Introduction

Introduction

The Birth of Electromagnetism

For over two millennia, humanity observed two distinct natural phenomena — electricity and magnetism — without realizing they were intimately connected. Ancient Greeks knew that amber, when rubbed, could attract light objects (static electricity), while lodestones could pull iron (magnetism). Yet these seemed like separate mysteries of nature. It wasn't until the summer of 1820 that a Danish physicist named Hans Christian Oersted accidentally stumbled upon a discovery that would revolutionize physics forever.

During a lecture demonstration, Oersted was showing his students various electrical experiments when he noticed something unexpected. A compass needle placed near a wire carrying electric current began to deflect! This was extraordinary — until that moment, no one had ever observed electricity affecting a magnet. The compass needle, which normally points north due to Earth's magnetic field, was being influenced by the moving charges in the wire.

{{VISUAL: photo: Hans Christian Oersted conducting his famous 1820 experiment with a compass needle deflecting near a current-carrying wire during a lecture demonstration}}

{{KEY: type=concept | title=Oersted's Groundbreaking Discovery | text=Moving electric charges (electric current) produce a magnetic field in the surrounding space. This was the first experimental evidence linking electricity and magnetism, two phenomena previously thought to be completely independent.}}

What Oersted Observed

Oersted's careful investigation revealed several fascinating patterns:

The compass needle's behavior:

• The needle aligned itself tangentially to an imaginary circle around the wire
• This circle had the wire as its centre and its plane perpendicular to the wire
• The deflection was most noticeable when the current was large and the needle was close to the wire (so Earth's weak magnetic field could be ignored)

The effect of current direction:

• Reversing the current's direction completely reversed the needle's orientation
• Increasing the current strength caused greater deflection
• Bringing the needle closer to the wire also increased the deflection

{{VISUAL: diagram: overhead view showing compass needles arranged in a circle around a current-carrying wire perpendicular to the page, with needles pointing tangentially clockwise when current emerges out}}

When Oersted sprinkled iron filings around the current-carrying wire, they arranged themselves in beautiful concentric circles with the wire at the center. This visual confirmation proved that the magnetic field had a specific geometric pattern — it wasn't random but followed a circular symmetry around the wire.

{{KEY: type=points | title=Key Features of Oersted's Observation | text=- Magnetic field lines form concentric circles around a straight current-carrying wire.

• The direction of the magnetic field reverses when current direction reverses.
• Field strength increases with current magnitude and decreases with distance from wire.
• The effect occurs only with moving charges, not static charges.}}

The Unification of Electricity and Magnetism

Oersted's discovery triggered intense experimental work across Europe. Scientists realized that the relationship between electricity and magnetism was deeper than anyone had imagined. Over the next few decades, brilliant minds like André-Marie Ampère, Michael Faraday, and others conducted systematic experiments to understand this connection.

The crowning achievement came in 1864 when James Clerk Maxwell formulated a unified theory — the laws of electromagnetism. Maxwell's equations elegantly described how electric and magnetic fields are generated and how they interact. But Maxwell's theory predicted something even more profound: accelerating charges should produce waves that travel through space at a specific speed. When Maxwell calculated this speed, it matched the known speed of light! He concluded that light itself is an electromagnetic wave — a brilliant theoretical prediction.

{{VISUAL: diagram: timeline showing key discoveries from 1820 Oersted's experiment through 1864 Maxwell's equations to 1888 Hertz's radio waves, with portrait sketches and key contributions}}

The Technological Revolution

The understanding of electromagnetism didn't remain confined to textbooks. By the end of the 19th century:

• Heinrich Hertz experimentally discovered radio waves in 1888
• Jagadish Chandra Bose and Guglielmo Marconi developed practical radio communication systems
• The foundation was laid for the entire telecommunications industry

The 20th century witnessed an explosion of electromagnetic technology: radio, television, radar, microwave ovens, mobile phones, Wi-Fi, and countless other devices that define modern life. All of this stems from understanding how moving charges create magnetic fields and how changing magnetic fields can induce electric currents.

{{KEY: type=exam | title=Historical Context in CBSE Exams | text=CBSE often asks 1-2 mark questions about Oersted's experiment — what he observed, the shape of magnetic field lines around a straight wire, and the effect of reversing current. Be prepared to sketch the compass needle arrangement and explain the circular field pattern.}}

What You'll Learn in This Chapter

This chapter explores the fascinating world of moving charges and the magnetic fields they create. We'll see how magnetism isn't just a property of permanent magnets, but emerges naturally whenever charges move.

