Which of the following statements are true with regard to resistance?

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Examoogle · Accepted Answer

Resistance is directly proportional to a length of the wire

Explanation: The electrical resistance $R$ of a uniform conductor is directly proportional to its length $l$ and inversely proportional to its cross-sectional area $A$. This relationship is defined by the formula $R = \rho \frac{l}{A}$, where $\rho$ is the resistivity of the material.

Working Principle: As the length of a wire increases, the number of electron collisions with ions in the crystal lattice increases, leading to higher resistance. Conversely, increasing the cross-sectional area provides more paths for charge carriers, thereby reducing the effective resistance of the material.

Key Formulas: $R = \rho \frac{l}{A}$ — Resistance formula where $\rho$ is resistivity, $l$ is length, and $A$ is area.; $\sigma = \frac{1}{\rho}$ — Conductivity is the reciprocal of resistivity.

Key Points: Resistance increases linearly with length ($R \propto l$).; Resistance decreases as cross-sectional area increases ($R \propto \frac{1}{A}$).; Resistivity ($\rho$) is a property of the material and does not change with geometry.

Advantages: Predictable circuit behavior; Uniform current distribution in homogeneous materials

Disadvantages: Resistance changes with temperature ($\alpha$ coefficient); Skin effect at high frequencies reduces effective cross-sectional area

Applications: Resistor manufacturing; Power transmission lines design

Additional Info: Option B is incorrect because $R \propto \frac{1}{A}$.. Option C is incorrect because $R$ is directly, not inversely, proportional to length.. Option D is incorrect because $R$ is directly proportional to resistivity ($\rho$).

Answer: A is correct — Resistance is directly proportional to the length of the wire, consistent with the fundamental relation $R = \rho \frac{l}{A}$.

Exam Tip: Always remember the mnemonic 'Longer wires have more resistance, fatter wires have less resistance' for quick analysis in DC circuit problems.

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