Two pieces of conductor of same material and of equal length are connected in series with a cell. One of the two pieces has cross-sectional area double that of the other. Which one of the following statements is correct in this regard ?

1. Electric Current and Potential Difference (basic)

Imagine a pipe filled with water. If the water is sitting still, there is no 'flow.' In physics, electric current is the rate at which electric charges move through a conductor, like a copper wire Science, Class X, Chapter 11, p. 171. In metallic circuits, these charges are carried by electrons. However, because electricity was studied long before electrons were discovered, we use a 'conventional' direction for current—it is taken as flowing from the positive terminal to the negative terminal, which is opposite to the actual direction of electron flow Science, Class X, Chapter 11, p. 192.

To mathematically express this, if a net charge (Q) flows across any cross-section of a conductor in time (t), then the current (I) is given by: I = Q/t. The SI unit of electric current is the Ampere (A). A continuous and closed path through which this current flows is known as an electric circuit; if this path is broken anywhere, the current stops immediately Science, Class X, Chapter 11, p. 171.

But what makes the charges move in the first place? Electrons don't just flow on their own; they need a 'push' or a difference in electrical pressure. This is called Electric Potential Difference (V). In a circuit, this pressure is provided by a cell or a battery. We define the potential difference between two points as the work done to move a unit charge from one point to the other Science, Class X, Chapter 11, p. 172. It is measured in Volts (V). Without this potential difference, there can be no sustained flow of current.

Sources: Science, Class X, Chapter 11: Electricity, p.171; 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 electricity lies a fundamental relationship discovered by Georg Simon Ohm. Ohm’s Law states that the electric current (I) flowing through a metallic conductor is directly proportional to the potential difference (V) across its ends, provided its temperature remains constant. Mathematically, this is expressed as V = IR, where R is the constant of proportionality called Resistance Science, Class X (NCERT 2025 ed.), Chapter 11, p. 192. Think of Voltage as the "push" and Resistance as the "friction" that opposes the flow of charges. The SI unit of resistance is the ohm (Ω). If a potential difference of 1 Volt allows 1 Ampere of current to flow, we say the resistance is 1 Ω Science, Class X (NCERT 2025 ed.), Chapter 11, p. 176.

But what determines how much resistance a wire offers? It isn't just a random number; it depends on the physical geometry and the material of the conductor. Specifically, 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). This gives us the 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.

Factor	Relationship with Resistance	Practical Logic
Length (L)	Directly Proportional (R ∝ L)	A longer path means more collisions for electrons, increasing resistance.
Area (A)	Inversely Proportional (R ∝ 1/A)	A thicker wire (wider area) allows charges to flow more easily, decreasing resistance.
Material (ρ)	Depends on Nature	Silver is a great conductor (low ρ); rubber is an insulator (high ρ).

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

3. Factors Affecting Resistance and Resistivity (intermediate)

To understand how electricity flows, we must look at what hinders it. Resistance (R) is the property of a conductor to oppose the flow of electric current. Through precise experiments, it has been established that the resistance of a uniform metallic conductor depends on three primary physical factors: its length (l), its area of cross-section (A), and the nature of its material Science, Chapter 11, p.178.

Mathematically, resistance is directly proportional to length (R ∝ l) and inversely proportional to the area of cross-section (R ∝ 1/A). When we combine these, we get the fundamental formula: R = ρ(l/A). Here, ρ (rho) is the constant of proportionality called electrical resistivity. While resistance changes if you stretch or thicken a wire, resistivity is an intrinsic property of the material itself. For instance, a thin copper wire and a thick copper block have different resistances, but their resistivity remains identical because they are both copper Science, Chapter 11, p.181.

The choice of materials in our daily appliances is dictated by these properties. Metals like copper and aluminium have very low resistivity, making them ideal for transmission wires. In contrast, alloys like Nichrome have higher resistivity than their constituent metals and do not oxidize (burn) easily at high temperatures. This is precisely why the coils of electric irons and toasters are made of alloys rather than pure metals Science, Chapter 11, p.181.

Factor	Relationship with Resistance (R)	Physical Logic
Length (l)	Directly Proportional (R ∝ l)	A longer path offers more collisions for electrons.
Area (A)	Inversely Proportional (R ∝ 1/A)	A wider 'pipe' allows charges to flow more easily.
Material (ρ)	Depends on Nature	Determined by how strongly the atoms hold onto electrons.

