Question

a wire has resistance r it is bent to form a circle what will be the effective resistance between two points on circle which subtends an angle of 120 at the centre
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Answer 1

Given : A wire of resistance r it is bent to form a circle. two point A and B on circle such that points subtend an angle of 120° at the centre of circle as shown in figure.

To find : effective resistance between points A and B.

solution : from formula of resistance,

R = ρl/A

here, ρ, A are constant so, R is directly proportional to l

I.e., R₁/R₂ = l₁/l₂......(1)

but l = Rθ [ where R is radius of circle , θ is angle subtend by arc of length l. ]

so, l₁/₂ = Rθ₁/Rθ₂

⇒l₁/l₂ = 120°/(360° - 120°) = 120°/240° = 1/2

now from equation (1) we get,

R₁/R₂ = 1/2

so, R₁ = r/3 and R₂ = 2r/3

now effective resistance, Req = R₁R₂/(R₁ + R₂)

= (r/3)(2r/3)/(r/3 + 2r/3)

= (2r²/9)/r

= 2r/9

Therefore effective resistance is 2r/9