A battery is connected between two points `A and B` on the circumference of a uniform conducting ring of radius `r` and resistance `R` . One of the arcs `AB` of the ring subtends an angle `theta` at the centre . The value of the magnetic induction at the centre due to the current in the ring is

Options

A. Proportonal to `2(180^(@) - theta)`
B. Inversely proportional to `r`
C. Zero, only if `theta=180^(@)`
D. Zero for all values of `theta`

Answer

D

Explanation

Directions of currents in two parts are different, so directions of magnetic fields due to these currents are opposite. Also applying `Ohm's` law across `AB`
`i_(1) R_(1) = i_(2) R_(2) implies i_(1) l_(2) = i_(2) l_(2)`
`( :' R = rho (1)/(A))`
Also `B_(1) = (mu_(0))/(4pi) xx (i_(1) l_(1))/(r^(2))` and `B_(2) = (mu_(0))/(4pi) xx (i_(2) l_(2))/(r^(2)) ( :' l = r theta)`
`:. (B_(2))/(B_(1)) = (i_(1) l_(1))/(i_(2) l_(2)) = 1`
Hence, two field induction's are equal but of opposite direction. SO, resultant magnetic induction at the centre is zero and is independent of `theta`.