Question

a wire of length l and resistance R is bent in the form of a ring. The resistance between two points which are separated by angle theeta is ​

Answer 1

we know, resistance, $R=\rho\frac{l}{A}$

here it is clear that resistance of wire is depends on length of wire.

wire of length l and resistance R is bent in the form of a ring.

we have to find resistance between two points which are separated by angle θ.

length of wire which is made an angle θ with centre of ring , $l_1$= rθ, where r is radius of ring.

e.g,. l = 2πr => r = l/2π

so, $l_1=\frac{l\theta}{2\pi}$

now, resistance of 1st part, $R_1=\rho\frac{l_1}{A}$

similarly length of 2nd apart of wire, $l_2=\frac{l(2\pi-\theta)}{2\pi}$

resistance of 2nd part, $R_2=\rho\frac{l_2}{A}$

because both part are joined in parallel combination.

so, equivalent resistance, R = $\frac{R_1R_2}{R_1+R_2}$

= $\frac{\rho l_1l_2}{(l_1+l_2)A}$

= $\rho\frac{\frac{l\theta}{2\pi}\frac{l(2\pi-\theta)}{2\pi}}{\frac{l}{2\pi}(\theta+2\pi-\theta)A}$

= $\left(\rho\frac{l}{A}\right)\frac{(2\pi-\theta)\theta}{4\pi^2}$

= $\left(\frac{(2\pi-\theta)\theta}{4\pi^2}\right)R$