Core Topics Covered

Magnetic forces on moving charges:

• How magnetic fields exert forces on moving charged particles (electrons, protons, ions)
• The Lorentz force — the combined effect of electric and magnetic fields on a charge
• Why magnetic force is always perpendicular to both velocity and field direction

Motion of charged particles:

• Circular paths of charges in uniform magnetic fields
• The cyclotron — a device that accelerates particles to extremely high energies
• Applications in particle physics and medical imaging

Magnetic fields produced by currents:

• The Biot-Savart law — calculating magnetic fields from current elements
• Magnetic fields of straight wires, circular loops, and solenoids
• Ampère's circuital law — an elegant way to find fields in symmetric situations

Force between current-carrying wires:

• How parallel currents attract or repel each other
• The definition of the ampere (unit of current) based on magnetic force

The moving coil galvanometer:

• How currents and voltages are detected using magnetic forces
• The principle behind analog meters and early measurement devices

{{VISUAL: diagram: concept map showing chapter structure with five main branches - Lorentz force, particle motion, magnetic field sources, current interactions, and galvanometer - with key formulas and applications under each}}

Convention for Diagrams

Throughout this chapter (and in electromagnetism generally), we use a standard convention to represent directions perpendicular to the page:

This notation will appear frequently in our diagrams, so familiarize yourself with it now.

The discovery that electricity and magnetism are two aspects of the same fundamental phenomenon ranks among the greatest intellectual achievements in physics — it unified our understanding of nature and unlocked technologies that transformed civilization.

{{KEY: type=definition | title=Magnetic Field B | text=A magnetic field is a vector field that exists at each point in space around moving charges or currents. It describes the magnetic influence at that point and determines the magnetic force experienced by other moving charges placed there.}}

In the sections ahead, we'll develop these ideas mathematically and see how moving charges and magnetic fields interact in beautiful and practical ways. The journey from Oersted's simple compass deflection to particle accelerators and smartphones is a testament to the power of scientific inquiry and mathematical reasoning.

Sources and fields

Sources and Fields

The Two-Stage Model of Field Interaction

In Chapter 1, we encountered the revolutionary idea that charges interact with each other not directly, but through an invisible intermediary — the electric field. This two-stage model of interaction fundamentally changed how physicists understand forces. Before we introduce the concept of a magnetic field B, let us revisit the electric field framework.

Recapitulation: The Electric Field

Consider a source charge Q placed in space. According to Coulomb's law, this charge does not simply "reach across" space to push or pull another charge. Instead, the charge Q creates an electric field E at every point in the surrounding space. The field at a distance r from the charge is given by:

{{FORMULA: expr=E = Q r̂ / (4πε₀r²) | symbols=E:electric field (N/C), Q:source charge (C), r̂:unit vector along r (dimensionless), ε₀:permittivity of free space (8.85×10⁻¹² C²/N·m²), r:distance from charge (m)}}

Here, r̂ is the unit vector pointing radially outward from Q. Notice that the electric field is a vector field — it has both magnitude and direction at every point in space.

{{VISUAL: diagram: labeled vector diagram showing source charge Q at the center with electric field vectors E pointing radially outward at various points, labeled with r̂ directions}}

Once the field is established, a test charge q placed at any point r experiences a force given by:

F = q E = q Q r̂ / (4πε₀r²)

{{KEY: type=concept | title=The Physical Reality of the Electric Field | text=The electric field E is not merely a mathematical convenience. It is a real physical entity that can carry energy and momentum. Crucially, the field takes finite time to propagate — changes in the field do not spread instantaneously but travel at the speed of light. This was Faraday's great insight and became central to Maxwell's theory of electromagnetism.}}

The Principle of Superposition

What happens when there are multiple charges creating electric fields? Experiment shows that electric fields obey the principle of superposition — the total electric field at any point is the vector sum of the electric fields produced by each individual charge acting alone.

If charges Q₁, Q₂, Q₃, ... are present, the total electric field at a point is:

E_total = E₁ + E₂ + E₃ + ...

where each Eᵢ is calculated using the formula above. This simple addition rule (vector addition, of course) makes the electric field extraordinarily powerful as a calculational tool.

