Understanding Conductor Resistance Proportionality

The question asks us to determine what factor the resistance of a conductor is directly proportional to, given its length 'L' and cross-sectional area 'A'. Let's explore the relationship between these properties.

Physics Principles of Resistance

Electrical resistance (R) is a measure of how much an object opposes the flow of electric current. For a conductor, resistance depends on several factors:

• The material of the conductor (its resistivity, represented by the Greek letter rho, \(\rho\)).
• The length of the conductor (L).
• The cross-sectional area of the conductor (A).

The relationship is defined by the formula:

\( R = \rho \frac{L}{A} \)

From this formula, we can see how resistance changes with length and area:

• Length (L): Resistance is directly proportional to the length of the conductor. If you double the length, the resistance also doubles (assuming area and material stay the same). This is because electrons have to travel a longer path, encountering more obstacles.
• Cross-sectional Area (A): Resistance is inversely proportional to the cross-sectional area. If you double the area, the resistance is halved (assuming length and material stay the same). A larger area provides more pathways for the current to flow, reducing opposition.

Therefore, the resistance 'R' is directly proportional to the ratio \(\frac{L}{A}\).

Analyzing the Options

Let's examine each option based on the formula \(R = \rho \frac{L}{A}\):

• Option 1: 'A' (Area). Resistance is inversely proportional to area, not directly. So, this is incorrect.
• Option 2: '\(L^2\)' (Length squared). Resistance is directly proportional to length (L), not its square (\(L^2\)). So, this is incorrect.
• Option 3: 'L/A' (Length divided by Area). According to the formula \(R = \rho \frac{L}{A}\), resistance is directly proportional to the term \(\frac{L}{A}\) (since \(\rho\) is constant for a given material). This matches our derivation.
• Option 4: 'A/L' (Area divided by Length). This represents the inverse relationship compared to the correct one. So, this is incorrect.

Conclusion on Proportionality

The resistance of a conductor is directly proportional to its length (L) and inversely proportional to its cross-sectional area (A). This means the resistance is directly proportional to the term \(\frac{L}{A}\).