Question

Resistance of a metal wire of length 1 m is 26 ohm at 20°C. If the
diameter of the wire is 0.3 mm, what will be the resistivity of the
metal at that temperature?​

Answer 1

★Given:-

• Resistance = 26Ω
• Length = 1m
• Diameter = 0.3mm
• Temperature = 20°c

★To find:-

• Resistivity of the  metal at that temperature = ?

★Solution:-

Given,

Diameter=0.3mm

               = 0.3 × 10⁻³m

We know:

$\leadsto \underline{\boxed{\sf Radius = \frac{diameter}{2} }}$

Putting the values,

$\implies \sf R = \dfrac{0.3\times 10^{-3}}{2} \\\\\implies \sf R = \underline{\bold{0.15 \times 10^{-3}m}}$

The base area of the wire:-

We know:

$\leadsto \underline{\boxed{\sf A = \pi r^2}}$

Putting the values,

$\implies \sf A = 3.14\times (0.15\times 10^{-3} )^2\\\\\implies \underline{\bold{A = 0.071\times 10^{-6}m^2 }}$

Using the formula:

$\leadsto \underline{\boxed{\sf R = \frac{\rho \ L}{A} }}$

Where,

R = Radius

L = length

A = area

ρ = resistivity

Putting the values,

$\implies \sf \rho = \dfrac{RA}{L} \\\\\implies \sf \rho = \dfrac{26\times 0.071 \times 10^{-6}}{1} \\\\\implies \underline{\boxed{\sf \rho =1.84 \times 10^{-6} }}$

Hence,

The resistivity is 1.84×10⁻⁶ Ωm at 20°C

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