{{VISUAL: diagram: three point charges Q1, Q2, Q3 arranged in a triangle with electric field vectors E1, E2, E3 at a common point P, and the resultant vector E_total shown using the parallelogram law}}

Introducing the Magnetic Field

Static Charges vs. Moving Charges

We now know that static charges produce electric fields. But what happens when charges move? Experiment shows that moving charges produce an additional field — the magnetic field, denoted by B(r).

Just as the electric field E is defined at each point in space, so too is the magnetic field B. Both can, in principle, vary with time, but in this chapter we will assume steady conditions where fields do not change with time.

{{KEY: type=definition | title=Magnetic Field | text=The magnetic field B is a vector field produced by moving charges (electric currents). It is defined at every point in space and obeys the principle of superposition — the magnetic field due to multiple current sources is the vector sum of the individual magnetic fields.}}

The discovery that currents produce magnetic fields came from Hans Christian Oersted's famous 1820 experiment. Oersted noticed that a compass needle deflected when placed near a wire carrying electric current. This was the first experimental evidence linking electricity and magnetism — two phenomena previously thought to be entirely separate.

{{VISUAL: photo: historical recreation of Oersted's experiment showing a compass needle deflecting near a current-carrying wire with visible magnetic field lines indicated by iron filings on paper}}

Basic Properties of the Magnetic Field

The magnetic field B shares several fundamental features with the electric field E:

• It is a vector field: At each point in space, B has both magnitude and direction.
• It obeys superposition: If multiple currents produce magnetic fields B₁, B₂, B₃, ... then the total field is B_total = B₁ + B₂ + B₃ + ... (vector sum).
• It can vary in space and time: Though we focus on static fields in this chapter, time-varying magnetic fields are central to electromagnetic induction (Chapter 6).

{{KEY: type=points | title=Key Features of the Magnetic Field | text=- Produced by moving charges (currents), not by stationary charges.

• Vector field defined at every point in space.
• Obeys the principle of superposition exactly like the electric field.
• Can carry energy and momentum, just like the electric field.}}

The Two Fields Working Together

In general, a charged particle may find itself in a region where both electric and magnetic fields are present. How does the particle respond? This brings us to one of the most important force laws in all of physics.

The Lorentz Force: Unifying Electric and Magnetic Effects

Suppose a point charge q is moving with velocity v and is located at position r at time t. If both an electric field E(r) and a magnetic field B(r) are present at that point, the total electromagnetic force on the charge is:

F = q [E(r) + v × B(r)]

This is called the Lorentz force, named after the Dutch physicist H.A. Lorentz. It has two parts:

1. Electric force: F_electric = q E(r) — depends only on the charge and the electric field.
2. Magnetic force: F_magnetic = q (v × B(r)) — depends on the charge, its velocity, and the magnetic field.

{{FORMULA: expr=F = q [E + v × B] | symbols=F:total electromagnetic force (N), q:charge of particle (C), E:electric field (N/C), v:velocity of particle (m/s), B:magnetic field (T)}}

{{KEY: type=concept | title=The Lorentz Force Law | text=The Lorentz force is the complete description of the electromagnetic force on a moving charged particle. It combines the electric force (parallel to E) and the magnetic force (perpendicular to both v and B). This single equation governs the motion of charged particles in all electromagnetic devices — from cathode ray tubes to particle accelerators.}}

Why the Cross Product?

The magnetic force involves a vector cross product v × B. This gives the magnetic force three critical properties:

• Direction: The force is perpendicular to both v and B, as determined by the right-hand rule or screw rule.
• Zero force when parallel: If the velocity is parallel (or anti-parallel) to the magnetic field, sin θ = 0 and the magnetic force vanishes.
• No work done: Since the force is always perpendicular to the velocity, the magnetic force does no work on the particle — it can change the direction of motion but not the speed.

