Refer to figure, where `c=3a` and `altbltc`, the magnetic field induction at point O will be zero due to current I in the various portion of the network when

Options

A. `b=5/7a`
B. `b=(7a)/(9)`
C. `b=(9a)/(13)`
D. `b=(13a)/(15)`

Explanation

Magnetic field induction at O will be
`vecB=(mu_0)/(4pi)I/axx(3pi)/(2)hatn+(mu_0)/(4pi)I/b(3pi)/(2)(-hatn)+(mu_0)/(4pi)I/c2pi(hatn)`
As per question, `B=0`.
`:. 0=(mu_0)/(4pi)I/a(3pi)/(2)-(mu_0)/(4pi)I/b(3pi)/(2)+(mu_0)/(4pi)I/c2pi`
or `0=(mu_0)/(4pi)Ipi[(3)/(2a)-(3)/(2b)+(2)/(c)]`
or `(3)/(2a)-(3)/(2b)+(2)/(3a)=0` (as `c=3a`)
or `(3)/(2a)+(2)/(3a)=(3)/(2b)`
On solving, we get , `b=(9a)/(13)`