If three resistors of 1 Ohm each connect in parallel to each other the resultant resistance is

1. Electric Current and Potential Difference (basic)

To understand electricity, we must first visualize what is happening inside a wire. Imagine a copper wire: it is full of electrons. However, these electrons don't just flow on their own; they need a "push." Electric Current is essentially this flow—specifically, a stream of electrons moving through a conductor. By convention, we say current flows from the positive terminal to the negative terminal, even though electrons (which are negatively charged) actually move in the opposite direction Science, Class X (NCERT 2025 ed.), Chapter 11, p.192. The strength of this flow is measured in Amperes (A).

But why do they move? This brings us to Electric Potential Difference. Think of it as "electrical pressure." Just as water only flows from a high-pressure tank to a low-pressure one, charges only move when there is a difference in electric potential between two points. We define this formally as the work done to move a unit charge from one point to another. The formula is V = W/Q, where V is potential difference, W is work done in Joules, and Q is the charge in Coulombs Science, Class X (NCERT 2025 ed.), Chapter 11, p.173. The unit for this "push" is the Volt (V), named after Alessandro Volta.

To keep this current flowing, we use a device like a cell or a battery. Inside the battery, chemical reactions create this potential difference at its terminals, acting like a pump that continuously pushes electrons through the circuit Science, Class X (NCERT 2025 ed.), Chapter 11, p.192. Without this potential difference, the net flow of electrons is zero, and the current stops.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.173, 192

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

At the heart of understanding how electricity behaves lies Ohm’s Law. Imagine an electric circuit like a water pipe system: the pressure pushing the water is the Potential Difference (V), and the flow of water itself is the Current (I). Ohm’s Law tells us that if you increase that pressure, the flow increases at a steady, predictable rate. Specifically, the potential difference across the ends of a metallic wire is directly proportional to the current flowing through it, provided its physical conditions, like temperature, remain constant Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.176.

This relationship is mathematically expressed as V ∝ I, which leads us to the famous formula: V = IR. Here, R represents the Resistance of the conductor. Resistance is a fundamental property of a material that describes how much it "resists" the flow of electric charges. Think of it as the friction or narrowness in our water pipe; the higher the resistance, the harder it is for current to flow for a given voltage Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.176.

The SI unit of resistance is the ohm, symbolized by the Greek letter Ω. Based on the formula R = V/I, we define 1 ohm as the resistance of a conductor when a potential difference of 1 Volt produces a current of 1 Ampere Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.176. It is crucial to remember that Ohm's Law is not a universal law for all materials, but it holds true for most metals under stable thermal conditions. If the temperature changes, the resistance R will also change, which would break the linear relationship between V and I Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.192.

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

3. Resistivity and Factors Affecting Resistance (intermediate)

When we talk about Resistance (R), we are essentially looking at how much a specific object opposes the flow of electric current. However, not all wires are created equal. Through experimentation, it has been observed 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.

Think of it like water flowing through a pipe. A longer pipe offers more friction to the water (Resistance is directly proportional to length, R ∝ l). Conversely, a wider pipe allows water to flow more freely (Resistance is inversely proportional to the area of cross-section, R ∝ 1/A). When we combine these observations, we get the fundamental relationship: R = ρ (l/A).

The constant ρ (rho) is called electrical resistivity. Unlike resistance, which changes if you cut a wire in half, resistivity is an intrinsic property of the material itself. It tells us how strongly a material opposes current regardless of its shape. The SI unit of resistivity is Ω m (ohm-meter) Science, Chapter 11, p.178. Metals like copper have very low resistivity (making them great conductors), while insulators like rubber have incredibly high resistivity.

Interestingly, alloys (like nichrome) usually have higher resistivity than their constituent pure metals. They also don't "burn" or oxidize easily at high temperatures, which is why your toaster or electric iron uses alloy coils instead of pure copper Science, Chapter 11, p.179. Finally, keep in mind that both resistance and resistivity are not strictly permanent; they vary with temperature.

Sources: Science (NCERT 2025 ed.), Chapter 11: Electricity, p.178; Science (NCERT 2025 ed.), Chapter 11: Electricity, p.179

4. Joule’s Heating Effect and Electric Power (intermediate)

When an electric current flows through a conductor, the conductor inevitably gets hot. This is known as the heating effect of electric current. At a microscopic level, as electrons drift through a conductor, they collide with the atoms and ions of the material. Each collision transfers some kinetic energy to the atoms, causing them to vibrate more vigorously, which we perceive as a rise in temperature Science, Class VIII, NCERT (Revised ed 2025), p.53. While this heating is often seen as a loss of energy in devices like electric fans or computers, it is the fundamental principle behind electric irons, kettles, and heaters Science, Class X (NCERT 2025 ed.), Chapter 11, p.190.

To quantify this, we look at Electric Power (P), which is the rate at which electrical energy is consumed or dissipated in a circuit. The basic relationship is P = VI. By applying Ohm’s Law (V = IR), we can derive other useful forms of this equation. Joule’s Law of Heating states that the heat (H) produced in a resistor is directly proportional to the square of the current (I²), the resistance (R), and the time (t) for which the current flows: H = I²Rt Science, Class X (NCERT 2025 ed.), Chapter 11, p.188.

