For a given conducting material at a fixed temperature, the electrical resistance of a conductor is inversely proportional to which of the following quantities?

Introduction / Context:

Resistance is a property of a conductor that opposes the flow of electric current. It depends on the material as well as on the geometry of the conductor. This question checks whether you remember how resistance varies with length and cross sectional area for a uniform conductor made of a given material.

Given Data / Assumptions:

• The conductor is made of a fixed material, so its resistivity is constant.
• Temperature is held constant, so resistivity does not change with heating.
• We examine the relationship between resistance and geometric factors such as length and area of cross section.

Concept / Approach:

The fundamental relation for resistance of a uniform conductor is R = rho * L / A, where R is resistance, rho is resistivity of the material, L is length, and A is area of cross section. From this expression, for constant rho and constant temperature, resistance is directly proportional to length and inversely proportional to area of cross section. Thus, if area is increased, resistance decreases, and if area is reduced, resistance increases.

Step-by-Step Solution:

Verification / Alternative check:

In practical wiring, thick wires are used for high current because they have large cross sectional area and therefore lower resistance. Thin wires, which have small area, have higher resistance. This real world observation supports the inverse relationship between resistance and area of cross section.

Why Other Options Are Wrong:

Option A: Length is directly proportional to resistance. Doubling length doubles resistance, so it is not an inverse relation.

Option B: Resistivity is a material property, not a geometric factor. For a given material it is constant and does not have an inverse relation to resistance in this context.

Option C: Temperature is held constant in the question. In many materials resistance increases with temperature, but this is not the simple inverse proportionality being asked about.

Common Pitfalls:

Sometimes students mistakenly think that both length and area have the same kind of relation with resistance. It is essential to remember that length appears in the numerator of R = rho * L / A, while area is in the denominator. Visualising a long thin wire versus a short thick wire can help: a long thin wire has high resistance, whereas a short thick wire has low resistance.

Final Answer:

Resistance is inversely proportional to its area of cross section.