The torque on a rectangular coil placed in a uniform magnetic field is large when the

1. Basics of Magnetic Fields and Lorentz Force (basic)

Welcome to the beginning of your journey into electromagnetism! To understand how complex machines like motors or MRI scanners work, we must first master the magnetic field. Think of a magnetic field as an invisible "zone of influence" around a magnet or a current-carrying wire. We visualize this using magnetic field lines. These lines are closed curves that emerge from the North pole and enter the South pole of a magnet; interestingly, inside the magnet, they travel from South to North Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.197. The density of these lines tells us the relative strength of the field: the closer the lines, the stronger the magnetic force in that region.

One of the most profound discoveries in physics is that electricity and magnetism are inseparable. When an electric current flows through a metallic wire, it generates its own magnetic field around it Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.206. For a straight wire, this field takes the shape of concentric circles. If you bend that wire into a circular loop, the magnetic field lines at the center actually appear as straight lines because the circular paths from different parts of the loop interact and straighten out Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.200.

Finally, we arrive at the concept of the Lorentz Force. When a conductor carrying an electric current is placed inside an external magnetic field, it doesn't just sit there—it experiences a physical mechanical force Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.206. This happens because the magnetic field of the wire interacts with the external magnetic field. This push or pull is the fundamental principle behind every electric motor you have ever used.

Property	Magnetic Field Lines
Direction	North to South (Outside); South to North (Inside)
Intersection	Lines never cross each other
Strength	Strongest where lines are most crowded

Sources: Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.197; Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.206; Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.200

2. Magnetic Effect of Electric Current (basic)

At the heart of modern technology lies a simple yet profound discovery: electricity and magnetism are not separate forces, but two sides of the same coin. In 1820, a Danish scientist named Hans Christian Oersted noticed something peculiar. While demonstrating a circuit, he saw a nearby compass needle deflect whenever the current was switched on Science, Class VIII NCERT (Revised ed 2025), Electricity: Magnetic and Heating Effects, p.48. This proved that a wire carrying an electric current produces a magnetic field around it. This accidental observation laid the foundation for everything from radio to fiber optics Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.195.

When we take this current-carrying wire and shape it into a rectangular coil placed within an external magnetic field, a fascinating interaction occurs. The magnetic field produced by the current interacts with the external magnetic field, creating a mechanical force. In a coil, these forces act in opposite directions on different sides, creating a turning effect known as Torque (τ). This is the fundamental principle that makes electric motors spin.

The magnitude of this torque isn't random; it follows a precise relationship defined by the formula: τ = NIAB sin θ. To understand how to increase the "turning power" of a motor or a galvanometer, we look at these variables:

• N (Number of turns): Adding more loops of wire to the coil multiplies the total force. Torque is directly proportional to the number of turns.
• I (Current): Increasing the flow of electrons strengthens the magnetic field of the wire.
• A (Area): A larger coil area experiences a greater total magnetic influence.
• B (Magnetic Field Strength): Using stronger external magnets increases the "push" on the coil.
• θ (Angle): The torque is strongest when the magnetic field is perpendicular to the normal of the coil's plane (sin 90° = 1). If the coil's normal is parallel to the field, the torque drops to zero.

Factor	Relationship to Torque	Effect of Increasing Factor
Number of Turns (N)	Directly Proportional	Torque Increases
Current (I)	Directly Proportional	Torque Increases
Area of Coil (A)	Directly Proportional	Torque Increases

Sources: Science, Class VIII NCERT (Revised ed 2025), Electricity: Magnetic and Heating Effects, p.48; Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.195

3. Magnetic Dipole Moment (intermediate)

When we move from studying a simple straight wire to a circular loop, something fascinating happens: the magnetic field lines, which are concentric circles around the wire, begin to bunch up at the center of the loop. At the very center, these lines appear almost straight and uniform Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.200. This configuration effectively turns the loop into a tiny magnet with a distinct North and South pole. The Magnetic Dipole Moment (μ) is the mathematical measure of this "magnet strength."

