Question

Two wires A and B of the same material have their lengths in the ratio 1:2 and cross-section areas in the ratio 2:3.If the resistance of wire A be 12ohm then calculate the resistance of wire B​

Answer 1

$\large{\underline{\underline{\red{\sf{\hookrightarrow Given :-}}}}}$

• Two wires A and B are of same material.
• Their ratio of lenght is 1 : 2 .
• Ratio of their cross - Sectional area is 2 : 3.
• Resistance of wire A is 12Ω.

$\large{\underline{\underline{\red{\sf{\hookrightarrow To\:Find:- }}}}}$

• The resistance of wire B .

$\large{\underline{\underline{\red{\sf{\hookrightarrow Formula\:Used:- }}}}}$

We know that when length and area are given , we can calculate resistance by using formula ,

$\large\pink{\underline{\boxed{\purple{\tt{ \dag \:\: Resistance\:\:=\:\:\rho\:\dfrac{l}{a}}}}}}$

✒️ Where ,

• l is lenght.
• a is area .
• $\rho$ is resistivity or specific resistance.

$\large{\underline{\underline{\red{\sf{ \hookrightarrow Calculation:-}}}}}$

Let us take the ratio of length of two wires be x : 2x .

And , the ratio of their cross - sectional areas be 2y : 3y .

Let us denote ,

• ❒ $\tt{R_1\:as\: resistance\: of\:wire\:A.}$
• ❒ $\tt{R_2\:as\: resistance\: of\:wire\:B.}$

Now , Resistance of Wire A :-

$\tt:\implies Resistance =\rho\dfrac{l}{a}$

$\pink{\tt:\longmapsto R_1 = \rho\dfrac{x}{2y} \:\:.............(i)}$

Similarly, Resistance of Wire B :-

$\tt:\implies Resistance =\rho\dfrac{l}{a}$

$\pink{\tt:\longmapsto R_2 = \rho\dfrac{2x}{3y} \:\:.............(ii)}$

Now divide equⁿ (i) and equⁿ (ii) ,

$\tt:\implies \dfrac{R_1}{R_2}=\dfrac{\cancel{\rho}\dfrac{x}{2y}}{\cancel{\rho}\dfrac{2x}{3y}}$

$\tt:\implies \dfrac{R_1}{R_2}=\dfrac{\cancel{x}\:\:\times\:\:3\cancel{y}}{2\cancel{y}\:\:\times\:\:2\cancel{x}}$

$\tt:\implies \dfrac{R_1}{R_2}=\dfrac{1\:\:\times\:\:3}{2\:\:\times\:\:2}$

$\tt:\implies \dfrac{12\Omega}{R_2}=\dfrac{3}{4}$

$\tt:\implies R_2=\dfrac{\cancel{12}^{4}\times4}{\cancel{3}}$

$\underline{\boxed{\red{\tt{\longmapsto\:\:R_2\:\:=\:\:16\Omega}}}}$

$\boxed{\bf{\green{\dag Hence\:the\: resistance\:of\:wire\:B\:is\:16\: \Omega.}}}$