Three resistance coils of 1 Q, 2 Q and 3 Q are connected in series. If the combination is connected to a battery of 9 V, what is the potential drop across the resistance coil of 3 fi ?

1. Fundamentals of Electric Current and Potential (basic)

To understand electricity, we must first look at the tiny particles called electrons. An electric current is essentially a stream of electrons moving through a conductor, such as a copper wire. By historical convention, we say current flows from the positive terminal to the negative terminal, which is opposite to the actual direction of electron flow Science, Class X, Electricity, p.192. The strength of this flow is measured in Amperes (A). However, electrons do not move on their own; they require a 'push' or pressure. This pressure is known as the Electric Potential Difference (V). Imagine moving a heavy box from one point to another—it requires work. Similarly, potential difference is defined as the work done to move a unit charge from one point to another (V = W/Q). Its SI unit is the Volt (V), named after Alessandro Volta Science, Class X, Electricity, p.173. For instance, a 6 V battery provides 6 Joules of energy to every Coulomb of charge passing through it. Finally, every material offers some degree of 'friction' to this flow, called Resistance (R), measured in Ohms (Ω). The relationship between these three—Current, Voltage, and Resistance—is captured by Ohm’s Law, which states that V = IR Science, Class X, Electricity, p.176. In a simple series circuit, where components are connected one after another, the same current must pass through every component, but the total 'push' (voltage) from the battery is shared among them based on their individual resistance Science, Class X, Electricity, p.183.

Sources: Science, Class X, Electricity, p.192; Science, Class X, Electricity, p.173; Science, Class X, Electricity, p.176; Science, Class X, Electricity, p.183

2. Ohm’s Law: The V-I Relationship (basic)

Imagine you are trying to push water through a garden hose. The harder you push (the pressure), the faster the water flows. In the world of electricity, Ohm’s Law describes this exact relationship. In 1827, German physicist Georg Simon Ohm discovered 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 Science, Class X (NCERT 2025 ed.), Electricity, p.176.

Mathematically, this relationship is expressed as V ∝ I. To turn this proportionality into an equation, we introduce a constant called Resistance (R). Thus, the fundamental formula becomes V = IR. Resistance is the inherent property of a conductor to resist or oppose the flow of electric charges through it Science, Class X (NCERT 2025 ed.), Electricity, p.176. If you keep the resistance the same and double the voltage, the current will also double. Conversely, if you double the resistance for the same voltage, the current will be halved.

When we plot this relationship on a V–I graph, it results in a straight line passing through the origin. This linear graph proves that the ratio of Voltage to Current (V/I) is always a constant value for a given material at a specific temperature Science, Class X (NCERT 2025 ed.), Electricity, p.176. In more complex circuits, such as those where multiple resistors are connected in series, the total resistance is simply the sum of individual resistances, but the fundamental law (V = IR) still applies to the circuit as a whole and to each component individually Science, Class X (NCERT 2025 ed.), Electricity, p.185.

Sources: Science, Class X (NCERT 2025 ed.), Electricity, p.176; Science, Class X (NCERT 2025 ed.), Electricity, p.185

3. Resistivity and Material Properties (intermediate)

To understand how electricity flows, we must distinguish between Resistance—which is a property of a specific object—and Resistivity—which is an inherent characteristic of the material itself. Imagine water flowing through a pipe: the resistance depends on how long or narrow the pipe is, but the "slickness" of the pipe's inner surface is like resistivity. According to the principles of physics, 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 relationship is expressed by the formula: R = ρ (l / A) Science, Class X (NCERT 2025 ed.), Electricity, p.178.

In this equation, the Greek letter ρ (rho) represents the electrical resistivity. While resistance changes if you stretch or thin out a wire, resistivity remains constant for a specific material at a constant temperature. It is measured in ohm-meters (Ω m). Materials are classified based on this property: conductors (like copper and aluminium) have very low resistivity ($10^{–8}$ Ω m to $10^{–6}$ Ω m), while insulators (like rubber and glass) have incredibly high resistivity, often reaching $10^{12}$ to $10^{17}$ Ω m Science, Class X (NCERT 2025 ed.), Electricity, p.179.

Feature	Resistance (R)	Resistivity (ρ)
Nature	Property of the specific component/object.	Inherent property of the material.
Dependencies	Depends on length, area, material, and temperature.	Depends only on the nature of the material and temperature.
SI Unit	Ohm (Ω)	Ohm-meter (Ω m)

An important practical application of these properties is found in alloys (like Nichrome). Alloys generally have higher resistivity than their constituent pure metals and, crucially, do not oxidise (burn) readily at high temperatures. This makes them perfect for heating elements in electric irons and toasters. In contrast, pure metals like tungsten are used for bulb filaments due to their high melting points, and copper is used for transmission lines because of its exceptionally low resistivity, which minimizes energy loss Science, Class X (NCERT 2025 ed.), Electricity, p.179.

