Question

A metal wire of length 4m has resistance 40 ohm at room temperature 25 degree Celsius . If diameter of wire is 0.2 mm. Then what will be the resistivity of the metal of the wire at the same temperature

Answer 1

resistivity is

$3.14 \times {10}^{ - 7} ohm - m$

Answer 2

Given: the length of the wire, l = 4m

           resistance of the wire, R = 40 ohm

           diameter of the wire, d = 0.2 mm

To Find: the resistivity of the wire, ρ.

Solution:

To calculate ρ, the formula used:

• R = ρ ( L/A)

       here, R is the resistance of the wire

       ρ is the resistivity

       L is the length of the wire

       A is the area of cross-section of the wire

Applying the above formula:

R   = ρ ( L/A)                                

40 = ρ ( 4 /A)                             ⇒1

And, Area of the wire, A = π x (radius of the wire, r)²

or, A = π r²                                  ⇒2

And, radius = diameter of the wire / 2

        r = d / 2

           = 0.2 / 2

       r  = 0.1 mm

Now convert 0.1 mm into m:

1 m = 1000 mm

1000mm = 1m

1 mm = 1 / 1000

0.1 mm = ( 1 / 1000) x 0.1

            =1 / 1000 x10

            = 1 / 10000

            = 0.0001 m

Putting the above value in equation 2:

A = π x (0.0001)²

   = 3.14 x 0.0001 x 0.0001

   = 3.14 x 1 / (10000 x 10000)

   = 3.14 x 10⁻⁸ m²

A  = 3.14 x 10⁻⁸ m²

Putting the value of A, in equation 1:

40 = ρ ( 4 / 3.14 x 10⁻⁸ )

40 x 3.14 x 10⁻⁸ =   ρ x 4

(40 x 3.14 x 10⁻⁸) / 4 = ρ

ρ = (40 x 3.14 x 10⁻⁸) / 4

  = 10 x 3.14 x 10⁻⁸

  = 3.14 x  10⁻⁷

ρ = 3.14 x  10⁻⁷ ohm.m

Hence, the resistivity of the wire is 3.14 x  10⁻⁷ohm.m