{{VISUAL: diagram: right-hand rule illustration showing thumb pointing along velocity v, fingers along magnetic field B, and palm indicating force direction F for a positive charge, with separate panel showing opposite force for negative charge}}

{{ZOOM: title=The Right-Hand Rule in Practice | text=To find the direction of F = q(v × B) for a positive charge, point your fingers along v, curl them toward B, and your thumb points along F. For a negative charge, the force is exactly opposite. This geometric rule is indispensable for sketching particle trajectories in magnetic fields.}}

{{KEY: type=exam | title=Magnetic Force — Common Exam Scenarios | text=CBSE exam questions often ask you to find the direction of magnetic force using the right-hand rule, or to explain why a charged particle moves in a circular path in a uniform magnetic field. Remember: magnetic force is zero if the particle is at rest or moving parallel to B.}}

Summary of Key Ideas

In this section, we have laid the conceptual groundwork for understanding magnetism:

• The electric field E is produced by charges (static or moving); the magnetic field B is produced only by moving charges.
• Both fields obey the principle of superposition.
• The Lorentz force F = q[E + v × B] describes the total electromagnetic force on a moving charge.
• The magnetic component of the Lorentz force is perpendicular to both the velocity and the magnetic field, leading to circular or helical motion in many practical situations.

The concept of a field — stressed by Faraday and mathematically formalized by Maxwell — is not just a calculational tool. It is the physical intermediary through which forces act, and it carries the energy and momentum that make electromagnetism a complete, self-consistent theory.

Magnetic Field, Lorentz Force

Magnetic Field, Lorentz Force

The Combined Force on a Moving Charge

Imagine a charged particle — say, an electron or a proton — moving through space. If it encounters only an electric field, it experiences a force along the direction of that field (or opposite, depending on the sign of the charge). But what if a magnetic field is also present?

The Dutch physicist H. A. Lorentz combined extensive experimental observations by André-Marie Ampère and others to formulate a single, elegant expression for the total force on a moving charge in the presence of both electric and magnetic fields. This force is now universally known as the Lorentz force.

{{KEY: type=definition | title=Lorentz Force | text=The total electromagnetic force on a point charge q moving with velocity v in the presence of an electric field E(r) and a magnetic field B(r) is given by F = q[E(r) + v × B(r)], which is the vector sum of the electric force and the magnetic force.}}

In the expression above, F = q[E(r) + v × B(r)], the first term q E(r) is the familiar electric force, which acts on the charge whether it is moving or at rest. The second term q(v × B(r)) is the magnetic force, and it has some remarkable properties that distinguish it sharply from the electric force.

{{VISUAL: diagram: labeled vector diagram showing a charged particle q with velocity vector v, magnetic field vector B, and the resulting Lorentz force F perpendicular to both v and B}}

Key Features of the Magnetic Force

The magnetic component of the Lorentz force reveals itself through experiment and observation. Let us examine its defining characteristics:

1. Dependence on Charge, Velocity, and Magnetic Field

The magnetic force depends on three factors:

• q, the magnitude and sign of the charge
• v, the velocity of the charge
• B, the magnetic field at the location of the charge

Importantly, if you reverse the sign of the charge (from positive to negative), the direction of the magnetic force reverses too. A proton and an electron moving with the same velocity in the same magnetic field will experience forces in opposite directions.

2. The Cross Product: A Force Perpendicular to Motion

The magnetic force is given by the vector cross product F_magnetic = q(v × B). This mathematical operation has a geometric consequence: the force is always perpendicular to both the velocity v and the magnetic field B.

• If v and B are parallel or anti-parallel (moving along or exactly against the field lines), then sin θ = 0, and the magnetic force vanishes.
• The maximum force occurs when the velocity is perpendicular to the magnetic field (θ = 90°).

The direction of the force follows the right-hand rule (or screw rule): point your fingers in the direction of v, curl them toward B, and your thumb points along the direction of v × B. For a positive charge, the force is in this direction. For a negative charge, it is opposite.

{{VISUAL: diagram: step-by-step illustration of the right-hand rule showing fingers along velocity v, curling toward magnetic field B, and thumb pointing in the direction of force F for a positive charge}}

{{KEY: type=points | title=Properties of Magnetic Force | text=- The magnetic force is zero if the charge is stationary (v = 0).

• The force is zero if velocity and magnetic field are parallel or anti-parallel.
• The force acts perpendicular to both the velocity and the magnetic field.
• The direction is given by the right-hand rule for the cross product v × B.
• For a negative charge, the force reverses direction.}}

3. Only Moving Charges Feel the Magnetic Force

A stationary charge placed in a magnetic field experiences no magnetic force. This is strikingly different from the electric force, which acts on a charge whether it is moving or not. The magnetic force is a consequence of motion through the magnetic field — it is a relativistic effect at its core, though we treat it classically in most introductory contexts.

