Q125 / 125 · BITSAT 2017 · Physics

What is the magnetic field at the centre of arc in the figure below?

A
$\frac{μ_{0}}{4 π} \cdot \frac{2 I}{r} [\sqrt{2} + π]$
Correct answer
B
$\frac{μ_{0}}{4 π} \cdot \frac{2 I}{r} [\sqrt{2} + \frac{π}{4}]$
C
$\frac{μ_{0}}{4 π} \cdot \frac{I}{r} [\sqrt{2} + π]$
D
$\frac{μ_{0}}{4 π} \cdot \frac{I}{r} [\sqrt{2} + \frac{π}{4}]$

Solution

Distance of straight path from $O$

$= \frac{r}{\sqrt{2}}$

Hence, $B = B_{1} + B_{2}$

$= \frac{μ_{0}}{4 π} \cdot \frac{2 I}{\frac{r}{\sqrt{2}}} + \frac{μ_{0}}{4 π} \frac{2 π I}{r} (\frac{1}{4})$

$= \frac{μ_{0}}{4 π} \cdot \frac{2 I}{r} [\sqrt{2} + \frac{π}{4}]$

Hence, the correct answer is option: 'B'.