`A` and `B` are two points on a uniform ring of resistance `R` the `/_ASCB=theta`, whre `C` is the centre of the sign. The equivalent reisrtance between `A` and `b`

Options

A. `(Rtheta)/(2pi)`
B. `(R(2pi-theta))/(4pi)`
C. `R(1-(theta)/(2pi))`
D. `(Rtheta)/(4pi^(2))(2pi-theta)`

Answer

D

Explanation

Let r be the radius of the ring.
Resistance per unit length of ring, `rho=R//(2pir)`
Lengths of sections ADB and ACD are `rtheta and r(2pi-theta)`.
Resistance of section ADB,
`R_(1)=rhortheta=(R)/(2pir)rtheta=(Rtheta)/(2pi)`
Resistance of section ACD,
`R_(2)=(R)/(2pir)r(2pi-theta)=(R(2pi-theta))/(2pi)`
Here `R_(1)andR_(2)` are in parallel between A and B, their equivalent resistance
`R_(eq)=(R_(1)R_(2))/(R_(1)+R_(2))=((Rtheta)/(2pi)xx(R(2pi-theta))/(2pi))/((Rtheta)/(2pi)+(R(2pi-theta))/(2pi))=(Rtheta(2pi-theta))/(4pi^(2))`