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A Helmholtz coil has pair of loops, each with $N$ turns and radius $R$. They are placed coaxially at distance $R$ and the same current $I$ flows through the loops in the same direction. The magnitude of magnetic field at $P$, midway between the centres $A$ and $C$, is given by (Refer to figure):
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$\frac{8 N \mu_0 I}{5^{3 / 2} R}$
$\frac{8 N \mu_0 I}{5^{3 / 2} R}$
Point $P$ is situated at the mid-point of the line joining the centres of the circular wires which have same radii (R). The magnetic fields $(\vec{B})$ at $\mathrm{P}$ due to the currents in the wires are in same direction.
Magnitude of magnetic field at point, $P$
$B=2\left\{\frac{\mu_0 N I R^2}{2\left(R^2+\frac{R^2}{4}\right)^{3 / 2}}\right\}$
$=\frac{\mu_0 N I R^2}{\frac{5^{3 / 2}}{8}}=\frac{8 \mu_0 N I}{5^{3 / 2} R}$
Magnitude of magnetic field at point, $P$
$B=2\left\{\frac{\mu_0 N I R^2}{2\left(R^2+\frac{R^2}{4}\right)^{3 / 2}}\right\}$
$=\frac{\mu_0 N I R^2}{\frac{5^{3 / 2}}{8}}=\frac{8 \mu_0 N I}{5^{3 / 2} R}$
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