Think of the Magnetic Dipole Moment as the "magnetic identity" of the loop. It is determined by three specific factors: the amount of current (I) flowing, the area (A) enclosed by the loop, and the number of turns (N) in the coil. Because the magnetic field of each individual turn adds up in the same direction, a coil with multiple turns creates a much stronger magnetic effect than a single loop Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.201. The formula for this strength is: μ = NIA. This vector always points perpendicular to the plane of the loop, following the Right-Hand Thumb Rule.

The real-world importance of the dipole moment becomes clear when the loop is placed inside an external magnetic field (B). The field exerts a Torque (τ) on the loop, trying to twist it so that its dipole moment aligns with the external field. This twisting force is calculated as τ = NIAB sin θ, where θ is the angle between the magnetic field and the normal (the perpendicular line) to the loop's surface. If the loop is oriented such that its face is perpendicular to the field (θ = 0°), the torque disappears because the dipole is already "happy" and aligned.

Factor	Effect on Dipole Moment (μ)	Reasoning
Current (I)	Increases	Stronger flow creates a stronger field.
No. of Turns (N)	Increases	Fields from each turn add up cumulatively.
Loop Area (A)	Increases	Larger geometry spreads the magnetic influence.

Sources: Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.200; Science, Class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.201

4. Measuring Current: The Moving Coil Galvanometer (intermediate)

To understand how we measure current, we must first look at the fundamental interaction between electricity and magnetism. As we've learned, a metallic wire carrying an electric current has a magnetic field associated with it Science, Class X, Magnetic Effects of Electric Current, p.206. When you take this current-carrying wire and place it inside an external magnetic field, it experiences a mechanical force. In a Moving Coil Galvanometer, we take a rectangular coil with several turns of wire and suspend it between the poles of a strong magnet. When current flows through the coil, the magnetic field exerts forces on the vertical sides of the rectangular loop, creating a torque (a twisting force) that makes the coil rotate.

The magnitude of this torque (τ) is determined by a specific set of physical factors, represented by the formula: τ = NIAB sin θ. Here, N is the number of turns in the coil, I is the current we want to measure, A is the area of the coil, and B is the strength of the external magnetic field. Because the torque is directly proportional to these factors, increasing the number of turns (N) or the area (A) will result in a larger torque for the same amount of current. This is why high-sensitivity galvanometers often use coils with a large number of turns of fine insulated copper wire wrapped around a soft iron core Science, Class X, Magnetic Effects of Electric Current, p.206.

The term sin θ in the formula is crucial. Here, θ is the angle between the magnetic field lines and the normal (a line perpendicular) to the plane of the coil. If the plane of the coil is perpendicular to the magnetic field, the angle θ is 0°, making sin 0° = 0; in this position, the magnetic field produces zero torque. To ensure the galvanometer is accurate and has a linear scale, engineers use concave-shaped magnets to create a radial magnetic field. This clever design ensures that the magnetic field is always parallel to the plane of the coil (θ = 90°) as it rotates, making sin θ = 1 and keeping the torque at its maximum possible value for a given current.

In practice, the rotation of the coil is resisted by a small spring. As the coil turns, the spring winds up and exerts a restoring torque. The needle attached to the coil stops moving when the magnetic torque exactly balances the spring's resistance. This equilibrium allows us to translate the physical deflection of the needle into a precise measurement of the current (I) flowing through the circuit.

Sources: Science, Class X, Magnetic Effects of Electric Current, p.206

5. Electromagnetic Induction and Electric Motors (basic)

Imagine electricity and magnetism as two sides of the same coin. When current flows through a wire, it creates an invisible magnetic field around it, effectively turning that wire into a temporary magnet Science, Class VIII, Electricity: Magnetic and Heating Effects, p.58. This fundamental discovery led to the invention of the Electric Motor, a device that converts electrical energy into mechanical energy. In a motor, a current-carrying coil is placed inside a fixed magnetic field. Because the coil is now 'magnetic' itself, it experiences a physical force—a push or pull—that causes it to rotate Science, Class X, Magnetic Effects of Electric Current, p.202.

