Question

Radius of current carrying coil is ${}^{\prime }{R}^{\prime }$. If fractional decreases in field value with respect to centre of the coil for a near by axial point is $1$% then find axial position of that point.

Answer 1

Fractional decrease in magnetic field along the axis at a distance $x$ from the center of coil is $1$%.

Magnetic field on the axis $\frac{{\mu }_{o}i{R}^2}{2{(x^2+R^2)}^{3/2}}$

Magnetic field on the center, $B_{center}=\frac{{\mu }_{o}i}{2R}$

$\Delta B=1$%$B_{center}$

$\frac{{\mu }_{o}i}{2R}-\frac{{\mu }_{o}i{R}^2}{2{(x^2+R^2)}^{3/2}}=\frac{1}{100}\frac{{\mu }_{o}i}{2R}$

$0.99\frac{{\mu }_{o}i}{2R}=\frac{{\mu }_{o}i{R}^2}{2{(x^2+R^2)}^{3/2}}$

$\frac{R^3}{0.99}={(x^2+R^2)}^{3/2}$

${\left(\frac{R^3}{0.99}\right)}^{2/3}=x^2+R^2$

$x^2=\frac{R^2}{{0.99}^{2/3}}-R^2$

$x=\pm 0.082R$

At distance $x=\pm 0.082R$ from center Magnetic field change is $1$% with respect to magnetic field at center.