Sources: Science, Class X (NCERT 2025 ed.), Electricity, p.178; Science, Class X (NCERT 2025 ed.), Electricity, p.179

4. Heating Effect of Electric Current (intermediate)

When electric current flows through a conductor, it doesn't move perfectly freely. It encounters resistance, which you can think of as a form of internal friction. As electrons drift through the wire, they repeatedly collide with the atoms and ions of the conducting material. These collisions transfer kinetic energy to the atoms, making them vibrate more vigorously. This internal agitation is what we perceive as an increase in temperature—a phenomenon known as the heating effect of electric current Science, Class VIII (NCERT), Electricity: Magnetic and Heating Effects, p.58.

This relationship is mathematically defined by Joule’s Law of Heating. The law states that the heat (H) produced in a resistor is directly proportional to three factors: the square of the current (I²), the resistance (R) of the conductor, and the time (t) for which the current flows. Written as H = I²Rt, this formula reveals a critical insight: doubling the current through a circuit doesn't just double the heat; it increases it by four times (2² = 4). This is why electrical systems must be carefully designed to manage heat, especially when high currents are involved Science, Class X (NCERT), Electricity, p.189.

While heating is often seen as a "loss" of energy in transmission lines, it is also highly useful. We intentionally harness this effect in domestic appliances like electric irons, kettles, and toasters. Even the traditional electric bulb works on this principle—the filament is heated to such an extreme temperature that it begins to emit light Science, Class X (NCERT), Electricity, p.190. Beyond the home, industrial electric furnaces use high-intensity heating to melt and recycle scrap steel into new products Science, Class VIII (NCERT), Electricity: Magnetic and Heating Effects, p.54.

Context	Examples
Useful Applications	Electric laundry iron, geysers, fuses, light bulb filaments, industrial furnaces.
Undesirable Effects	Energy loss in power lines, overheating of computer processors, melting of wire insulation.

Sources: Science, Class VIII (NCERT), Electricity: Magnetic and Heating Effects, p.54, 58; Science, Class X (NCERT), Electricity, p.189-190

5. Electric Power and Energy Consumption (intermediate)

In our study of electricity, we often ask how much work a circuit can do. Electric Power is defined as the rate at which electrical energy is consumed or dissipated in a circuit Science, class X (NCERT 2025 ed.), Electricity, p.191. Think of it as the 'speed' at which energy is being used. If you have two bulbs, one glowing brighter than the other, it is likely consuming energy at a higher rate, meaning it has higher power. Mathematically, power (P) is the product of potential difference (V) and current (I), expressed as P = VI. By applying Ohm's Law (V = IR), we can also express power in terms of resistance: P = I²R or P = V²/R Science, class X (NCERT 2025 ed.), Electricity, p.193. These variations are crucial; for instance, if you know the voltage of your home supply is constant, you can see that power is inversely proportional to resistance (P ∝ 1/R).

While the SI unit of power is the Watt (W)—defined as the power consumed by a device carrying 1 A of current at 1 V—this unit is often too small for practical use. In our homes and industries, we use the Kilowatt (kW), which is 1000 Watts. However, when we pay our electricity bills, we aren't paying for power (the rate), but for Electrical Energy (the total quantity). Energy is the product of power and time (E = P × t). The commercial unit of energy is the kilowatt-hour (kWh), popularly known as a 'unit' Science, class X (NCERT 2025 ed.), Electricity, p.192. One kWh represents the energy consumed by a 1000 W appliance running for one hour, which equals 3.6 × 10⁶ Joules.

From a broader developmental perspective, electricity is more than just physics; it is a 'clean' and transportable energy source that drives modern economies. Interestingly, the per capita consumption of electricity is a vital indicator of a nation's socio-economic development Geography of India, Majid Husain, Energy Resources, p.17. For context, while the global average is around 1000 kWh per person, India's consumption has historically been lower (around 350 kWh in previous decades), highlighting the ongoing need for infrastructure growth to match developed nations like the USA, where consumption exceeds 7000 kWh per head.

