Precise definition: In circuit theory, how is electrical resistance best defined?

Introduction / Context:
Resistance is a fundamental concept relating voltage, current, and energy dissipation. It underpins Ohm’s law, thermal design, and noise analysis. A crisp definition helps prevent confusion with related but distinct ideas such as conductance, impedance, and analogies used in pedagogy.

Given Data / Assumptions:

• Linear resistor behavior is assumed for small signals and constant temperature.
• Units: ohm (Ω) for resistance.
• Real resistors dissipate power as heat when current flows.

Concept / Approach:
Resistance R characterizes how a component opposes current for a given applied voltage: V = I * R. When current passes through a resistive element, electrical energy converts to heat at a rate P = I^2 * R = V^2 / R = V * I. This thermal effect is inseparable from resistance in passive materials (aside from superconductivity and idealizations).

Step-by-Step Solution:

Verification / Alternative check:
Measure a resistor’s temperature rise under load; infrared imaging or simple touch (carefully) confirms heat generation proportional to I^2 * R and validates the definition in practice.

Why Other Options Are Wrong:

• Directly proportional to conductance: Conductance G = 1 / R, so they are inversely proportional.
• Directly proportional to current and voltage: R is the ratio V / I, not proportional to both independently.
• Fluid flow analogy: Useful for intuition, but it is not a formal definition.

Common Pitfalls:
Assuming resistance is constant regardless of temperature or frequency; many resistors change value with temperature (tempco) and show parasitic inductance/capacitance at high frequency.

Final Answer:
the opposition to current flow accompanied by the dissipation of heat