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

Given

  • Resistivity (ρ) = 1.6 × 10-8 Ω m
  • Resistance (R) = 10 Ω
  • Diameter (d) = 0.5 mm

d = 5 × 10⁻⁴ m

Hence, we will get radius

Radius (r) = 0.25 mm

r = 0.25 × 10⁻³ m

r = 2.5 × 10⁻⁴ m

We need to find the area of cross-section

A = πr2

A = (22/7)(2.5 × 10⁻⁴)2

A = (22/7)(6.25×10⁻⁸)

A = 1.964 × 10-7 m2

Find out

We have to find the length of the wire

Let the length of the wire be L

Formula

We know that

R = ρ (L) / (A)

L = (R × A) / ρ

Substituting the values in the above equation we get

L = (10 × 1.964 × 10⁻⁷) / 1.6 × 10⁻⁸ m

L = 1.964×10-6 /1.6 × 10-8

L = 122.72 m

If the diameter of the wire is doubled, the new diameter = 2 × 0.5 = 1mm = 0.001m
Let new resistance be Rʹ

R = ρ (L) / (A)

R’ = ρ (L) / (4A)

R’ = ρ (L) X 1/(4A)

Hence, if the diameter doubles, resistance becomes 1/4 times.

Hence, new resistance =2.5Ω

Answer

Therefore, the length of the wire is 122.7 m and the new resistance becomes 1/4 times.

Check out the video below to know more about the factors that affect resistance

Further Reading

  1. Judge the equivalent resistance when the following are connected in parallel
  2. The maximum resistance which can be made using four resistors each of 2 Ω is (a) 2 Ω (b) 4Ω (c) 8Ω (d) 16 Ω

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