Question

In the loop shown, the magnetic induction at the point 'O' is

A. $\dfrac{{{\mu _0}I}}{8}\left( {\dfrac{{{R_1} - {R_2}}}{{{R_1}{R_2}}}} \right)$
B. $\dfrac{{{\mu _0}I}}{8}\left( {\dfrac{{{R_1} + {R_2}}}{{{R_1}{R_2}}}} \right)$
C. $\dfrac{{{\mu _0}I}}{8}\left( {\dfrac{{{R_1}{R_2}}}{{{R_1} + {R_2}}}} \right)$
D. Zero

Answer 1

Hint: In this question we will place the Biot-Savart law. Through which we can conclude that the magnetic field along the line of a straight current carrying conductor is zero. Then we need to know the magnetic fields due to sections AB, CD, EF & FG are zero at point O. Then by placing the formula of Magnetic field at the center of a current carrying loop we can find our basic answer.

Complete step-by-step answer:

The magnetic field in the lines of a straight current conveying conductor may be concluded with Biot-Savart 's rule to be zero.
Hence, magnetic fields due to sections AB, CD, EF & FG are zero at point O.
Magnetic field at the center of a current carrying loop is represented by:
$\Rightarrow B = \dfrac{{{\mu _0}I}}{{2R}}$
Now, the magnetic field due to the section BC would be
$\Rightarrow {B_{BC}} = \dfrac{1}{4} \times \dfrac{{{\mu _0}I}}{{2R}}$ Inside the plane of paper (by using right hand thumb rule)
We can see that the factor $\dfrac{1}{4}$ appears as only one fourth of the total circumference contributes to the magnetic field.
Now similarly, magnetic field due to section DE would be
$\Rightarrow {B_{DE}} = \dfrac{{{\mu _0}I}}{{8R}}$Which is inside the plane of paper (by using right hand thumb rule)
Now the total magnetic field at point O is
$\Rightarrow B = {B_{AB}} + {B_{BC}} + {B_{CD}} + {B_{DE}} + {B_{EF}} + {B_{FG}}$
$\Rightarrow B = \dfrac{{{\mu _0}I}}{{8{R_1}}} + \dfrac{{{\mu _0}I}}{{8{R_2}}}$ Inside the plane of paper
$\Rightarrow B = \dfrac{{{\mu _0}I}}{8}\left( {\dfrac{{{R_1} + {R_2}}}{{{R_1}{R_2}}}} \right)$
Hence, the magnetic induction at the point 'O' is calculated to be-
$\Rightarrow B = \dfrac{{{\mu _0}I}}{8}\left( {\dfrac{{{R_1} + {R_2}}}{{{R_1}{R_2}}}} \right)$
Thus, option B will be the correct option.

Note- The development of an electromotive (i.e. voltage) force by an electrical conductor in a shifting magnetic field is electromagnetic or magnetic induction. The theory of induction by Michael Faraday in 1831 was widely attributed and James Clerk Maxwell mathematically represented it as the rule of induction by Faraday.