Term	Definition	Unit
Electric Power (P)	Rate of energy consumption (VI)	Watt (W) or J/s
Electric Energy (E)	Total energy used over time (P × t)	kilowatt-hour (kWh) or Joule (J)

Sources: Science, class X (NCERT 2025 ed.), Electricity, p.191-193; Geography of India, Majid Husain, Energy Resources, p.17

6. Domestic Wiring and Safety Systems (exam-level)

In our homes, electricity is supplied through a system of three distinct wires: the Live wire (positive), the Neutral wire (negative), and the Earth wire. In India, the potential difference between the Live (usually red insulation) and Neutral (black insulation) wires is standardized at 220 V Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.204. These wires enter the house through a main fuse and an electricity meter, eventually branching into separate parallel circuits. We use parallel circuits domestically because they allow each appliance to operate at the same voltage (220 V) and ensure that switching off one device does not interrupt the power to others. Safety is paramount in domestic wiring to prevent fire and electric shocks. Two primary hazards are Overloading and Short-circuiting. Overloading occurs when too many high-power appliances are used simultaneously or when there is a sudden hike in supply voltage. Short-circuiting happens when the Live and Neutral wires come into direct contact, often due to damaged insulation. In both cases, the current in the circuit increases abruptly and heavily Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.205. To mitigate these risks, we use two critical safety mechanisms:

• Electric Fuse: A safety device with a low melting point. When the current exceeds a safe limit, Joule heating causes the fuse wire to melt, breaking the circuit and protecting appliances from damage Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.205.
• Earthing: The Earth wire (green insulation) provides a low-resistance path for current. It is connected to the metallic body of appliances. If a fault causes the Live wire to touch the metal casing, the current flows into the earth rather than through a person touching the appliance, preventing a fatal shock Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.207.

Feature	Short-Circuiting	Overloading
Cause	Direct contact of Live and Neutral wires.	Connecting too many appliances to one socket/circuit.
Effect	Resistance becomes near-zero; current spikes.	Total current exceeds the wire's capacity.

Sources: Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.204; Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.205; Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.207

7. Resistors in Series and Parallel (exam-level)

When we connect multiple resistors in a circuit, we can arrange them in two fundamental ways: Series or Parallel. Understanding how these configurations affect the flow of electricity is crucial for designing everything from simple flashlights to complex home wiring systems. As defined in Ohm's Law, the potential difference across a resistor is directly proportional to the current through it (V = IR), provided the temperature remains constant Science, Class X (NCERT 2025 ed.), Electricity, p.192.

In a Series Connection, resistors are joined end-to-end like a single-track railway. There is only one path for the current to flow, meaning the current (I) remains identical through every resistor in the circuit. However, the total potential difference (voltage) provided by the battery is divided among the resistors. The Equivalent Resistance (Rₑ) is simply the sum of all individual resistances: Rₑ = R₁ + R₂ + R₃ + ... Science, Class X (NCERT 2025 ed.), Electricity, p.185. Because the current is constant, the resistor with the highest resistance will always have the largest voltage drop across it.

Conversely, in a Parallel Connection, resistors are connected across the same two points, creating multiple paths for the current. Here, the potential difference (V) is the same across every resistor, but the total current divides among the branches. Interestingly, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances: 1/Rₚ = 1/R₁ + 1/R₂ + 1/R₃ + ... Science, Class X (NCERT 2025 ed.), Electricity, p.186. This mathematical relationship means that the equivalent resistance in a parallel circuit is always less than the smallest individual resistance in the group.

Feature	Series Circuit	Parallel Circuit
Current (I)	Same through all components.	Divides; more current takes the path of least resistance.
Voltage (V)	Divided; proportional to resistance.	Same across all components.
Total Resistance	Increases (Rₑ = ΣR).	Decreases (1/Rₚ = Σ1/R).

Sources: Science, Class X (NCERT 2025 ed.), Electricity, p.185; Science, Class X (NCERT 2025 ed.), Electricity, p.186; Science, Class X (NCERT 2025 ed.), Electricity, p.192

8. Solving the Original PYQ (exam-level)

Now that you have mastered Ohm’s Law and the properties of Series Circuits, this question serves as the perfect synthesis of those building blocks. In a series connection, you must remember two golden rules: first, the Equivalent Resistance (R_eq) is simply the sum of all individual resistances; second, the Current (I) remains constant through every component in the chain. This question tests your ability to move from the 'macro' view of the whole circuit to the 'micro' view of a single resistor.

To solve this, think like a coach: first, find the total resistance by adding 1 Ω, 2 Ω, and 3 Ω to get 6 Ω. Using the total voltage of 9 V, you apply Ohm’s Law (I = V/R) to find a total current of 1.5 A. Since this current is uniform across the series, the potential drop across the 3 Ω resistor is simply its resistance multiplied by that shared current (3 Ω × 1.5 A), leading us directly to the correct answer: (C) 4.5 volt. You can verify this by noting that 4.5 V is exactly half of the 9 V supply, matching the fact that 3 Ω is exactly half of the 6 Ω total resistance.

UPSC often includes "distractor" options to catch students who stop halfway or lose focus. Option (A) and (B) are the potential drops for the 1 Ω and 2 Ω resistors, respectively; a common trap is calculating the correct current but applying it to the wrong resistor. Option (D) is a numerical trap designed for those who might perform the division incorrectly. Success in the Preliminary Exam depends on precision in application—always ensure your final calculation matches the specific component requested in the prompt.

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