The resistance of a wire of length / and area of cross-section a is x ohm. If the wire is stretched to double its length, its resistance would become:

1. Basics of Electric Current and Potential Difference (basic)

To understand electricity, we must first look at what happens inside a wire. Imagine a copper wire: it is filled with tiny particles called electrons. When these electrons move together in a specific direction, we call this flow an Electric Current. Formally, electric current is the rate of flow of electric charge through a cross-section of a conductor Science, Class X, Chapter 11, p.192. If a net charge (Q) flows across any cross-section of a conductor in time (t), then the current (I) is represented by the formula I = Q / t. The SI unit used to measure this flow is the Ampere (A), named after the French scientist André-Marie Ampère Science, Class X, Chapter 11, p.172.

Now, why do these electrons move at all? They don't move on their own, just as water doesn't flow in a perfectly level pipe. To make charges move, we need a "pressure difference," which in electricity is called Potential Difference (or Voltage). This is the work done to move a unit charge from one point to another. In a practical circuit, this "push" is provided by a cell or a battery. Chemical action within the cell generates this potential difference across its terminals, even when no current is being drawn Science, Class X, Chapter 11, p.192. The SI unit for potential difference is the Volt (V).

It is important to note a historical quirk regarding the direction of current. When electricity was first discovered, electrons were unknown, so current was assumed to be the flow of positive charges. Therefore, by convention, the direction of electric current is taken as opposite to the direction of the flow of electrons, which are negatively charged particles Science, Class X, Chapter 11, p.192.

Feature	Electric Current (I)	Potential Difference (V)
Definition	The rate of flow of electric charges.	The work done to move a unit charge.
Analogy	The flow of water in a pipe.	The pressure difference that causes flow.
SI Unit	Ampere (A)	Volt (V)

Sources: Science, Class X, Chapter 11: Electricity, p.172; Science, Class X, Chapter 11: Electricity, p.192

2. Ohm's Law and the Concept of Resistance (basic)

At the heart of understanding how electricity flows through a circuit lies Ohm's Law. Imagine water flowing through a pipe; the pressure pushing the water is like voltage (V), the flow of water is the current (I), and the narrowness of the pipe that restricts flow is the resistance (R). Ohm's Law states that the potential difference (V) across the ends of a metallic wire is directly proportional to the current (I) flowing through it, provided its temperature remains constant. Mathematically, this is expressed as V = IR Science, Class X (NCERT 2025 ed.), Chapter 11, p.176. Resistance is the inherent property of a conductor to resist the flow of charges, and its SI unit is the ohm (Ω).

But what actually determines how much a wire will resist current? Resistance isn't just a random number; it depends on the physical dimensions and the material of the conductor. The resistance (R) of a uniform metallic conductor is directly proportional to its length (l) and inversely proportional to its area of cross-section (A). We combine these into the fundamental formula: R = ρ(l/A), where ρ (rho) is the electrical resistivity, a characteristic property of the material itself Science, Class X (NCERT 2025 ed.), Chapter 11, p.192.

A classic conceptual trap involves changing the shape of a wire. If you take a wire and stretch it to double its length, its resistance doesn't just double—it quadruples! This happens because as the wire gets longer (l becomes 2l), it must also become thinner to keep the total volume constant. Consequently, the cross-sectional area (A) halves (becomes A/2). When you plug these new values into our formula, the 2 from the length and the 2 from the denominator of the area multiply, resulting in four times the original resistance Science, Class X (NCERT 2025 ed.), Chapter 11, p.178.

Factor	Relationship with Resistance (R)	Effect of Increasing the Factor
Length (l)	Directly Proportional (R ∝ l)	Resistance increases
Area (A)	Inversely Proportional (R ∝ 1/A)	Resistance decreases (thicker wire = less resistance)
Temperature	Directly Proportional (for metals)	Resistance increases as atoms vibrate more

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.176; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.178; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.192

3. Factors Affecting Electrical Resistance (intermediate)

To understand what determines the electrical resistance (R) of a conductor, we must look at how electrons navigate through the material. Imagine a crowd moving through a corridor: the longer the corridor, the more collisions occur; the wider the corridor, the easier it is to pass through. Similarly, scientific observations show 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) Science, Class X (NCERT 2025 ed.), Chapter 11, p.178.

This relationship is expressed by the fundamental formula: R = ρ (l / A). Here, ρ (rho) represents the electrical resistivity of the material. While resistance depends on the shape and size of the object, resistivity is an intrinsic property of the material itself. For instance, at a constant temperature, a thick copper wire and a thin copper wire have the same resistivity, but the thin wire will have a higher resistance because its cross-sectional area (the path for electrons) is smaller Science, Class X (NCERT 2025 ed.), Chapter 11, p.181.

