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. Proportional 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

Direction of currents in two parts are different, so directions of magnetic field 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=rhol/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`.