A current I flows through an infinitly long wire having infinite brands as shown. The radius of the first curved section is a and the radii of the successive curved portions each increases by a factor `eta`. Find magnetic field at O.

Explanation

The magnetic field at O due to arc AB is `B_(1)=(mu_(0)itheta)/(4pia)odot` The field at O due to arc CD and other arcs are `B_(2)=(mu_(0)itheta)/(4pietaa),B_(3)=(mu_(0))/(4pi)(itheta)/(eta^(2)a)` ,………. Etc
Therefore resultant field is `B=B_(1)+B_(2)+.......`
`B=(mu_(0)i)/(4pia)[1-1/eta+1/(eta^(2))-1/(eta^(3))......]`
The resultant field is the sum of above fields which is a geometric progression with common ratio `(-1)/eta`
Therefore resultant field is `B=((mu_(0)itheta)/(4pia))(1+1/eta)=(mu_(0)ithetaeta)/(4pia(eta+1))`