The magnetic field at the center of a current carrying loop of radius \( 0.1 \mathrm{~m} \) is \( 5 \sqrt{5} \) times that at
a point along its axis. The distance of this point from the centre of the loop is

Magnetic field at the centre of current carrying loop is given \(\operatorname{as} B_{C}=\frac{\mu_{0}}{2} \frac{I}{r}\)
Magnetic field at the axis is given as \(B_{A}=\frac{\mu_{0}}{2} \frac{I r^{2}}{\left(\chi^{2}+r^{2}\right)^{3 / 2}}\)
Given, \(B_{C}=5 \sqrt{5} B_{A}\)
\(\Rightarrow \frac{\mu_{0}}{2} \frac{I}{r}=5 \sqrt{5} \frac{\mu_{0}}{2} \frac{I r^{2}}{\left(\chi^{2}+r^{2}\right)^{3 / 2}}\)
\(\Rightarrow\left(x^{2}+r^{2}\right)^{3 / 2}=r^{3} 5 \sqrt{5}\)
\(\Rightarrow\left(x^{2}+r^{2}\right)^{3 / 2}=r^{3} \times 5^{3 / 2}\)
\(\Rightarrow \chi^{2}+r^{2}=r^{2} \times 5\)
\(\Rightarrow \chi^{2}=5 r^{2}-r^{2}\)
\(\Rightarrow \chi^{2}=4 r^{2}\) or \(x=2 r\)
Given, \(r=0.1 \mathrm{~m}\)
\(\Rightarrow \chi=0.2 m\)