Question

Uniform metal wire of specific resistance 64 × 10^-6 Ωm and length 1.98m has a resistance of 7Ω. What is the diameter of the wire?

Answer 1

$\bigstar$ $\sf{\blue{\underline{\underline{\red{CONCEPT:-}}}}}$

• When Specific Resistance, Length of metal wire and Resistance is given and to ask Radius or Diameter of the wire, then put the below formula .

• $\tt{\red{\boxed{R\:=\:{\dfrac{\rho\:l}{A}}\:}}}$

Where,

• R = Resistance
• $\tt{\rho\:=\:Specific\:Resistance}$
• l = Length of metal wire
• A = Area of metal wire = $\tt{\pi\:r^2}$

$\bigstar$ $\sf{\blue{\underline{\underline{\red{To\:Find:-}}}}}$

• The Diameter(D) of the wire .

$\bigstar$ $\sf{\blue{\underline{\underline{\red{SOLUTION:-}}}}}$

♻️ GIVEN:-

• $\tt{Specific\:Resistance\:({\rho})\:=\:64\times10^{-6}\:Ohm.m\:}$
• Resistance (R) = 7 Ohm
• Length (l) = 1.98 m

♻️ We have know that,

$\tt{\:R\:=\:{\dfrac{\rho\:l}{A}}\:}$

$\tt{\implies\:A\:=\:{\dfrac{\rho\:l}{R}}\:}$

$\tt{\implies\:{\pi\:r^2}\:=\:{\dfrac{\rho\:l}{R}}\:}$

$\tt{\implies\:r\:=\:{\sqrt{\dfrac{\rho\:l}{\pi\:R}}}\:}$

$\tt{\implies\:r\:=\:{\sqrt{\dfrac{(64\times10^{-6})\times\:1.98}{3.14\times7}}}\:}$

$\tt{\implies\:radius\:=\:2.401\times10^{-3}\:m}$

♻️ So, Diameter of Metal wire is,

$\tt{\implies\:Diameter\:=\:2\times\:radius\:}$

$\tt{\red{\boxed{\implies\:Diameter\:=\:4.802\times10^{-3}\:m\:}}}$