Magnetic Field At The Centre Of A Circular Coil

For a tightly wound circular coil, each turn contributes equally at the centre, so the Biot-Savart result for one loop is multiplied by the number of turns: $B=\frac{{\mu }_{0}NI}{2R}$. The field is uniform in direction along the axis at the centre and adds up turn by turn. Substituting $N=50$, $I=2\text{A}$, $R=0.10\text{m}$ and ${\mu }_0=4\pi \times 10^{-7}$, we obtain $B=\frac{(4\pi \times 10^{-7})(50)(2)}{2(0.10)}=\frac{4\pi \times 10^{-5}}{0.2}=6.28\times 10^{-4}\text{T}$. The option $1.26\times 10^{-5}\text{T}$ omits the factor of $N$, treating it as a single turn. The option $3.14\times 10^{-4}\text{T}$ mistakenly uses $4R$ in the denominator. The option $2.50\times 10^{-3}\text{T}$ confuses this geometry with a solenoid formula. This relation is the standard NCERT expression for the centre of a current loop, distinct from the axial-point formula. A magnitude check shows $10^{-4}\text{T}$, reasonable for fifty turns carrying a few amperes. Physically, this turn-by-turn addition is what makes coils so useful as compact field sources in instruments and electromagnets. Because the centre-of-loop field scales inversely with radius, a smaller coil with the same current and turns produces a stronger central field, a trade-off engineers exploit when designing deflection coils and laboratory magnets. The direction of the central field again follows the right-hand rule applied to the circulating current around the loop.