{{ZOOM: title=Why only moving charges? | text=The magnetic force arises from the interaction of a moving charge with the magnetic field, which itself is produced by moving charges (currents). At a deeper level, electric and magnetic fields are two aspects of the electromagnetic field, unified in Einstein's theory of relativity. What appears as a purely magnetic force in one frame may have electric components in another frame moving relative to it.}}

Defining the Unit of Magnetic Field: The Tesla

The expression for the magnetic force allows us to define the unit of magnetic field strength in the SI system. Consider the magnitude of the magnetic force:

F = q v B sin θ

where θ is the angle between v and B. If we take special cases where q = 1 C, v = 1 m/s, θ = 90° (so sin θ = 1), and the force F = 1 N, then:

B = F / (q v) = 1 N / (1 C × 1 m/s) = 1 N s / (C m)

This unit is called the tesla, symbol T, named in honour of the inventor and electrical engineer Nikola Tesla.

{{FORMULA: expr=B = F / (q v sin θ) | symbols=B:magnetic field (T), F:magnetic force (N), q:charge (C), v:velocity (m/s), θ:angle between v and B}}

{{KEY: type=definition | title=Tesla (T) | text=One tesla is the magnetic field strength that exerts a force of one newton on a charge of one coulomb moving perpendicular to the field at a speed of one metre per second. 1 T = 1 N/(C·m/s) = 1 N/(A·m).}}

The tesla is a fairly large unit. For comparison:

• The Earth's magnetic field at the surface is about 3.6 × 10⁻⁵ T (or 0.36 gauss).
• A small bar magnet produces a field of roughly 10⁻² T near its poles.
• Powerful electromagnets in research labs can reach several teslas; superconducting magnets in MRI machines operate around 1.5 to 3 T.

A smaller, non-SI unit called the gauss (G) is also widely used, especially in geophysics and astrophysics:

1 gauss = 10⁻⁴ tesla

So the Earth's field is approximately 0.36 G.

{{VISUAL: chart: table comparing magnetic field strengths in tesla for Earth's field, bar magnet, MRI machine, and laboratory electromagnet}}

{{KEY: type=exam | title=Common Exam Trap | text=Students often confuse the direction of the force on positive and negative charges. Remember: apply the right-hand rule for positive charges; for negative charges, the force is in the opposite direction. Also, magnetic force does NO work on a charged particle because it is always perpendicular to the velocity — it changes direction, not speed.}}

Summary and Physical Insight

The Lorentz force is the cornerstone of electromagnetism. It unifies electric and magnetic interactions into a single framework, revealing that:

• Electric fields exert force on any charge, moving or stationary.
• Magnetic fields exert force only on moving charges, and that force is always perpendicular to the motion.

This perpendicularity has profound consequences: the magnetic force does no work on the particle (since work = F · v and F ⊥ v implies zero dot product). The magnetic field can change the direction of a particle's velocity — bending its path into a circle or helix — but it cannot change the particle's speed or kinetic energy.

{{VISUAL: photo: curved trajectory of charged particles in a bubble chamber showing spiral paths due to magnetic force}}

The magnetic force steers; it does not speed up or slow down. It is the choreographer of motion, not the engine.

This principle underlies many modern technologies: cyclotrons and synchrotrons (particle accelerators), mass spectrometers (separating ions by mass), and the auroras (charged particles from the Sun spiralling along Earth's magnetic field lines). In the next section, we will extend this analysis from a single moving charge to a current-carrying conductor — a collection of countless charges drifting together.

Magnetic force on a current-carrying conductor

Magnetic Force on a Current-Carrying Conductor

So far, we've explored how a single moving charge experiences a force in a magnetic field — the Lorentz force. But what happens when we have millions of charges flowing together as an electric current through a wire? Can we extend the same physics to understand the force on a conductor carrying current?

The answer is yes, and the result is beautifully simple: a current-carrying wire in a magnetic field experiences a force that we can predict and calculate. This principle is the foundation of electric motors, galvanometers, and loudspeakers — devices you use every day.

From Moving Charges to Current-Carrying Conductors

The Big Picture

When current flows through a wire, we have a drift of charge carriers (usually electrons in metals) moving with an average velocity. Each moving charge experiences the magnetic Lorentz force F = q(v × B). Since there are billions of such charges in a conductor, the net force on the wire is the vector sum of forces on all these individual carriers.

