If a heater coil is cut into two equal parts and only one part is used in the heater, the heat generated will be

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

To understand the foundation of electricity, we must first visualize what is happening inside a wire. Imagine a pipe filled with water. For the water to flow from one end to the other, there must be a pressure difference—water naturally flows from a high-pressure zone to a low-pressure zone. In the world of physics, Electric Current (I) is the actual flow of electric charges (electrons), while Potential Difference (V) is the electrical "pressure" that forces those charges to move.

Electric Current is defined as the rate of flow of electric charge through a cross-section of a conductor. However, charges do not move on their own. They require work to be done on them. This brings us to the concept of Electric Potential Difference. We define it as the work done (W) to move a unit charge (Q) from one point to another in an electric circuit. Mathematically, it is expressed as:

V = W / Q

The SI unit of potential difference is the Volt (V), named after the Italian physicist Alessandro Volta. One volt is specifically the potential difference between two points when 1 Joule of work is done to move a charge of 1 Coulomb from one point to the other Science, Class X (NCERT 2025 ed.), Chapter 11, p.173.

To keep this concept clear, consider this comparison table:

Feature	Electric Current (I)	Potential Difference (V)
Core Meaning	The flow of charges.	The cause of the flow (electrical pressure).
SI Unit	Ampere (A)	Volt (V)
Formula	I = Q / t	V = W / Q

In a practical circuit, such as a heater or a bulb, the potential difference is provided by a battery or a power source. When you increase this "pressure" (voltage), the current (flow) typically increases as well, provided the path remains the same Science, Class X (NCERT 2025 ed.), Chapter 11, p.180.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.173; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.180

2. Ohm's Law and Electrical Resistance (basic)

Imagine an electric circuit as a water pipe system. The potential difference (Voltage, V) is the pressure pushing the water, and the current (I) is the flow rate of the water itself. Ohm’s Law states that the current flowing through a conductor is directly proportional to the potential difference across its ends, provided physical conditions like temperature remain constant Science, Class X (NCERT 2025 ed.), Chapter 11, p.176. This relationship is defined by the formula: V = IR.

The constant R in this equation stands for Resistance. It is the inherent property of a material to resist the flow of electric charges. Not all materials offer the same resistance; for example, metals like copper have low resistance (good conductors), while materials like rubber have extremely high resistance (insulators). In practical appliances like heaters or toasters, we often use alloys because they have higher resistivity than pure metals and do not oxidize (burn) easily at high temperatures Science, Class X (NCERT 2025 ed.), Chapter 11, p.181.

Crucially, resistance is not fixed for a specific material—it depends on the geometry of the conductor. Think of a hallway: a longer hallway is harder to get through, and a wider hallway is easier. Similarly, the resistance of a wire is directly proportional to its length (l) and inversely proportional to its cross-sectional area (A). If you shorten a wire, its resistance decreases; if you make it thicker, its resistance also decreases.

Factor Change	Effect on Resistance (R)	Reasoning
Increasing Length (l)	Increases	Electrons collide with more atoms over a longer path.
Increasing Thickness (Area)	Decreases	More space/paths available for electrons to flow.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.176; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.181

3. Factors Affecting Resistance: Length and Area (intermediate)

Welcome back! Now that we understand what resistance is, let's look at the physical geometry of a conductor. Think of resistance not just as a fixed number, but as a result of the 'path' an electron must travel. There are two primary physical factors that determine how much a wire will oppose the flow of current: its length and its cross-sectional area.

First, consider the length (l). Imagine you are walking through a crowded corridor; the longer the corridor, the more people you will bump into. Similarly, as the length of a wire increases, electrons encounter more atoms to collide with. Precise experiments show that the resistance (R) of a uniform metallic conductor is directly proportional to its length. If you double the length of a wire, you double its resistance Science, Chapter 11, p.178.

