Question

# A and B are two points on a uniform ring of resistance R made of material of resistivity ρ as shown in the figure. Find the equivalent resistance between A and B interms of R.

A
3R4
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B
3R16
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C
R4
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D
R16
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Solution

## The correct option is B 3R16Let, l be the circumference of ring, A be the area of cross-section and R be the resistance of the uniform ring. We know that, Resistance depends upon the length of the material. Since the area of cross-section is uniform, we can say that R∝l Equivalent diagram of the given figure is as given below, From the diagram, it is evident that, R1 and R2 are parallel to each other and the equivalent resistance Req is given by 1Req=1R1+1R2 Since, R∝l we can write that, R1∝l1 and R2∝l2. Thus, Req=kl1l2l1+l2 where , k=ρA From the figure , l1+l2=l also l1=l4 and l2=3l4 Substituting the data in Req we get, Req=k×l4×3l4l ⇒Req=ρA×3l16 ⇒Req=3R16 Hence, option (b) is the correct answer. Alternate method: we know , Req=Rθ(2π−θ)4π2 Since, A and B are perpendicular, θ=900 or π2. Thus, Req=R×π2×(2π−π2)4π2 ⇒Req=3R16.

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