Question

Find magnetic field b at the centre of square loop of side a carrying a current i

Answer 1

each side of the square makes angle of 90° at its centre.
so, alpha = beta = 45°.

Answer 2

"According to the given data, the centre of the square loop will have an angle of 90 degree.

sin $\alpha$ and $\beta$ will be 45 degree as the diagonal from the two corners will divide the angle equally.

Now, as each side of length of a will be conducting the current in the square loop, it will be multiplied by 4 and thereby we get B .

$B\quad =\quad 4\quad \times \quad \frac { { \mu  }_{ \o  }i }{ 4\pi a } \left( sin\quad \alpha \quad +\quad sin\quad \beta  \right)$

$B\quad =\quad \frac { { \mu  }_{ \o  }i }{ \pi a } \left( sin\quad 45^{ \circ  }+sin\quad 45^{ \circ  } \right)$

$=\quad \frac { { \mu  }_{ \o  }i }{ \pi a } \left( \frac { 1 }{ \sqrt { 2 }  } +\frac { 1 }{ \sqrt { 2 }  }  \right)$

$B\quad =\quad \frac { \sqrt { 2 } { \mu  }_{ \o  }i }{ \pi a }$"