Question

Complete the given statement. The resistance of a wire increases if:
A. The length is increased
B. The diameter is increased
C. The cross sectional area is increased
D. The length is decreased
E. The resistivity is decreased

Answer 1

Hint: The resistance of a wire is the measure of the opposition offered by it to the flow of electric current across it. The resistance depends directly upon the length and resistivity of the wire and inversely upon the cross sectional area of the wire.

Formula used:
The resistance $R$ of a wire of length $l$ and cross sectional area $A$ is given by
$R=\rho \dfrac{l}{A}$
where $\rho$ is the resistivity or specific resistance of the wire. Resistivity of a material is the resistance of a wire of unit length and unit cross sectional area made of that material.
Cross sectional area $A$ of a cylindrical wire of diameter $D$ is given by
$A=\pi {{\left( \dfrac{D}{2} \right)}^{2}}=\dfrac{\pi {{D}^{2}}}{4}$

Complete step by step answer:
Resistance is the amount of opposition that a body offers to the flow of current when a potential difference is applied across it. At a constant temperature, it usually depends on the shape, nature and dimensions of the body.
The resistance $R$ of a wire of length $l$ and cross sectional area $A$ is given by
$R=\rho \dfrac{l}{A}$ --(1)
where $\rho$ is the resistivity or specific resistance of the wire. Resistivity of a material is the resistance of a wire of unit length and unit cross sectional area made of that material.
Cross sectional area $A$ of a cylindrical wire of diameter $D$ is given by
$A=\pi {{\left( \dfrac{D}{2} \right)}^{2}}=\dfrac{\pi {{D}^{2}}}{4}$ --(2)
Hence, from (1), we can see that the resistance increases if the length of the wire increases or the resistivity increases or the cross sectional area decreases.
Now from (2), we can see that the cross sectional area decreases when the diameter of the wire decreases.
Hence, the correct option is A) the length is increased.

Note: Students must also remember that resistance depends on temperature also, since the specific resistance or resistivity depends on temperature of the body.
For conductors, specific resistance increases with increase in temperature. Therefore, the resistance also increases with increase in temperature. However, for semiconductors, the specific resistance decreases with increase in temperature. Hence, the resistance of semiconductors decreases with increase in temperature.