A copper wire has diameter 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

Hence we will get radius

Radius r = 0.25 mm = 0.25 × 10⁻³ m = 2.5 × 10⁻⁴ m

Area of cross-section  A = 6.25 π × 10⁻⁸ 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 × 6.25 π × 10⁻⁸ ) / 1.6 × 10⁻⁸ m

L = 62.5 π / 1.6

L = 122.72 m

If the diameter of the wire is doubled, 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 diameter doubles, resistance becomes 1/4 times.

Therefore new resistance=  2.5 Ohm

Answer

Therefore, the length of the wire is 122.7 m and the new resistance is 2.5 Ohm

 

 

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