Question

A piece of wire of resistance $4\Omega$ is bent through 180° at its midpoint and the two halves are twisted together. Then the resistance is
A. $8\Omega$
B. $1\Omega$
C. $2\Omega$
D. $5\Omega$

Answer 1

Hint: To solve this problem, use the relation between cross-sectional area of the conductor, length of the conductor and resistance. From the above relation, we can infer that the length and area of cross-section of the conductor are proportional to each other. So, when the wire is bent, the length of the wire gets halved and the area of cross-section doubles. Substitute these values in the formula for resistance and find the resistance of the wire after bending.
Formula used:
$R=\rho \dfrac { l }{ A }$

Complete answer:
Given: Resistance of wire (R)= $4\Omega$
The relation between length, area of cross-section and resistance of a conductor is given by,
$R=\rho \dfrac { l }{ A }$ …(1)
Where, l is the length of the wire
            A is the area of cross-section of the wire
            $\rho$ is the resistivity of the material of the wire
When the wire is bent through 180°, its length becomes half and the area of cross-section doubles.
${ l }^{ ' }=\dfrac { l }{ 2 }$
${ A }^{ ' }=2A$
Now, the formula for resistance becomes,
${ R }^{ ' }=\rho \dfrac { { l }^{ ' } }{ { A }^{ ' } }$ …(2)
Substituting these values in the equation. (2) we get,
${ R }^{ ' }=\rho \times \dfrac { \dfrac { l }{ 2 } }{ 2A }$
$\Rightarrow { R }^{ ' }=\rho \times \dfrac { { l } }{ { 4A } }$
Substituting equation. (1) in above equation we get,
${ R }^{ ' }=\dfrac { R }{ 4 }$
Now, substituting the value of R we get,
${ R }^{ ' }=\dfrac { 4 }{ 4 }$
$\Rightarrow { R }^{ ' }= 1\Omega$
Hence, the resistance is $1\Omega$.

So, the correct answer is “Option B”.

Note:
To solve these types of formula, students should know the relation between resistance, area of cross-section and the length of a conductor. They should also understand how the change in one parameter of the formula changes the resistance of the conductor. From the equation. (1) we can infer that the resistance of a conductor is directly proportional to its length. So, if we want to increase the resistance then we can just increase the length and this will cause increment in the resistance of the conductor.