Factor	Relationship with Resistance (R)	Practical Effect
Length (l)	Directly Proportional (R ∝ l)	Doubling the length doubles the resistance.
Area (A)	Inversely Proportional (R ∝ 1/A)	A thicker wire (larger A) has lower resistance.
Material (ρ)	Depends on Nature of Substance	Silver is a better conductor than Iron due to lower resistivity.

An intermediate concept often tested is the stretching of a wire. When a wire is stretched to increase its length, its volume (V = l × A) remains constant. If you stretch a wire to double its length (l' = 2l), its cross-sectional area must simultaneously decrease to half (A' = A/2) to keep the volume same. Substituting these new values into our formula (R' = ρ × 2l / (A/2)), we find that the new resistance becomes four times (4x) the original. This is because you are simultaneously making the "corridor" twice as long and half as wide.

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

4. Combination of Resistors: Series and Parallel (intermediate)

In our journey through electricity, we often encounter circuits with multiple components. To simplify these, we look for an equivalent resistance—a single value that could replace the entire group without changing the current or voltage in the rest of the circuit. Resistors can be combined in two primary ways: Series and Parallel. Each arrangement governs how current and potential difference (voltage) behave across the components.

When resistors are connected in series, they are joined end-to-end, providing only one path for the electric charge. Consequently, the current (I) remains the same through every resistor. However, the total potential difference (V) is distributed across them. By applying Ohm’s law, we find that the total resistance (Rs) is simply the sum of individual resistances: Rs = R₁ + R₂ + R₃. This results in a total resistance that is greater than any individual resistor in the chain Science, Class X (NCERT 2025 ed.), Chapter 11, p. 184.

Conversely, in a parallel circuit, resistors are connected across the same two points, creating multiple branches. Here, the potential difference (V) is the same across each resistor, but the total current divides among the branches. The mathematics here is reciprocal: the reciprocal of the equivalent resistance (1/Rp) equals the sum of the reciprocals of the individual resistances: 1/Rp = 1/R₁ + 1/R₂ + 1/R₃. This arrangement always results in an equivalent resistance that is smaller than the smallest resistor in the group Science, Class X (NCERT 2025 ed.), Chapter 11, p. 186.

Feature	Series Combination	Parallel Combination
Current (I)	Same through all resistors	Splits across branches
Voltage (V)	Splits across resistors	Same across all branches
Total Resistance	Increases (Rs = ΣR)	Decreases (1/Rp = Σ1/R)

It is also fascinating to consider the geometry of a single conductor. If you stretch a wire to double its length (l), its cross-sectional area (A) must decrease to half to keep the volume constant. Since resistance is directly proportional to length and inversely proportional to area (R = ρl/A), doubling the length while halving the area actually quadruples the resistance (4x) Science, Class X (NCERT 2025 ed.), Chapter 11, p. 178. You can think of stretching a wire as adding more "lengthwise" resistance in series while losing the "thickness" that parallel paths (area) provide!

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.178; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.184; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.186

5. Heating Effect of Electric Current (intermediate)

When an electric current flows through a conductor, it isn't a frictionless journey. Think of electrons as runners trying to sprint through a crowded marketplace; they constantly collide with the atoms and ions of the material. These collisions transfer kinetic energy from the electrons to the particles of the conductor, causing them to vibrate more vigorously. This increased internal energy manifests as an increase in temperature, a phenomenon we call the Heating Effect of Electric Current. This transformation of electrical energy into heat is an inevitable consequence of resistance in any circuit.

To quantify this, we look to Joule’s Law of Heating. It states that the heat (H) produced in a resistor is directly proportional to three specific factors: the square of the current (I²), the resistance (R), and the time (t) for which the current flows. This is expressed by the formula H = I²Rt Science, Class X (NCERT 2025 ed.), Chapter 11, p.189. For a UPSC aspirant, the "square of the current" is the most critical part—it means if you double the current passing through a wire, the heat generated doesn't just double; it quadruples. This is why high-power appliances require thicker wires to manage the heat and prevent insulation from melting.

While heating is often a loss of energy (like in computers or phone chargers), we harness it intentionally in many devices. Appliances like electric irons, toasters, and kettles use high-resistance alloys (like Nichrome) to maximize heat production Science, Class X (NCERT 2025 ed.), Chapter 11, p.190. This effect is also utilized in safety devices like the electric fuse. A fuse wire is designed with a specific melting point; when an accidental surge or short-circuit causes the current to rise abruptly, the Joule heating melts the fuse, breaking the circuit and protecting your home from fire Science, Class X (NCERT 2025 ed.), Chapter 12, p.205. Beyond the home, industrial electric furnaces use this principle to melt and recycle scrap steel at massive scales Curiosity — Textbook of Science for Grade 8, Chapter 4, p.54.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.189-190; Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p.205; Curiosity — Textbook of Science for Grade 8, Chapter 4: Electricity: Magnetic and Heating Effects, p.54

