Question

Math List I with List II

List I (Current configuration)		List II(Magnetic field at point O)
A		I	$B_0=\frac{{\mu }_{0}I}{4\pi r}[\pi +2]$
B		II	$B_0=\frac{{\mu }_0}{4}\frac{I}{r}$
C		III	$B_0=\frac{{\mu }_{0}I}{2\pi r}[\pi -1]$
D		IV	$B_0=\frac{{\mu }_{0}I}{4\pi r}[\pi +1]$

Choose the correct answer from the option given below:

• A. A-III, B-IV, C-I, D-II
• B. A-I, B-III, C-IV, D-II
• C. A-III, B-I, C-IV, D-II
• D. A-II, B-I, C-IV, D-III

Answer 1

Answer: (C)

(A)
$B_{ab}=\frac{{\mu }_0}{4\pi }\frac{I}{r}$ (out of the plane)
$B_{bcd}=\frac{{\mu }_0}{4\pi }\frac{I}{r}(2\pi )$ (in the plane)
$B_{de}=\frac{{\mu }_0}{4\pi }\frac{I}{r}$ (out of the plane)
Hence magnetic field at $O$ is
$B_0=-\frac{{\mu }_0}{4\pi }\frac{I}{r}+\frac{{\mu }_0}{4\pi }\frac{I}{r}(2\pi )-\frac{{\mu }_0}{4\pi }\frac{I}{r}$
$B_0=\frac{{\mu }_0}{2\pi }\frac{I}{r}(\pi -1)\dots \dots ..\text{ (III) }$

$B_{ab}=\frac{{\mu }_0}{4\pi }\frac{I}{r}$ (out of the plane)
$B_{bcd}=\frac{{\mu }_0}{4\pi }\frac{I}{r}(\pi )$ (out of the plane)
$B_{de}=\frac{{\mu }_0}{4\pi }\frac{I}{r}$ (out of the plane)
Hence magnetic field at $O$ is
$B_0=\frac{{\mu }_0}{4\pi }\frac{I}{r}+\frac{{\mu }_0}{4\pi }\frac{I}{r}(\pi )+\frac{{\mu }_0}{4\pi }\frac{I}{r}$
$B_0=\frac{{\mu }_0}{4\pi }\frac{I}{r}(\pi +2)\dots \text{ (I) }$

$B_{ab}=\frac{{\mu }_0}{4\pi }\frac{I}{r}$ (in the plane)
$B_{bcd}=\frac{{\mu }_0}{4\pi }\frac{I}{r}(\pi )$ (in the plane)
$B_{de}=0$ (at the axis)
Hence magnetic field at $O$ is
$B_0=\frac{{\mu }_0}{4\pi }\frac{I}{r}(1+\pi )\dots \text{ (IV) }$

$B_{ab}=0$ (at the axis)
$B_{bcd}=\frac{{\mu }_0}{4\pi }\frac{I}{r}(\pi )$ (out of the plane)
$B_{de}=0$ (at the axis)
Hence magnetic field at $O$ is
$B_0=\frac{{\mu }_{0}I}{4r}\dots$