The magnetic field due to a current carrying circular loop of radius $$3\,cm$$  at a point on the axis at a distance of $$4\,cm$$  from the centre is $$54\,\mu T.$$  What will be its value at the centre of loop ?

Solution :
$$\eqalign{ & B = \frac{{{\mu _0}i{a^2}}}{{2{{\left( {{x^2} + {a^2}} \right)}^{\frac{3}{2}}}}} \cr & B' = \frac{{{\mu _0}i}}{{2a}} = \frac{{{\mu _0}i{a^2}}}{{2a{{\left( {{x^2} + {a^2}} \right)}^{\frac{3}{2}}}}}\left( {\frac{{{{\left( {{x^2} + {a^2}} \right)}^{\frac{3}{2}}}}}{{{a^2}}}} \right) \cr & B' = \frac{{B.{{\left( {{x^2} + {a^2}} \right)}^{\frac{3}{2}}}}}{{{a^3}}} \cr & {\text{Put}}\,\,x = 4\,\& \,a = 3 \Rightarrow B' = \frac{{54\left( {{5^3}} \right)}}{{3 \times 3 \times 3}} = 250\mu T \cr} $$