6. Electric Power and Commercial Energy Units (intermediate)

To understand electric power, we must first look at the rate of doing work. In an electrical context, Electric Power (P) is the rate at which electrical energy is consumed or dissipated in a circuit. From first principles, we know that power is the product of potential difference (V) and current (I). This gives us the foundational formula P = VI Science, Class X, Chapter 11, p.191. Since power is essentially energy per unit time, its SI unit is the Watt (W), where 1 Watt is equal to 1 Joule per second (1 J/s). One watt of power is consumed when 1 Ampere of current flows through a circuit at a potential difference of 1 Volt Science, Class X, Chapter 11, p.192. Depending on the known variables in a circuit, we can express power in different ways using Ohm’s Law (V = IR). These alternative formulas are crucial for solving complex problems:

• P = VI: Used when voltage and current are known.
• P = I²R: Useful for series circuits where current (I) is constant.
• P = V²/R: Useful for parallel circuits (like household wiring) where voltage (V) is constant.

While the 'Watt' is the standard unit, it is far too small for commercial billing purposes. For instance, a single household consumes millions of Joules every day. To make these numbers manageable, we use the Commercial Unit of Electrical Energy, known as the Kilowatt-hour (kWh), which is commonly referred to as a 'unit' on your electricity bill Science, Class X, Chapter 11, p.192. One kWh represents the energy consumed by a 1000-Watt appliance running for exactly one hour.

To convert this commercial unit back into the SI unit (Joules), we calculate the energy as Power × Time:

1 kWh = 1000 W × 3600 seconds = 3,600,000 Joules = 3.6 × 10⁶ J Science, Class X, Chapter 11, p.192.

Term	Definition	Standard Unit
Electric Power	Rate of energy consumption	Watt (W) or J/s
Electric Energy	Total work done (Power × Time)	Joule (J) or kWh

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.191-192

7. Geometry of Resistance: Stretching and Volume Conservation (exam-level)

To understand how stretching a wire impacts its electrical resistance, we must first look at the geometry of a conductor. Every uniform wire has a length (l) and a cross-sectional area (A). According to the fundamental law of resistance, the resistance (R) is directly proportional to length and inversely proportional to the area of cross-section (Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p. 178). This gives us the standard formula: R = ρ(l/A), where ρ (rho) is the resistivity, a constant determined solely by the material and temperature.

The critical concept often missed in competitive exams is Volume Conservation. A wire is essentially a cylinder with a volume V = l × A. When you stretch a wire to increase its length, you are not adding more metal; you are simply redistributing the existing material. Therefore, the total volume must remain constant. If you stretch a wire to double its length (l' = 2l), the cross-sectional area must decrease to half (A' = A/2) to keep the product (l × A) the same. If the wire becomes longer and thinner simultaneously, both factors work together to increase the resistance significantly.

Let's look at the math of doubling the length: the new resistance R' = ρ(2l / (A/2)). When we simplify this, the 2 in the denominator's denominator flips to the top, giving us R' = 4 × ρ(l/A), or 4R. This explains why the ammeter reading (current) decreases when a wire's length is increased or when a thinner wire is used (Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p. 178). In general, if a wire is stretched to 'n' times its original length, its resistance increases by a factor of n².

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.178; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.193

8. Solving the Original PYQ (exam-level)

Excellent work on mastering the fundamentals of electricity! This question perfectly bridges the gap between theoretical formulas and practical application. To solve this, you must synthesize two critical concepts you just studied: the relationship between resistance and geometry ($R = \rho l/A$) and the law of conservation of mass/volume. In the UPSC context, a common mistake is looking at length in isolation. However, as a coach, I want you to remember that when a material is physically stretched, its total volume remains constant. As you increase the length ($l$), the wire must simultaneously become thinner, meaning the area of cross-section ($A$) decreases.

Let’s walk through the logic: the question states the wire is stretched to double its length ($l' = 2l$). Because the volume ($V = l \times A$) is constant, doubling the length forces the area to be reduced by half ($A' = A/2$). When you plug these new values into our primary formula, the numerator doubles and the denominator is halved. Mathematically, this creates a factor of four ($2 \div 0.5 = 4$), leading us to the conclusion that the new resistance is (C) 4 x ohm. This quadrupling effect is a classic example of how resistance is directly proportional to length and inversely proportional to the area of cross-section working in tandem, as detailed in Science, class X (NCERT 2025 ed.).

UPSC often uses specific "distractors" to test the depth of your conceptual clarity. Option (A) 2 x ohm is the most common trap; it targets students who only account for the doubling of length but forget the thinning of the wire. Option (B) 0.5 x ohm assumes you confused the proportionality, thinking resistance decreases with length. By understanding that both length and area change during stretching, you avoid these pitfalls and demonstrate the precise thinking required for the Civil Services Examination.