Let's build this step-by-step using a straight conducting rod.

{{VISUAL: diagram: a straight cylindrical conducting rod of length l and cross-sectional area A, with electrons drifting inside showing current direction I and external magnetic field B perpendicular to the rod}}

Setting Up the Model

Consider a straight rod of:

• Uniform cross-sectional area A
• Length l
• Number density of mobile charge carriers n (number of carriers per unit volume)
• Charge on each carrier q (for electrons, q = -e)
• Average drift velocity of carriers v_d

The total number of mobile carriers in the rod is:

N = n × V = n × A × l

where V = A × l is the volume of the rod.

{{KEY: type=concept | title=Drift Velocity and Current | text=The drift velocity v_d is the average velocity with which charge carriers move through a conductor when a steady current flows. It is typically very small (around 10^-4 m/s in metals), but the large number density n makes the current appreciable.}}

Deriving the Force Formula

Step 1: Force on All Moving Charges

In the presence of an external magnetic field B, each carrier experiences a force. The total force on all N carriers in the rod is:

F = N × q × (v_d × B)
F = (n × A × l) × q × (v_d × B)

Rearranging:

F = (n × q × v_d) × (A × l) × B

Step 2: Introducing Current Density

Recall from Chapter 3 that current density j is defined as:

j = n × q × v_d

This is a vector pointing in the direction of flow of positive charge. The magnitude of j gives the current per unit area.

The total current I through the rod is:

I = j × A = |n × q × v_d| × A

{{KEY: type=definition | title=Current Density | text=Current density j is the current per unit cross-sectional area, given by j = n q v_d, where n is the number density of charge carriers, q is the charge per carrier, and v_d is their drift velocity. It is a vector quantity.}}

Step 3: The Final Formula

Substituting j × A back into our force expression:

F = (j × A) × l × B = I × l × B

But we want this in vector form. Define a length vector l of magnitude l, pointing in the direction of the current I. Then:

{{FORMULA: expr=F = I l × B | symbols=F:magnetic force on conductor (N), I:current (A), l:length vector of conductor (m), B:magnetic field (T)}}

This is the force on a straight current-carrying conductor in a uniform magnetic field.

{{VISUAL: diagram: vector diagram showing current I along a rod, magnetic field B perpendicular to it, and the resulting force F = I l × B pointing in the direction given by the right-hand rule}}

{{KEY: type=points | title=Key Features of the Force F = I l × B | text=- The force is perpendicular to both the current direction and the magnetic field.

• Maximum force occurs when l and B are perpendicular (sin θ = 1).
• Zero force when current is parallel to the magnetic field (sin θ = 0).
• Direction is given by the right-hand rule or the cross product.}}

Direction of the Force: The Right-Hand Rule

To find the direction of F = I l × B, use the right-hand rule:

1. Point your fingers in the direction of current I (along l).
2. Curl them toward the direction of magnetic field B.
3. Your thumb points in the direction of the force F.

{{VISUAL: diagram: right-hand showing fingers along current direction, curling toward magnetic field, and thumb pointing in the direction of force}}

{{ZOOM: title=Why transfer the vector sign from j to l? | text=In the derivation, we write j as a vector because it has direction. But when we express force in terms of current I (a scalar) and length l, we make l a vector to preserve the directional information. This is purely a notational convenience — the physics remains the same.}}

Worked Example: Wire Suspended in a Magnetic Field

Let's apply this formula to a real situation.

Example 4.1 (from NCERT): A straight wire of mass 200 g and length 1.5 m carries a current of 2 A. It is suspended in mid-air by a uniform horizontal magnetic field B. What is the magnitude of the magnetic field?

{{VISUAL: photo: a horizontal wire suspended in mid-air with upward magnetic force balancing downward gravitational force, showing current direction and magnetic field perpendicular to it}}

Solution

For the wire to be suspended in mid-air, the upward magnetic force must balance the downward gravitational force:

F_magnetic = F_gravity
I × l × B = m × g

Solving for B:

B = (m × g) / (I × l)
B = (0.2 kg × 9.8 m/s²) / (2 A × 1.5 m)
B = 1.96 / 3
B = 0.65 T

Key observation: We only need the mass per unit length m/l to solve this — the absolute mass and length can vary as long as their ratio stays the same. Also, Earth's magnetic field (≈ 4 × 10⁻⁵ T) is negligible compared to 0.65 T, so we ignore it.

