Helmholtz coil

Problem

Two identical circular coils of radius $R = 10$ cm, each with $N = 100$ turns, are separated on the axis by a distance equal to $R$, and carry the same current $I = 1$ A in the same direction. Calculate the magnitude of the field at the midpoint on the axis, using for each coil the formula $B_1 = \mu_0 N I R^2/[2(R^2+z^2)^{3/2}]$.

Solution

At the midpoint each coil is at a distance $z=R/2=0{,}05$ m from the point $P$. The field of a single coil is: $B_1=\frac{\mu_0 N I R^2}{2(R^2+z^2)^{3/2}}$

With $R^2=0{,}01\;\text{m}^2$ and $z^2=0{,}0025\;\text{m}^2$: $R^2+z^2=0{,}0125\;\text{m}^2,\qquad (0{,}0125)^{3/2}\approx1{,}40\cdot10^{-3}$

Numerator: $\mu_0 N I R^2=4\pi\cdot10^{-7}\cdot100\cdot1\cdot0{,}01=1{,}257\cdot10^{-6}$

So for one coil: $B_1=\frac{1{,}257\cdot10^{-6}}{2\cdot1{,}40\cdot10^{-3}}\approx4{,}5\cdot10^{-4}\;\text{T}$

The two coils, with currents in the same direction, add up their contributions: $B=2B_1\approx9\cdot10^{-4}\;\text{T}$

$\ev{B \approx 0{,}9\;\text{mT}}$

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Topics: Magnetismo Concepts: Campo magnetico