Question

“The resistance of any conductor is directly proportional to the length and inversely proportional to the area of cross-section of the substance”. From this statement, we can conclude that
(P) Value of resistance increases with an increase in the length of the conductor.
(Q) Value of resistance decreases with an increase in the length of the conductor.
(R) Value of resistance decreases with an increase in the area of cross-section.
(S) Value of resistance decreases with a decrease in the area of cross-section.
Which of the above statements are correct:
A) Statements Q and S are true
B) Statement P and S are true
C) Statement P and R are true
D) Statement Q and R are true.

Answer 1

Ohm's Law states that when a voltage (V) source is applied between two points in a circuit, an electrical current will flow between them encouraged by the presence of the potential difference between these two points. The amount of electrical current which flows is restricted by the amount of resistance. In other words, the voltage encourages the current to flow (the movement of charge), but it is resistance that discourages it. The electrical resistance between two points can depend on many factors such as the conductor’s length, its cross-sectional area, the temperature, as well as the actual material from which it is made. The resistance of a conductor is given by $R = \rho \left( {\dfrac{L}{A}} \right)$ ohms, where $L$ is the length of the conductor, $A$ is the area of cross-section, and $\rho$ is the resistivity of the material. Since the resistivity of a material is constant for a specific material, then the resistance depends only on the length and the cross-section area of the conductor.

Complete answer:
Let us consider a conductor of length $L$ and the area of cross-section $A$ . When the potential difference is applied the current starts to flow. Assume that the resistance of the conductor is $R$ . If we connect the same two conductors of length $L$ and cross-section area $A$ in a series combination, that is end to end, we have effectively doubled the length of the conductor $2L$ , while the cross-section area remains the same. But as we doubled the length, we also have doubled the total resistance $R + R = 2R$ , as the conductors are connected in the series combination. Therefore we can see that the resistance of the conductor is directly proportional to its length, that is, if we increase the length of the conductor, the total resistance will increase.
Suppose we connect two conductors of length $L$ and cross-section area $A$ in a parallel combination. Here, by connecting the two conductors in a parallel combination, we have effectively doubled the total area giving $2A$ , while the conductor’s length, $L$ remains the same as the original single conductor. But as we doubled the are of the conductor by connecting them in the parallel combination, the resistance will be $\dfrac{1}{{{R_{final}}}} = \dfrac{1}{R} + \dfrac{1}{R}$ . Or we can write ${R_{final}} = \dfrac{R}{2}$ . we can see that the resistance is inversely proportional to the cross-section area of the conductor, that is if we increase the cross-section area the resistance of the conductor will decrease.

Therefore statements P and R are true. Hence the option C is correct.

Note: The resistance of a conductor depends on many factors like length, cross-section area, temperature, etc. When we talk about the length and cross-section area of a conductor, the other factors are supposed to be constant. When two variables are directly proportional to each other they increase or decrease together and when two variables are inversely proportional to each other, then if one increases the other decreases.