Question

Two wire A and B of the same material have their length in the ratio 1:5 and diameter in the ratio 3:2 if the resistance of the wire B is 180 ohm find the resistance of the wire A​

Answer 1

Given :

➳ Ratio of length = 1 : 5

➳ Ratio of diameter = 3 : 2

(Ratio of diameter = Ratio of radius)

➳ Resistance of wire B = 180Ω

To Find :

⟶ Ratio of wire A.

SoluTion :

➻ We know that, Resistance of a conductor is directly proportional to the length of conductor and inversely proportional to the area of cross section of conductor.

Mathematically,

$\bigstar\:\boxed{\bf{R\propto\:\dfrac{L}{A}}}$

➝ Area of cross section = π r²

$\implies\tt\:\dfrac{R_A}{R_B}=\dfrac{L_A}{L_B}\times \dfrac{\pi(R_B)^2}{\pi(R_A)^2}$

$\implies\tt\:\dfrac{R_A}{180}=\dfrac{1}{5}\times \dfrac{(2)^2}{(3)^2}$

$\implies\tt\:\dfrac{R_A}{180}=\dfrac{4}{45}$

$\implies\tt\:R_A=\dfrac{180\times 4}{45}$

$\implies\boxed{\bf{R_A=16\Omega}}$

Answer 2

$\green{\bold{\underline{\underline{Given}}}}$

$\bigstar \: \orange{\bold{{{Ratio \: of \: length = 1 : 5}}}}$

$\bigstar \: \red{\bold{{{Ratio \: of \: diameter = 3 : 2}}}}$

$\bigstar \: \blue{\bold{{{Resistance \: of \: the \: wire \: B = 180Ω}}}}$

$\green{\bold{\underline{\underline{To \: Find}}}}$

$\bigstar \: \pink{\bold{{{The \: resistance \: of \: the \: wire \: A}}}} </p><p>$

$\green{\bold{\underline{\underline{Formula}}}}$

$\boxed{\bf{R\propto\:\dfrac{L}{A}}}$

$\green{\bold{\underline{\underline{Solution}}}}$

We know that,

★ Area of cross section = π r²

$\implies\dfrac{R_A}{R_B}=\dfrac{L_A}{L_B}\times \dfrac{\pi(R_B)^2}{\pi(R_A)^2}</p><p>$

$\implies\dfrac{R_A}{180}=\dfrac{1}{5}\times \dfrac{(2)^2}{(3)^2}$

$\implies\dfrac{R_A}{180}=\dfrac{4}{45}$

$\implies\ R_A=\dfrac{180\times 4}{45}$

$\implies{\bf{R_A=16\Omega}}$