A battary is connected between two points A and B on the circumference of a uniform conducting ring of radius r and resistance R as shown in Fig. One of the arcs AB of the ring subtends angle `theta` at the centre. Show that that the magnetic field at the centre of the coil is zero and independent `theta`.

Explanation

(a) As the field due to arc at the centre is given by
`B=(mu_(0))/(4pi)(iphi)/(r)`
`therefore B=(mu_(0))/(4pi)(i_(1)theta)/(r)ox+(mu_(0))/(4pi)(i_(2)(2pi-theta))/(r)odot`
But `(V_(A)-V_(B))=i_(1)R_(1)=i_(2)B_(2)`
or, `i_(2)=i_(1)(R_(1))/(R_(2))=i_(1)(L_(1))/(L_(2))" "[therefore RpropL]`
`i_(2)=i_(1)(theta)/((2pi-theta))" "[thereforeL=rtheta]`
`therefore B_(R)=(mu_(0))/(4pi)(i_(1)theta)/(r)ox+(mu_(0))/(4pi)(i_(1)theta)/(r)odot=0`
i.e., the field at the centre of the coil is zero and is independent of `theta`.