At what distance on the axis of a current loop is the magnetic field half the strength of the field at the center of the loop? Give your answer as a multiple of R.

we should describe at what distance on the axis of a current loop is the magnetic field half the strength of the field at the center of the loop? Give your answer as a multiple of R.

Each segment on the loop applies a magnetic field, So the total magnetic field exerted by bone loop at any point z is given by

$B_{loop}=\frac{{\mu }_o}{2}\frac{I{R}^2}{{\left(z^2+R^2\right)}^{3/2}}$ $\left(1\right)$

At the center of the loop we have $z<<Rorz=0$so the magnetic field at the center is

$B_{center}=\frac{{\mu }_{o}I}{2R}$ $\left(2\right)$

Our target is to find z when the magnetic field of the loop is half the magnetic field at the center

$B_{center}=\frac{1}{2}{B}_{loop}$ $\left(3\right)$

We use the expression of each magnetic fie field and find z in terms of R by

$$\begin{gathered} B_{center}=\frac{1}{2}{B}_{loop} \\ \frac{{\mu }_{o}I}{2R}=\frac{1}{2}\left(\frac{{\mu }_o}{2}\frac{I{R}^2}{{\left(z^2+R^2\right)}^{3/2}}\right) \\ 2R^3={\left(z^2+R^2\right)}^{3/2} \\ {\left(2R^3\right)}^{\frac{2}{3}}=\left(z^2+R^2\right) \\ 1.58R^{2=}{z}^2+R^2 \\ {z}^2=0.58R^2 \\ z=0.76R \end{gathered}$$