We are given three copper wires of different lengths and different areas of cross-section. Which one of the following would have highest resistivity ?

1. Electric Current and Ohm's Law (basic)

To understand electricity, we must first understand the "push" that makes charges move. This push is called the Electric Potential Difference (V). Imagine moving a unit charge between two points in a circuit; the work done to achieve this is the potential difference (Science, Chapter 11, p.173). It is measured in Volts (V), named after Alessandro Volta. Mathematically, V = W/Q, where W is work done and Q is charge.

Once this potential difference is applied, current (I) flows. In 1827, Georg Simon Ohm discovered a fundamental relationship between the two, known as Ohm’s Law. It states that the potential difference across a metallic wire is directly proportional to the current flowing through it, provided the temperature remains constant (Science, Chapter 11, p.176). This gives us the famous formula: V = IR.

Here, R represents Resistance, the property of a conductor to oppose the flow of charges. While resistance changes based on the physical dimensions of the conductor—becoming higher if the wire is longer or thinner—the material itself has an underlying, constant property called Electrical Resistivity (ρ). Think of resistance as the "actual hurdle" and resistivity as the "nature of the track." Even if you change the length or thickness of a copper wire, its resistivity remains identical because the material hasn't changed (Science, Chapter 11, p.178).

Sources: Science (NCERT 2025 ed.), Chapter 11: Electricity, p.173; Science (NCERT 2025 ed.), Chapter 11: Electricity, p.176; Science (NCERT 2025 ed.), Chapter 11: Electricity, p.178

2. Conductors, Insulators, and Semiconductors (basic)

To understand electricity, we must first look at the materials through which it travels. Imagine electric current as a flow of water through a pipe. Some materials act like wide-open pipes, while others are like pipes filled with sand, and some are completely blocked off. This ability to allow or resist the flow of electrons is what distinguishes conductors, insulators, and semiconductors.

Conductors are materials that allow electric current to flow through them with very little resistance. This is usually because they have a high density of "free electrons" that can move easily between atoms. Most metals are excellent conductors; for instance, silver, copper, and gold are among the best Science-Class VII, Electricity: Circuits and their Components, p.36. While silver is technically the most efficient, copper is the industry standard for household wiring because it is much more affordable and widely available Science-Class VII, Electricity: Circuits and their Components, p.36. In terms of physical properties, metals are generally hard, lustrous, and ductile, making them ideal for being drawn into the long, thin wires we see in our homes Science-Class VII, The World of Metals and Non-metals, p.48.

On the opposite end of the spectrum are Insulators (or poor conductors). These materials have electrons that are tightly bound to their atoms, making it nearly impossible for current to pass through. This resistance is vital for safety. For example, the plastic or rubber coating on a screwdriver or an electrical wire protects you from electric shocks by preventing the current from reaching your hand Science-Class VII, The World of Metals and Non-metals, p.48. It is important to note that "poor conductor" and "insulator" are often used to describe degrees of resistance—an insulator of the same size as a poor conductor will offer significantly higher resistance to the flow of electricity Science, Class X, Electricity, p.177.

Finally, we have Semiconductors (like Silicon and Germanium). These materials are the middle ground; they don't conduct as well as metals, but they aren't quite insulators either. Their unique ability is that their conductivity can be controlled—for example, by changing their temperature or adding small amounts of other elements. This "controllable" nature is exactly what makes modern electronics, like your smartphone and computer chips, possible.

Sources: Science-Class VII . NCERT(Revised ed 2025), The World of Metals and Non-metals, p.48; Science-Class VII . NCERT(Revised ed 2025), Electricity: Circuits and their Components, p.36; Science, class X (NCERT 2025 ed.), Electricity, p.177

3. Electrical Resistance and its Dependencies (intermediate)

When we talk about Electrical Resistance (R), we are describing the property of a conductor that opposes the flow of electric current. Think of it like friction for moving charges. Through careful observation, we find that the resistance of a uniform metallic conductor is not a random value; it is governed by its physical dimensions and its inherent nature. Specifically, resistance is directly proportional to its length (l) and inversely proportional to its area of cross-section (A) Science, Class X (NCERT 2025 ed.), Chapter 11, p.178. This means if you double the length of a wire, you double the obstacles the electrons face, thereby doubling the resistance. Conversely, a thicker wire (greater area) provides a wider path for electrons, making it easier for current to flow.

To tie these factors together, we use the fundamental formula: R = ρl/A. Here, ρ (rho) is the constant of proportionality known as electrical resistivity Science, Class X (NCERT 2025 ed.), Chapter 11, p.178. It is vital to distinguish between Resistance and Resistivity. While resistance changes if you stretch or thicken a wire, resistivity is an intrinsic property of the material itself. Whether you have a 1-meter copper wire or a 100-meter copper wire, their resistivity remains identical because the material (copper) hasn't changed.

Furthermore, the nature of the material plays a massive role in practical applications. For instance, the coils of electric toasters or irons are often made of alloys rather than pure metals because alloys generally have higher resistivity and do not oxidize (burn) easily at high temperatures Science, Class X (NCERT 2025 ed.), Chapter 11, p.181. This ensures the device generates heat efficiently without the heating element melting away.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.178; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.181

4. Heating Effect of Electric Current (intermediate)

When an electric current flows through a conductor, it isn't a frictionless journey. At the microscopic level, moving electrons constantly collide with the atoms or ions that make up the material. These collisions transfer kinetic energy from the electrons to the atoms, causing them to vibrate more vigorously, which we perceive as an increase in temperature. This phenomenon is known as the Heating Effect of Electric Current. While often seen as a loss of energy in transmission lines, this effect is the fundamental principle behind many household appliances like electric irons, toasters, and water heaters Science, class X (NCERT 2025 ed.), Chapter 11, p.190.