Second, we have the area of cross-section (A), which refers to the thickness of the wire. A wider pipe allows more water to flow through it with less pressure; likewise, a thicker wire (larger area) provides a 'wider path' for electrons, reducing resistance. Resistance is inversely proportional to the cross-sectional area. This means a thick wire has less resistance than a thin wire of the same material Science, Chapter 11, p.192.

Factor	Relationship with Resistance (R)	Physical Intuition
Length (l)	Directly Proportional (R ∝ l)	More distance = more collisions for electrons.
Area (A)	Inversely Proportional (R ∝ 1/A)	More width = more space for electrons to flow.

When we combine these observations, we get the fundamental formula: R = ρ(l/A), where ρ (rho) is the electrical resistivity, a constant that depends on the material itself Science, Chapter 11, p.178. A practical application of this is seen in heating coils. If you cut a heater coil exactly in half, you have halved its length. According to our rule of proportionality, the resistance of that half-coil will be exactly half of the original resistance.

Sources: Science, Electricity, p.178; Science, Electricity, p.192

4. Magnetic Effects of Electric Current (intermediate)

Welcome back! Now that we have mastered how electricity flows through a circuit, we are ready to explore one of the most profound discoveries in physics: Electromagnetism. In 1820, Hans Christian Ørsted noticed a compass needle deflect near a current-carrying wire, proving that electricity and magnetism are inextricably linked. Essentially, every time an electric current flows through a metallic conductor, it generates a magnetic field around it Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p. 206.

The geometry of this magnetic field depends entirely on the shape of the conductor. For a simple straight wire, the magnetic field lines form a series of concentric circles. To determine the direction of these lines, we use the Right-Hand Thumb Rule: imagine holding the wire with your right hand, thumb pointing in the direction of the current; your fingers will naturally curl in the direction of the magnetic field lines Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p. 198. As you move further from the wire, these circles get larger and the field strength weakens.

When we take that wire and wrap it into a tight coil of many circular turns, we create a solenoid. This is where things get interesting for engineering and technology. A current-carrying solenoid produces a magnetic field nearly identical to that of a bar magnet, with one end acting as a North pole and the other as a South pole. Crucially, inside the solenoid, the field lines are parallel straight lines. This indicates that the magnetic field is uniform—meaning it has the same strength at all points inside the coil Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p. 201. By placing a soft iron core inside this solenoid, we create an electromagnet, a temporary but powerful magnet that can be turned on and off at will.

Feature	Straight Conductor	Solenoid
Field Pattern	Concentric circles around the wire.	Similar to a bar magnet; parallel lines inside.
Field Strength	Decreases as distance from wire increases.	Uniform and constant at all points inside.
Direction Rule	Right-Hand Thumb Rule.	Polarity depends on the direction of current flow.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p.198, 201, 206

5. Domestic Circuits: Safety and Efficiency (exam-level)

In a domestic setting, the way we arrange our electrical components is a matter of both efficiency and safety. Unlike simple laboratory circuits, domestic circuits must power a variety of appliances—from a low-power LED bulb to a high-power air conditioner—simultaneously and independently. To achieve this, all domestic appliances are connected in parallel Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p.205.

Connecting in parallel offers two critical advantages. First, it ensures that every appliance receives the same potential difference (usually 220V in India), allowing them to operate at their designed power rating. Second, it provides independence; if one bulb fuses or is switched off, the rest of the circuit remains closed and functional. In a series circuit, if one component fails, the entire circuit breaks—a nightmare scenario for a household where one blown fuse in a toaster could turn off all the lights Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.187.

Feature	Series Circuit	Parallel Circuit (Domestic)
Voltage	Divided across components	Same for all (220V)
Current	Same throughout the circuit	Divided based on appliance need
Failure	One failure stops everything	Independent operation

Safety is managed through two primary devices: the fuse and the earth wire. A fuse is a safety wire with a specific melting point, always connected in series with the live wire. If a fault causes an excessive current (overloading or short-circuiting), the fuse wire melts due to Joule heating, safely breaking the circuit before the appliance is damaged or a fire starts Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.190. Meanwhile, the earth wire provides a low-resistance path to the ground, ensuring that any current leakage to a metallic casing doesn't give the user a dangerous shock Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p.206.