To make a motor powerful, we need to maximize the torque, which is the turning force that makes the coil spin. This force depends on several factors: the amount of current (I), the strength of the magnetic field (B), and the area (A) of the coil. However, one of the most practical ways to boost this force is by increasing the number of turns (N) in the wire coil. Each extra turn adds more 'magnetic' strength, resulting in a stronger push. This relationship is often expressed as τ = NIAB sin θ, where the torque is directly proportional to the number of turns.

While motors use electricity to create motion, the reverse is also true—a phenomenon known as Electromagnetic Induction (EMI). This is the process of generating an electric current by moving a magnet near a conductor or by changing the magnetic field around it Science, Class X, Magnetic Effects of Electric Current, p.195. This 'reverse' effect is what allows us to generate electricity in massive power plants, proving that motion and magnetism are just as capable of producing electricity as electricity is of producing motion.

Sources: Science, Class VIII (NCERT 2025), Electricity: Magnetic and Heating Effects, p.58; Science, Class X (NCERT 2025), Magnetic Effects of Electric Current, p.195, 202

6. Dynamics of Torque on a Current Loop (exam-level)

To understand why an electric motor spins or how a galvanometer needle moves, we must look at the Dynamics of Torque. When a current-carrying loop is placed in a uniform magnetic field, the magnetic forces acting on the opposite arms of the loop create a 'turning effect' known as torque (τ). While a single straight wire simply experiences a lateral force Science, class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.203, a loop experiences a pair of forces that push one side up and the other down, causing rotation.
The magnitude of this torque is governed by the equation τ = NIAB sin θ. Let’s break down these variables:

• N (Number of turns): Torque is directly proportional to the number of turns in the coil. More loops mean more 'push'.
• I (Current): Increasing the flow of charge increases the magnetic interaction.
• A (Area): A larger loop catches more magnetic flux, increasing the torque.
• B (Magnetic Field): A stronger external magnet results in a stronger rotational force.
• θ (Theta): This is the angle between the normal to the plane of the loop and the magnetic field lines.

Just as the Coriolis force varies with latitude (sin ϕ) Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309, the torque on a loop varies with its orientation. When the plane of the loop is parallel to the magnetic field, the angle θ between the normal and the field is 90°, making sin 90° = 1 (Maximum Torque). Conversely, if the loop is perpendicular to the field, the normal aligns with the field lines (θ = 0°), making sin 0° = 0 (Zero Torque). This explains why a magnetic needle or a loop naturally seeks an equilibrium position where the torque disappears, similar to how a compass needle aligns with Earth's magnetic dip Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.77.

Sources: Science, class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.203; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309; Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.77

7. Solving the Original PYQ (exam-level)

You have just explored how moving charges in a magnetic field experience force, which leads to the collective rotational behavior of a current-carrying loop. This question synthesizes the variables that define the magnetic dipole moment and its interaction with a uniform external field. The magnitude of the torque (τ) is determined by the fundamental relationship τ = NIAB sin θ, where the torque depends on the physical characteristics of the coil and its orientation. By understanding this formula, you can see how the building blocks of current, area, and magnetic flux converge to create mechanical work.

To identify the correct answer, we must examine which variable, when increased, directly maximizes the torque. Since τ is directly proportional to the number of turns (N), adding more turns effectively multiplies the magnetic effect of a single loop. Therefore, (A) number of turns is large is the correct choice because each additional turn contributes an equal amount of rotational force to the total system. Think of it as a force multiplier where the cumulative effect of many loops results in a significantly stronger twisting motion.

UPSC often includes distractors that test your precision regarding geometry and proportionality. Option (C) is a common trap: when the plane of the coil is perpendicular to the magnetic field, the normal vector of the coil is actually parallel to the field lines. This makes the angle θ = 0°, leading to sin 0° = 0 and resulting in zero torque. Similarly, because torque is directly proportional to the area (A), a smaller area (Option D) would decrease the torque rather than increase it. Mastering these relationships, as outlined in NCERT Physics Class XII, ensures you can navigate such conceptual traps with ease.