Question

Consider a circular loop of radius R on the xy-plane carrying a steady current anticlockwise. The magnetic field at the center of the loop is given by

• A. $\frac{\mu_0}{2R}i \hat{x}$
• B. $\frac{\mu_0}{2R} i \hat{y}$
• C. $\frac{\mu_0}{2R} i \hat{z}$
• D. $\frac{\mu_0}{R} i \hat{x}$
• E. $\frac{\mu_0}{R} i \hat{y}$

Answer 1

Answer: (C)

A circular loop of radius $R$ and current $I$ is shown in the figure below

From Biot-savart law, the magnetic field at some point in space at distance R is given as,
$d B=\frac{\mu_{0}}{4 \pi} i \frac{ d l \times P }{R^{3}}$
Since, the loop is circular in shape so,
$=2 \pi R$
Now integrating the field in whole length of wire loop
$\Rightarrow \int_\limits{0}^{B} d B=\frac{\mu_{0}}{4 \pi} \frac{i R}{R^{3}} \int_\limits{0}^{2 \pi R} d l $
$\Rightarrow B=\frac{\mu_{0}}{4 \pi} \frac{i}{R^{2}} 2 \pi R=\frac{\mu_{0} i}{2 R}$
Also, with help of right hand thumb rule, we can conclude, that the magnetic field is in $+ z$ direction.
$\Rightarrow B =\frac{\mu_{0} i}{2 R} \hat{ z }$