Question

An infinitely long straight conductor is bent into the shape as shown in the figure. It carries a current of i ampere and the radius of the circular loop is r metre. Then the magnetic induction at its centre will be

• A. $\frac { \mu _ { 0 } } { 4 \pi } \frac { 2 i } { r } ( \pi + 1 )$
• B. $\frac { \mu _ { 0 } } { 4 \pi } \frac { 2 i } { r } ( \pi - 1 )$
• C. Zero
• D. Infinite

Answer 1

Answer: (B)

$\frac { \mu _ { 0 } } { 4 \pi } \frac { 2 i } { r } ( \pi - 1 )$

Answer 2

The given shape is equivalent to the following diagram

The field at $O$ due to straight part of conductor is$B _ { 1 } = \frac { \mu _ { o } } { 4 \pi } \cdot \frac { 2 i } { r }$◉. The field at $O$due to circular coil is $B _ { 2 } = \frac { \mu _ { 0 } } { 4 \pi } \cdot \frac { 2 \pi i } { r } \otimes$. Both fields will act in the opposite direction, hence the total field at O.

i.e. $B = B _ { 2 } - B _ { 1 } = \left( \frac { \mu _ { o } } { 4 \pi } \right) \times ( \pi - 1 ) \frac { 2 i } { r } = \frac { \mu _ { o } } { 4 \pi } \cdot \frac { 2 i } { r } ( \pi - 1 )$