Question

If $B_c$ is the magnetic induction at the centre of a circular coil carrying current, then the magnetic induction at a point on the axis of the coil at a distance equal to the radius of the coil is

• A. $\frac{B_c}{2\sqrt{2}}$
• B. $\frac{B_c}{2}$
• C. $\frac{B_c}{4}$
• D. $\frac{B_c}{\sqrt{2}}$
• E. $\frac{B_C}{8}$

Answer 1

Answer: (A)

At a point situated at a distance $r$ from the centre of a current carrying circular coil along the axial line, the magnetic field is given as
$B=\frac{{\mu }_{0}i{R}^2}{2{\left(R^2+r^2\right)}^{3/2}}$
where, $i$ is the current in the coil and $R$ is the radius of the coil.
Given, $r=R$
$\Rightarrow B=\frac{{\mu }_{0}i{R}^2}{2{\left(R^2+R^2\right)}^{3/2}}$
$=\frac{{\mu }_{0}i}{2\cdot 2^{3/2}R}\dots \text{(i) }$
Since, magnetic field due to a current carrying circular coil at centre is given as
$B=\frac{{\mu }_{0}i}{2R}=B_c\text{ (given) ...(ii) }$
From Eqs. (i) and (ii), we get
$B=\frac{B_c}{2\sqrt{2}}$