3/2})$. This reduces to $\mu_0 I/(2R)$ at the center and approximates the magnetic dipole field $\mu_0 m/(2\pi x^3)$ (where $m = I\pi R^2$) at large distances. The axial field is always directed along the axis.

Critical problem-solving shortcuts: (1) Straight wire segments passing through the center contribute zero field ($d\vec{l} \parallel \hat{r}$). (2) For composite shapes, decompose into arcs and lines, compute each separately, then add vectorially. (3) Concentric coplanar loops — fields add or subtract based on current directions. (4) Winding the same wire into $N$ turns of radius $R/N$ multiplies the field by $N^2$. The right-hand rule determines direction: curl fingers along current, thumb gives field direction.