The mathematical relationship governing this is Joule’s Law of Heating. It states that the heat (H) produced in a resistor is directly proportional to: (i) the square of the current (I²) for a given resistance, (ii) the resistance (R) for a given current, and (iii) the time (t) for which the current flows. Formally, this is expressed as H = I²Rt Science, class X (NCERT 2025 ed.), Chapter 11, p.189. This implies that if you double the current passing through a wire, the heat generated doesn't just double—it quadruples!

In practical applications, we categorize this heating as either desirable or undesirable. For instance, in an electric bulb, we want the filament to reach such a high temperature that it emits light, whereas in a computer processor, heat is a harmful byproduct that must be dissipated by fans. A critical safety application is the electric fuse. A fuse is a wire with a specific melting point placed in series with a circuit. If the current exceeds a safe limit (due to overloading or a short-circuit), the Joule heating becomes intense enough to melt the fuse wire, breaking the circuit and preventing fire or damage to appliances Science, class X (NCERT 2025 ed.), Chapter 11, p.190.

Sources: Science, class X (NCERT 2025 ed.), Chapter 11: Electricity, p.189; Science, class X (NCERT 2025 ed.), Chapter 11: Electricity, p.190; Science, class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p.205

5. Temperature Dependence and Superconductivity (exam-level)

In our previous steps, we focused on how the physical shape of a conductor affects its resistance. Now, we must look deeper at what happens inside the material itself. Electrical resistivity (ρ) is an intrinsic property; it reflects the "DNA" of a material rather than its size or shape. However, this "DNA" is not static—it is highly sensitive to temperature. For most metals, as the temperature rises, the atoms inside the conductor vibrate more vigorously. These vibrations act like a crowded hallway where people are waving their arms wildly; it becomes much harder for electrons (the current) to pass through without bumping into something. Consequently, the resistivity of metals increases with temperature Science, Class X (NCERT 2025 ed.), Chapter 11, p. 179.

Interestingly, not all materials react to heat in the same way. Alloys, such as Nichrome or Manganin, are engineered to have very high resistivity that remains relatively stable even when they get hot. This is why alloys are the preferred choice for heating elements in irons and toasters—they can withstand high temperatures without oxidizing (burning) or significantly changing their electrical behavior Science, Class X (NCERT 2025 ed.), Chapter 11, p. 179. In contrast, semiconductors (like silicon) behave oppositely: their resistivity actually decreases as they get warmer because the heat provides enough energy to liberate more electrons for conduction.

The most fascinating boundary of this concept is superconductivity. In certain materials, when the temperature is lowered below a specific threshold called the Critical Temperature (T꜀), the resistivity suddenly drops to exactly zero. In this state, an electric current can flow through a loop indefinitely without any power source, as there is no resistance to dissipate energy as heat. While most known superconductors require extreme cooling (often using liquid helium or nitrogen), discovering "room-temperature superconductors" remains one of the "holy grails" of modern physics because it would revolutionize how we transport energy.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.176, 178-180

6. Resistivity: The Intrinsic Material Property (exam-level)

When we study how electricity flows through a conductor, we often focus on Resistance (R). However, to truly master this topic for the UPSC, you must distinguish between what a specific object does and what the material itself is made of. Imagine two blocks of copper—one long and thin, the other short and thick. They will offer different levels of resistance to current, but they are both still copper. This "inner essence" of the material is what we call Electrical Resistivity (ρ).

Through precise experiments, scientists found that the resistance of a uniform metallic conductor is directly proportional to its length (L) and inversely proportional to its area of cross-section (A). When we combine these observations into a single equation, we get R = ρ(L/A), where ρ (the Greek letter rho) is the constant of proportionality known as resistivity Science, Chapter 11, p.178. While resistance changes if you stretch, cut, or bend a wire, the resistivity remains constant for that material at a given temperature.

Resistivity is a characteristic property of the material. It tells us how strongly a material opposes the flow of electric current regardless of its shape Science, Chapter 11, p.178. Metals and alloys typically have very low resistivity (making them good conductors), while insulators like rubber have incredibly high resistivity. The SI unit for resistivity is the ohm-metre (Ω m).

Sources: Science, Chapter 11: Electricity, p.178

7. Solving the Original PYQ (exam-level)

Now that you have mastered the distinction between resistance and resistivity, this question serves as the perfect litmus test for your conceptual clarity. You've learned that while resistance is an extrinsic property—changing based on a conductor's shape and size—resistivity (ρ) is an intrinsic property. As highlighted in Science, class X (NCERT 2025 ed.), resistivity depends solely on the nature of the material and its temperature. Because the problem specifies that all three wires are made of copper, the material identity remains constant across all options, meaning the characteristic electrical property of the substance does not change.

To arrive at the correct answer, you must look past the specific measurements provided in the options. Whether a copper wire is 50 cm or 10 cm long, or whether it is thick or thin, the substance itself remains copper. Since the temperature is implicitly constant, the resistivity remains identical for all three specimens regardless of their physical dimensions. Therefore, the logical conclusion is (D) All the wires would have same resistivity. A coach's tip: whenever a question asks about resistivity, your first instinct should be to check if the material changes; if the material is the same, the resistivity is the same.

The trap here lies in the numerical data provided in options (A), (B), and (C). UPSC frequently provides specific dimensions like "50 cm length" or "0.5 mm diameter" to trigger a "calculation reflex" in students. The common mistake is confusing resistivity with resistance. If the question had asked for the highest resistance, you would indeed need to use the formula R = ρL/A to find which wire has the highest length-to-area ratio. However, by asking for resistivity, the examiners are testing whether you can distinguish between a material constant and a geometric variable. Don't let the distractors lure you into unnecessary computation!