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

Magnetic field at the centre of an arc is given by
`B=(mu_0I)/(2R)xx theta/(2pi)`
Megnetic field due to a smaller arc, `vecB_1=(mu_0I_1)/(2r)xx theta/(2pi)(-hatk)`
Megnetic field due to a larger arc, `vecB_2=(mu_0I_1)/(2r)xx (2pi-theta)/(2pi)(+hatk)`
Resultant magnetic field =`[-(mu_0I_1 theta)/(4pir)+(mu_0I(2pi-theta))/(4pir)](hatk)...(i)`
Two arcs form a parallel combination of resistors.
Here `l_1R_1=i_1R_2 implies(I_1)/(I_2)=(R_2)/(R_1)....(ii)`
Here `R_1 and R_2` are the resistances of smaller and bigger arcs,
respectively.
But `(R_2)/(R_1)=(l_2)/(l_1)=((2pi- theta)r)/(thetar)=(2pi- theta)/ theta.....(iii)`
From Eqs. (ii) and (iii), we get
`I_1 theta=I_2(2pi-theta).....(iv)`
Hence, from Eqs. (i) and (iv), the magnetic field at the centre
of ring will be zero.