Question

The ratio of the magnetic field at the centre of a current carrying circular wire and the magnetic field at the centre of a square coil made from the same length of wire will be

• A. $\frac { \pi ^ { 2 } } { 4 \sqrt { 2 } }$
• B. $\frac { \pi ^ { 2 } } { 8 \sqrt { 2 } }$
• C. $\frac { \pi } { 2 \sqrt { 2 } }$
• D. $\frac { \pi } { 4 \sqrt { 2 } }$

Answer 1

Answer: (B)

$\frac { \pi ^ { 2 } } { 8 \sqrt { 2 } }$

Answer 2

Circular coil Square coil

Length L = 2π r Length L = 4a

Magnetic field $B = \frac { \mu _ { 0 } } { 4 \pi } \cdot \frac { 2 \pi i } { r }$ $= \frac { \mu _ { 0 } } { 4 \pi } \cdot \frac { 4 \pi ^ { 2 } i } { r }$

$B = \frac { \mu _ { 0 } } { 4 \pi } \cdot \frac { 2 \sqrt { 2 } i } { a }$ $B = \frac { \mu _ { 0 } } { 4 \pi } \cdot \frac { 8 \sqrt { 2 } i } { a }$

Hence $\frac { B _ { \text {circular } } } { B _ { \text {square } } } = \frac { \pi ^ { 2 } } { 8 \sqrt { 2 } }$