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

4. Domestic Electric Circuits and Safety (intermediate)

To understand how electricity safely powers our homes, we must look at the Domestic Electric Circuit as a carefully engineered system. In India, power is supplied through two main wires: the Live wire (positive, usually with red insulation) and the Neutral wire (negative, with black insulation). The potential difference between these two is maintained at 220 V. A third wire, the Earth wire (green insulation), serves as a critical safety valve by providing a low-resistance path for leakage current to the ground Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p. 204.

Unlike simple battery circuits, domestic appliances are connected parallel to each other. This design ensures two things: first, every appliance receives the same 220 V potential difference; second, if one appliance is switched off or fails, the rest of the circuit remains functional Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p. 205. However, this system faces two major risks: Short-circuiting (when live and neutral wires touch directly due to damaged insulation) and Overloading (when too many high-power appliances are used simultaneously). In both cases, the current increases abruptly, leading to dangerous levels of heat.

The primary guardian against these risks is the Electric Fuse. Connected in series with the circuit, it consists of a wire with a low melting point. It operates on Joule’s Law of Heating (H = I²Rt). If the current (I) exceeds a safe limit, the heat produced increases rapidly, melting the fuse wire and breaking the circuit before damage occurs. Interestingly, for a given material and length, a thinner wire has a higher resistance (R = ρL/A). Because it is in series, the same current flows through it as the rest of the circuit, but the higher resistance causes it to produce more heat and melt first, protecting the thicker, more expensive house wiring Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p. 190.

Feature	Live Wire	Neutral Wire	Earth Wire
Color Code	Red	Black	Green
Function	Carries current to appliance	Completes the return path	Safety/Discharges leakage

Sources: Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p.204-205; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.190

5. Electric Power and Commercial Energy (intermediate)

In our journey through electricity, we now reach the concept of Electric Power—the rate at which work is done or electrical energy is consumed in a circuit. While potential difference tells us about the "push" and current tells us about the "flow," power tells us how fast the energy is being used up Science, Class X (NCERT 2025 ed.), Chapter 11, p. 191. Mathematically, Power (P) = V × I. By applying Ohm’s Law (V = IR), we can derive two other vital forms: P = I²R and P = V²/R. These equations are not just formulas; they are tools for engineers. For instance, if you connect two different wires in series, the same current flows through both. Since P = I²R, the wire with higher resistance (like a thinner wire) will dissipate more heat energy Science, Class X (NCERT 2025 ed.), Chapter 11, p. 185.

The SI unit of power is the Watt (W), defined as the power consumed by a device when 1 Ampere of current flows at a potential difference of 1 Volt. However, in our daily lives and industries, a Watt is a tiny unit. We instead use Kilowatts (kW), where 1 kW = 1000 W. It is crucial to distinguish between Power (the rate) and Energy (the total amount). Electrical Energy is the product of power and time (E = P × t). If you leave a 100W bulb on for 10 hours, you have consumed a specific amount of energy, which we measure commercially in Kilowatt-hours (kWh), often simply called a 'unit' Science, Class X (NCERT 2025 ed.), Chapter 11, p. 192.

Term	Unit	Physical Meaning
Electric Power	Watt (W) or J/s	How fast energy is used (Rate)
Electric Energy	Joule (J) or kWh	Total capacity of work done over time

From a socio-economic lens, electricity is more than just physics; it is a development indicator. The per capita consumption of electricity in a country reflects its industrial growth and the quality of life of its citizens. Currently, India's per capita consumption (around 350 kWh) is significantly lower than the global average and far behind developed nations like the USA, which consumes about 7000 kWh per head Geography of India, Majid Husain (McGrawHill 9th ed.), Energy Resources, p. 17. As we transition to cleaner sources like solar and wind, understanding the efficiency of power consumption becomes vital for sustainable growth.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.185, 191, 192; Geography of India, Majid Husain (McGrawHill 9th ed.), Energy Resources, p.17

