Question

In the figure shown there are two semicircles of radii $r _ { 1 }$ and $r _ { 2 }$ in which a current i is flowing. The magnetic induction at the centre O will be

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

Answer 1

Answer: (C)

$\frac { \mu _ { 0 } i } { 4 } \left( \frac { r _ { 1 } + r _ { 2 } } { r _ { 1 } r _ { 2 } } \right)$

Answer 2

The magnetic induction due to both semicircular parts will be in the same direction perpendicular to the paper inwards.

$\therefore B = B _ { 1 } + B _ { 2 } = \frac { \mu _ { 0 } i } { 4 r _ { 1 } } + \frac { \mu _ { 0 } i } { 4 r _ { 2 } } = \frac { \mu _ { 0 } i } { 4 } \left( \frac { r _ { 1 } + r _ { 2 } } { r _ { 1 } r _ { 2 } } \right) \otimes$