In practical applications like an electric bulb, the filament must be made of a material with a very high melting point, such as tungsten. The goal is to retain as much heat as possible so the filament becomes white-hot and emits light, rather than just melting Science, Class X (NCERT 2025 ed.), Chapter 11, p.190. The standard unit of power is the Watt (W), which represents an energy dissipation rate of one Joule per second Science, Class X (NCERT 2025 ed.), Chapter 11, p.191.

Sources: Science, Class VIII, NCERT (Revised ed 2025), Electricity: Magnetic and Heating Effects, p.53; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.188, 190, 191

5. Domestic Electric Circuits and Safety (intermediate)

When we talk about Domestic Electric Circuits, we are looking at how electrical energy is safely distributed throughout a home. In India, the electricity supplied to our homes through the mains is an alternating current (AC) with a potential difference of 220 V. This power is delivered via two main wires: the Live wire (usually with red insulation) and the Neutral wire (usually with black insulation). To ensure safety, a third wire—the Earth wire (green insulation)—is connected to a metal plate deep in the earth near the house Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p.204.

The most critical design feature of a home circuit is that appliances are connected in parallel. This is done for two vital reasons: first, it ensures that every appliance receives the full 220 V supply; second, it allows each appliance to have its own independent switch. If appliances were in series, turning one off would break the circuit for everything else! Within these circuits, we must guard against two major hazards:

• Short-circuiting: This occurs when the Live and Neutral wires come into direct contact (perhaps due to damaged insulation), causing resistance to drop to nearly zero and current to spike dangerously.
• Overloading: This happens when too many high-power appliances are switched on simultaneously, drawing a total current that exceeds the capacity of the wires Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p.205.

The Electric Fuse is the first line of defense. It consists of a wire with an appropriate melting point. According to the law of Joule heating (H = I²Rt), if the current (I) exceeds the fuse's rated value, the wire heats up rapidly and melts, safely breaking the circuit before the appliances are damaged Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.190.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p.204, 205, 207; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.190

6. Mathematical Modeling of Parallel Combinations (exam-level)

In a parallel combination, resistors are connected between two common points so that each resistor provides a separate path for the current. The fundamental principle here is that the potential difference (V) remains identical across all resistors, while the total current (I) is divided among the various branches (Science, Chapter 11, p.186).

By applying Ohm’s Law (I = V/R), we can derive the mathematical model for the equivalent resistance (Rp). Since the total current is the sum of the currents through each branch (I = I₁ + I₂ + I₃), we substitute the Ohm's Law relationship to find that 1/Rp = 1/R₁ + 1/R₂ + 1/R₃. In simpler terms, the reciprocal of the equivalent resistance is equal to the sum of the reciprocals of the individual resistances (Science, Chapter 11, p.192).

This mathematical structure leads to two very important observations for your exams:

• The "Smaller than Smallest" Rule: The equivalent resistance (Rp) is always less than the resistance of the smallest individual resistor in the combination (Science, Chapter 11, p.187).
• Identical Resistors: If you have n identical resistors, each with resistance R, the formula simplifies to Rp = R/n. For example, if you connect three 1 Ω resistors in parallel, the equivalent resistance becomes 1/3 Ω.

Sources: Science, Chapter 11: Electricity, p.182; Science, Chapter 11: Electricity, p.186; Science, Chapter 11: Electricity, p.187; Science, Chapter 11: Electricity, p.192

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental properties of electric circuits, this question invites you to apply the Parallel Combination principle. In your lessons, you learned that when resistors are arranged in parallel, the current is divided among multiple branches, which effectively increases the total area for the flow of charge. This physical reality leads to a crucial rule: the resultant resistance in a parallel circuit will always be less than the smallest individual resistor in the group. By connecting these concepts, you can immediately see that the total resistance must be lower than 1 Ohm.

To find the exact value, we apply the Reciprocal Formula: 1/R_p = 1/R₁ + 1/R₂ + 1/R₃. Substituting the values, we get 1/1 + 1/1 + 1/1 = 3. A common mistake in the high-pressure environment of the UPSC exam is to stop here and choose 3; however, you must remember to take the reciprocal of the sum to find R_p. This gives us the correct answer: (B) 1/3 Ohm. As a coaching shortcut for your toolkit, remember that for n identical resistors of resistance R in parallel, the formula is simply R/n.

UPSC often includes "trap" options to test the precision of your conceptual clarity. Option (C) 3 Ohm is the result if the resistors were connected in series, where resistances are simply added. Option (D) 9 Ohm is a mathematical distractor for those who might incorrectly square the values. Finally, Option (A) 1 Ohm represents the resistance of a single component, ignoring the additive effect of parallel paths. As emphasized in Science, Class X (NCERT), the goal of a parallel arrangement is to decrease the overall resistance of the circuit.