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Magnetic fields obey the superposition principle: the total field at any point is the vector sum of fields from all sources. This is the foundation for solving composite wire problems — JEE's favorite format in this chapter.

Strategy: (1) Decompose the wire shape into standard segments (straight lines, circular arcs). (2) Calculate each segment's field independently using standard formulas. (3) Determine directions using the right-hand rule. (4) Add vectorially (often all components point in the same direction for coplanar configurations).

Critical shortcuts: straight segments passing through the point of interest contribute zero. Semi-infinite segments from the point contribute $\mu_0 I/(4\pi d)$. Concentric semicircles of radii $R_1$ and $R_2$: fields add if currents create same-direction fields, subtract otherwise. Two parallel wires at the midpoint: fields cancel for same-direction currents, add for opposite-direction currents.

For field at the center of complex shapes: the answer almost always reduces to a combination of arc contributions and possibly semi-infinite wire contributions. Radial segments contribute nothing. The trick is correctly identifying which segments are arcs (use $\mu_0 I\theta/(4\pi R)$), which are straight through the center (zero), and which are semi-infinite ($\mu_0 I/(4\pi d)$).