Question

The magnetic field at the centre of a circular coil of radius $r$, due to current $I$ flowing through it, is $B$. The magnetic field at a point along the axis at a distance $\frac{r}{2}$ from the centre is:

• A. $B/2$
• B. $2B$
• C. ${\left(\frac{2}{\sqrt{5}}\right)}^{3}B$
• D. ${\left(\frac{2}{\sqrt{3}}\right)}^{3}B$

Answer 1

Answer: (C)

$B_C=\frac{{\mu }_{0}I}{2r},$
$B_a=\frac{{\mu }_{0}I{r}^2}{2{\left(x^2+r^2\right)}^{3/2}}$
At $x=\frac{r}{2}$
$B_a=\frac{{\mu }_{0}I{r}^2}{2{\left(\frac{r^2}{4}+r^2\right)}^{3/2}}$
$=\frac{{\mu }_{0}I{r}^2}{2{\left(\frac{5}{4}{r}^2\right)}^{3/2}}=\frac{{\mu }_{0}I}{2r}{\left(\frac{4}{5}\right)}^{3/2}$
$=\frac{{\mu }_{0}I}{2r}{\left(\frac{2}{\sqrt{5}}\right)}^3$