Diagram shows a potentiometer. The wire of potentiometer is of length ℓ and is of uniform cross-section A but its resistivity varies as ρ = ρ 0 ℓ x (x measured from end P ). When switch was opened

Question

Diagram shows a potentiometer. The wire of potentiometer is of length $ℓ$ and is of uniform cross-section A but its resistivity varies as $ρ = ρ_{0} \frac{x}{ℓ}$ (x measured from end P ). When switch was opened, balancing length was $ℓ_{1}$ while when switch was closed, balancing length was $ℓ_{2}$ . Choose the correct option(s) :

TestHub · Accepted Answer

Wire resistance up to length x: R ( x ) = ∫ 0 x ​ ℓ A ρ 0 ​ x ′ ​ d x ′ = 2 ℓ A ρ 0 ​ x 2 ​ Switch open: ε 1 ​ = R ( ℓ ) + r 0 ​ ε 0 ​ ​ R ( ℓ 1 ​ ) = 2 A ℓ r 0 ​ + ρ 0 ​ ℓ 2 ρ 0 ​ ε 0 ​ ℓ 1 2 ​ ​ (A correct). Switch closed, terminal voltage: ε 1 ​ V ​ = R + r R ​ = ( ℓ 1 ​ ℓ 2 ​ ​ ) 2 ⇒ r = R [ ( ℓ 2 ​ ℓ 1 ​ ​ ) 2 − 1 ] (C correct).

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Options:(select one or more)

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