Question

When the diameter of the conductor is doubled, then its resistance:
A) Decreases two times
B) Decreases four times
C) Decreases six times
D) Increases four times

Answer 1

Hint: Resistance of the conductor is directly proportional to the resistivity and length of the conductor, and inversely proportional to the area of the conductor.
Formula for resistance is given by:
$R = \rho \dfrac{l}{A}$ (R is the resistance, l is the length of the conductor, A is the area of the conductor, $\rho $ is the resistivity of the conductor)
Using the above relation we will find the change in resistor when diameter is doubled.

Complete step by step solution:
Let's discuss some points about resistors.
Resistance is the measure of the opposition of the current flow in an electrical circuit. Resistance is measured in ohms. Almost all the materials show some opposition to the flow of current which are classified as conductors and insulators.
Conductors are the material which has the maximum flow of current and insulators are the one which allow almost negligible flow of current through it.
Now we will perform the calculation part:
As we know resistance is directly proportional to the length of the conductor and inversely proportional to the area.
$R = \rho \dfrac{l}{A}$........................(1)
From this relation we can conclude that resistance is inversely proportional to area A of the conductor.
Area of the conductor is directly proportional to the diameter of the conductor, which is given by:
$A = \pi {(\dfrac{d}{2})^2}$
When diameter is doubled, then area becomes
$ \Rightarrow A = \pi {(\dfrac{{2d}}{2})^2}$
$ \Rightarrow A = \pi {(d)^2}$
Which is:
$\Rightarrow A = \pi {(2r)^2} $
$ \Rightarrow A = 4\pi {r^2} $
Area of the conductor becomes 4 times the area of the conductor without doubling the diameter.
Area of the conductor is inversely proportional to the resistance , therefore resistance will become
$R = \rho \dfrac{l}{{4A}}$
Thus, we can say that new resistance will decrease four times.

Hence, option (B) is correct.

Note: Resistance has the properties of limiting current which used in the electric motors such Induction motor and DC motors. Split phase induction motors use an extra resistance to limit the starting current in the motor and increase the starting torque. Similarly, DC series motors use resistance in the armature winding to control the starting current.