Two wires A and B are made up of the same material and have the same mass. Wire A has radius of 2.0 mm and wire B has radius of 4.0 mm. The resistance of wire B is 2 . The resistance of wire A is

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Solution

Q.
Two wires A and B are made up of the same material and have the same mass. Wire A has radius of 2.0 mm and wire B has radius of 4.0 mm. The resistance of wire B is 2 $Ω$ . The resistance of wire A is _______ $Ω$ . [2024]

Ans.
(32) Given: $r_{A} = 2 \times 10^{- 3} m,$ $r_{B} = 4 \times 10^{- 3} m,$ $R_{B} = 2 Ω$

We know $R = \frac{ρ l}{A}$

   $\frac{R_{A}}{R_{B}} = \frac{ρ_{A} l_{A}}{A_{A}} \times \frac{A_{B}}{ρ_{B} l_{B}} = \frac{l_{A}}{l_{B}} \frac{A_{B}}{A_{A}}$ ...(i)

  The volume of wire remains constant, $V_{A} = V_{B}$

$l_{A} A_{A} = l_{B} A_{B} \Rightarrow \frac{l_{A}}{l_{B}} = \frac{A_{B}}{A_{A}}$ ...(ii)

  from equation (i) and equation (ii)

$R_{A} = R_{B} {(\frac{A_{B}}{A_{A}})}^{2} = 2 {(\frac{π \times {(4 \times 10^{- 3})}^{2}}{π \times {(2 \times 10^{- 3})}^{2}})}^{2} = 2 \times 16 = 32 Ω$

$\Rightarrow R_{A} = 32 Ω$

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