Home » » A wire of 5Ω resistance is drawn out so that its new length is two times the origi...

# A wire of 5Ω resistance is drawn out so that its new length is two times the origi...

### Question

A wire of 5Ω resistance is drawn out so that its new length is two times the original length. If the resistivity of the wire remains the same and the cross-sectional area is halved, the new resistance is

A) 40Ω

B) 20Ω

C) 10Ω

D) 5Ω

### Explanation:

Let the original length = L1
Let the original resistance = R1 = 5Ω
Let the original resistivity = P1
Let the original area = a1
Let the new length = L2 = 2L1

let the new area = a2 = 1/(2a2)
Let the new resistance = R2
Let the new resistivity = P2
But since the resistivity remains the same,
=> P1 = P2

 ∴ P1 = R1 a1 _L1
 = P2 = R2 a2 _L2

But a2 = 1/2a1; and L2 = 2L1
 ∴ R1 a1 _L1
 = R2 2a1/2 _2L1
 => 5 x a1 _L1
 = R2 x a1 _4L1
 ∴R2 = 5 x a1 x 4L1 _a1 x L1

= 20Ω

## Dicussion (1)

• Let the original length = L1
Let the original resistance = R1 = 5Ω
Let the original resistivity = P1
Let the original area = a1
Let the new length = L2 = 2L1

let the new area = a2 = 1/(2a2)
Let the new resistance = R2
Let the new resistivity = P2
But since the resistivity remains the same,
=> P1 = P2

 ∴ P1 = R1 a1 _L1
 = P2 = R2 a2 _L2

But a2 = 1/2a1; and L2 = 2L1 ∴ R1 a1 _L1
 = R2 2a1/2 _2L1
 => 5 x a1 _L1
 = R2 x a1 _4L1
 ∴R2 = 5 x a1 x 4L1 _a1 x L1

= 20Ω