Finally, understanding the relationship between resistance and heat is vital for efficiency. For a heating appliance like a geyser or room heater, the heat produced (P) at a constant domestic voltage (V) is given by P = V²/R. This leads to a counter-intuitive fact: if you reduce the resistance (for example, by shortening the heater coil), the heat generated actually increases. Because resistance is directly proportional to length (R ∝ l), cutting a coil in half halves the resistance, which effectively doubles the power consumption and heat output Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.192.

Sources: Science, Class X (NCERT 2025 ed.), Chapter 12: Magnetic Effects of Electric Current, p.205-206; Science, Class X (NCERT 2025 ed.), Chapter 11: Electricity, p.187, 190, 192

6. Joule's Law of Heating and Electric Power (intermediate)

When electric current flows through a conductor, the moving electrons face resistance due to collisions with the atoms of the material. These collisions convert electrical energy into thermal energy—a phenomenon known as the heating effect of electric current. In many devices like electric irons, toasters, and heaters, we utilize this effect intentionally, whereas in others, it is an inevitable energy loss Science, Chapter 11, p.190.

The mathematical foundation for this is Joule’s Law of Heating. It 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. This gives us the core formula: H = I²Rt Science, Chapter 11, p.189.

Electric Power (P) is the rate at which this electrical energy is consumed or dissipated. While the SI unit is the Watt (W)—defined as the power consumed by a device carrying 1 A of current at 1 V—we use different formulas depending on whether the current or voltage is constant:

Scenario	Power Formula	Key Relationship
Constant Current (e.g., Series)	P = I²R	Power is directly proportional to Resistance.
Constant Voltage (e.g., Household)	P = V²/R	Power is inversely proportional to Resistance.

Understanding the P = V²/R relationship is vital for practical engineering. For instance, if you reduce the resistance of a heating element (like by shortening the wire) while keeping it connected to the same household voltage, the power consumption—and thus the heat output—actually increases. This is because the lower resistance allows a much larger current to flow through the circuit Science, Chapter 11, p.191-192.

Sources: Science (NCERT 2025 ed.), Chapter 11: Electricity, p.189; Science (NCERT 2025 ed.), Chapter 11: Electricity, p.190; Science (NCERT 2025 ed.), Chapter 11: Electricity, p.191

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental relationships between a conductor’s physical properties and its electrical behavior, you can see how those building blocks snap together to solve this classic UPSC problem. The first step is recognizing that the Resistance (R) of the heater coil is directly proportional to its length (l), a core concept found in Science, class X (NCERT 2025 ed.) > Chapter 11: Electricity. When the coil is cut into two equal parts, the length is halved, which means the resistance of the single part being used is also halved ($R' = R/2$).

As a coach, I want you to focus on the constant variable: the voltage ($V$) from the power socket remains the same. To find the heat generated, we use the power formula $P = V^2/R$. Because the resistance is in the denominator, it has an inverse relationship with the power. When you divide the resistance by two, the resulting power (heat) is multiplied by two ($P' = V^2 / (R/2) = 2P$). Therefore, the correct answer is (A) doubled. This happens because the reduction in resistance allows a much larger current to flow through the circuit, more than offsetting the fact that there is less wire to heat up.

UPSC often includes (D) halved as a trap to catch students who use "common sense" (less wire must mean less heat) instead of physics. This mistake usually occurs if a student uses the formula $H = I^2Rt$ and forgets that the current (I) changes when the resistance is modified. Options (B) four times and (C) one-fourth are designed to confuse those who might incorrectly square the resistance or misapply the proportions. Always identify your constant (Voltage) before choosing your formula to avoid these common pitfalls.