Important to note that, The magnetic field of a n-side polygon carrying current I is given by 

`B=(nmu_0I)/(2pid)sin(pi/n)`

Where d is the perpendicular distance from the each side.

Problem: A current I flows in the anticlockwise direction through a square loop of side a lying in the xoy plane with its center at the origin. The magnetic induction at the center of the square loop is: [GATE 2001]

(A)`(2sqrt2mu_0I)/(pia)hate_x`,    ✅(B)`(2sqrt2mu_0I)/(pia)hate_z`,     (C)`(2sqrt2mu_0I)/(pia^2)hate_z`,    (D)`(2sqrt2mu_0I)/(pia^2)hate_x`

Answer:

The square in the x-y pane so the magnetic field in z axis direction.

The magnetic field at the center of the square is 

`vecB=(4mu_0I)/(2pid)sin(pi/4)odot`

or, `vecB=(2sqrt2mu_0I)/(pia)odot`

or, `vecB=(2sqrt2mu_0I)/(pia)hate_z`

Option B  is correct.

Problem: A conducting wire is in the shape of a regular hexagon, which is inscribed inside an imaginary circle of radius R, as shown. A current I flows through the wire. The magnitude of the magnetic field at the centre of the circle is [JEST 2016]

(A)`(sqrt3mu_0I)/(2piR)`,    (B)`(mu_0I)/(2sqrt3piR)`,    ✅ (C) `(sqrt3mu_0I)/(piR)`,     (D) `(3mu_0I)/(2piR)`