A current I is flowing in an octagonal coil of side a figure. The magnetic field induction at the centre O of the coil will be

Options

A. `(sqrt5mu_0I)/(2pia)`
B. `(mu_0I)/(sqrt5pia)`
C. `(5sqrt2mu_0I)/(pia)`
D. `(5mu_0I)/(4pia)`

Answer

D

Explanation

The angle subtended by each side of octagonal at centre O is `45^@`, figure. So
`phi_1=22.5^@` and `phi_2=22.5^@`

Here, `r=OJ=BJ cot 22.5^@`
`=(a)/(2tan 22.5^@)=(a)/(2xx0*4142)`
Magnetic field induction at O due to current through all the eight sides of an octagonal will be
`H=8xx(mu_0)/(4pi)I/r(sin phi_1+sin phi_2)`
`=8xx(mu_0)/(4pi)(I)/((a//2xx0*4142))[sin 22.5^@+sin 22.5^@]`
`=8xx(mu_0)/(4pi)xxI/axx2xx0*4145xx2sin 22.5^@`
`=(5mu_0I)/(4pia)`