Chapter 11

NCERT
Class 10
Science
Solutions
6. A copper wire has a diameter of 0.5 mm and resistivity of 1.6 × 10–8  Ω. What will be the length of this wire to make its resistance 10 Ω? How much does the resistance change if the diameter is doubled?

Question:

A copper wire has a diameter of 0.5 mm and resistivity of 1.6 × 10–8 Ω m. What will be the length of this wire to make its resistance 10 Ω? How much does the resistance change if the diameter is doubled?

Answer:

d= diameter = 0.5 mm, ρ = 1.6 × 10–8 Ωm, R = 10Ω

Assume wire length as l

r = d/2

r = (0.5 x 10-3)/2                 // 1 m = 1000 mm

A = πd2/4

R =  ρ (l/A)

   = ( ρl)/(πd2/4)

⟹ l = (Rπd2)/4ρ

      l = 10 x 3.14 x ((0.5 x 10-3)/2 )2 / (4 x 1.6 × 10–8 )              // π = 3.14

      l = 10 x 3.14 x ((0.5 x 10-3)/2 )2 / (4 x 1.6 × 10–8 )  

      l = (10 x 314 x 10-2 x 25 x 10-6)/ (4 x 16 x 10-1 x 10-8)

      l = 3140 x 25 x 10-8/ 64 x 10-9

     l = 122.7 m

∴  Length of wire = 122.7 m

If diameter is doubled then d = 0.25mm

d = 25 x 10-2 x 10-3 m

d = 25 x 10-5m

and R = ( ρl)/(πd2/4)

R = (1.6 × 10–8 X 122.7) / ((3.14 X 25 x 10-5) / 4)

R = 490.5m

Thus, we can say,

Since R ∝ 1/d2

So, if d is doubled, then R becomes 1/4th time.

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