6. Resistors in Series and Parallel (intermediate)

In any electrical circuit, the way we arrange resistors determines how current and voltage behave across the system. When we connect resistors end-to-end so that the same current flows through each one in sequence, we call this a Series Circuit Science, Class X (NCERT 2025 ed.), Chapter 11, p. 182. In this configuration, the total resistance (Rₛ) is simply the sum of individual resistances (R₁ + R₂ + ...). This means adding more resistors in series always increases the total resistance and decreases the overall current in the circuit. Conversely, in a Parallel Circuit, resistors are connected between the same two points (X and Y), providing multiple paths for the current Science, Class X (NCERT 2025 ed.), Chapter 11, p. 185. Here, the potential difference (V) across each resistor remains identical, but the total current (I) divides among the branches. Interestingly, the reciprocal of the equivalent resistance (1/Rₚ) is the sum of the reciprocals of individual resistances. This results in an equivalent resistance that is always less than the smallest individual resistor in the combination Science, Class X (NCERT 2025 ed.), Chapter 11, p. 187. Understanding these arrangements is crucial for predicting Joule’s Heating (H = I²Rt). In a series circuit, because the current (I) is constant, the resistor with the highest resistance will produce the most heat. In a parallel circuit, however, because the voltage (V) is constant and I = V/R, the resistor with the lowest resistance will draw the most current and consequently generate the most heat.

Feature	Series Connection	Parallel Connection
Current (I)	Same through all components	Divided among branches
Voltage (V)	Divided across components	Same across all branches
Total Resistance	Rₛ = R₁ + R₂ + R₃	1/Rₚ = 1/R₁ + 1/R₂ + 1/R₃
Impact of failure	Circuit breaks (everything turns off)	Other branches continue to work

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.182; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.185; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.187

7. Joule's Law of Heating (exam-level)

When we talk about Joule’s Law of Heating, we are looking at the conversion of electrical energy into thermal energy. At a microscopic level, as electrons drift through a conductor, they constantly collide with the atoms of the material. These collisions transfer kinetic energy to the atoms, causing them to vibrate more vigorously, which we observe as a rise in temperature. This is known as the heating effect of electric current Science, Class VIII, Electricity: Magnetic and Heating Effects, p.53.

While this heat is often an "inevitable consequence" that leads to energy waste in power lines, it is the very principle that makes devices like electric irons, toasters, and water heaters work Science, Class X, Electricity, p.190. Joule's Law provides a precise mathematical way to calculate this heat (H). It states that the heat produced in a resistor is:

• Directly proportional to the square of the current (I²) for a given resistance.
• Directly proportional to the resistance (R) for a given current.
• Directly proportional to the time (t) for which the current flows Science, Class X, Electricity, p.189.

The resulting formula is H = I²Rt. This "squared" relationship with current is critical; it means that if you double the current passing through a wire, the heat generated doesn't just double—it quadruples.

Variable Change	Effect on Heat (H)	Reasoning
Current (I) is doubled	4x Heat	H ∝ I² (2² = 4)
Resistance (R) is doubled	2x Heat	H ∝ R (Linear)
Time (t) is doubled	2x Heat	H ∝ t (Linear)

A practical application often tested in exams involves conductors in series. In a series circuit, the current (I) remains constant through all components. Since H = I²Rt, the component with the highest resistance will generate the most heat. For example, if you have a thin wire and a thick wire of the same material in series, the thinner wire—which has a higher resistance due to its smaller cross-sectional area—will become much hotter than the thick one.

Sources: Science, Class VIII, Electricity: Magnetic and Heating Effects, p.53; Science, Class X, Electricity, p.189; Science, Class X, Electricity, p.190

8. Solving the Original PYQ (exam-level)

This question serves as a perfect synthesis of three fundamental pillars of electricity you have just studied: series circuit behavior, the geometry of resistance, and Joule’s Law of Heating. As your coach, I want you to see how the UPSC tests your ability to layer these concepts systematically. First, identify the primary constraint: the conductors are in a series connection. This fundamental rule, found in Science, class X (NCERT), dictates that the electric current (I) must be identical through every component in the chain. This single realization allows you to instantly discard options (A) and (B), which incorrectly suggest that current varies based on thickness.

Now, let's walk through the physical properties. Since the material and length are identical, resistance depends entirely on the cross-sectional area (A). Using the relationship R = ρL/A, we know that resistance is inversely proportional to the area; therefore, the thinner wire possesses a higher resistance. Finally, we apply Joule’s Law of Heating (H = I²Rt). Because the current (I) and time (t) are constant for both pieces, the heat produced is directly proportional to the resistance. Consequently, the piece with higher resistance—the thinner one—will generate more thermal energy, making (D) Same amount of electric current would pass through both the pieces producing more heat in the thinner one the only logical conclusion.

The common trap in this question lies in confusing series logic with parallel logic. In a parallel circuit, a thicker wire would indeed allow more current to pass (the path of least resistance). UPSC uses Option (C) to catch students who understand the current remains the same but fail to connect low area to high resistance. Always remember: in a series circuit, higher resistance leads to higher heat production because the current is forced through the more difficult path with the same intensity.