{{KEY: type=exam | title=Common Question Type | text=CBSE often asks for the condition of equilibrium of a current-carrying wire in a magnetic field. Remember to equate magnetic force I l B sin θ with gravitational force m g, and solve for the unknown — usually B, I, or the angle θ.}}

Extension to Arbitrary Shapes

What if the wire is not straight? For a wire of arbitrary shape, we can:

• Divide it into small linear elements dl_j
• Calculate the force on each element: dF_j = I dl_j × B
• Sum (or integrate) over all elements:

F = Σ (I dl_j × B) → ∫ I dl × B

This integral approach is powerful for calculating forces on coils, loops, and complex conductor geometries — topics you'll explore in later sections and in college physics.

Key Takeaways

The force on a current-carrying conductor in a magnetic field is a macroscopic manifestation of the Lorentz force acting on billions of moving charge carriers. The formula F = I l × B is central to the working of every electric motor and measuring instrument.

You've now bridged the gap between the motion of individual charges and the behaviour of conductors carrying steady currents. In the next section, we'll dive deeper into the motion of charged particles in magnetic fields — circular paths, helical trajectories, and the physics behind devices like the cyclotron.

Motion in a Magnetic Field

Motion in a Magnetic Field

When a charged particle enters a magnetic field, the force it experiences is always perpendicular to its velocity. This unique property gives rise to fascinating and predictable trajectories — from perfect circles to elegant helices — that are exploited in particle accelerators, mass spectrometers, and even the aurora borealis. Let us explore the beautiful geometry of motion in a magnetic field.

The Force Does No Work

Recall that the magnetic force on a moving charge is given by F = q(v × B). Since the cross product is always perpendicular to both v and B, the force acts at right angles to the velocity.

From mechanics, we know that work is done only when a force has a component along the displacement. Here, the force is perpendicular to the motion at every instant, so:

Work done by the magnetic force = 0

This has a profound consequence: the kinetic energy of the particle remains constant. The magnetic field can change the direction of velocity but never its magnitude.

The magnetic field steers the particle without speeding it up or slowing it down.

This is unlike the electric field, where F = qE can have a component parallel or antiparallel to velocity, thus transferring energy.

{{KEY: type=concept | title=Zero Work by Magnetic Force | text=Because the magnetic force is always perpendicular to velocity, it does no work on a charged particle. The speed remains constant; only the direction of motion changes. This is the fundamental difference between electric and magnetic forces.}}

Circular Motion: When v ⊥ B

Consider a uniform magnetic field B pointing into the page, and a charged particle with velocity v perpendicular to B.

{{VISUAL: diagram: top view of a charged particle entering a uniform magnetic field perpendicular to the page, with velocity vector v horizontal and magnetic field B into the page, showing the perpendicular Lorentz force F causing circular motion}}

The force F = qvB acts perpendicular to both v and B, pointing towards the center of a circle. This is precisely a centripetal force.

For circular motion, the centripetal force required is:

F_c = (m v²) / r

Equating the magnetic force to the centripetal force:

q v B = (m v²) / r

Solving for the radius of the circular path:

r = (m v) / (q B)

{{FORMULA: expr=r = m v / (q B) | symbols=r:radius of circular path (m), m:mass of particle (kg), v:speed of particle (m/s), q:charge of particle (C), B:magnetic field strength (T)}}

{{KEY: type=definition | title=Radius of Circular Path | text=The radius of the circular trajectory of a charged particle in a perpendicular magnetic field is r = m v / (q B). It is directly proportional to momentum (m v) and inversely proportional to charge and field strength.}}

Angular Frequency and Period

The particle completes one circle in time T, the period. Using v = ω r where ω is angular frequency:

ω = v / r = (q B) / m

The frequency of revolution:

ν = ω / (2π) = (q B) / (2π m)

And the period:

T = 1 / ν = (2π m) / (q B)

{{KEY: type=points | title=Key Properties of Circular Motion | text=- Angular frequency ω = q B / m is independent of speed and energy.

• Larger momentum → larger radius.
• Period T = 2π m / (q B) depends only on q, B, and m.
• This energy-independence is the basis of the cyclotron.}}

Notice something remarkable: the frequency is independent of the particle's speed or energy. A fast particle traces a large circle, a slow one a small circle, but both take the same time to complete one revolution. This principle is exploited in the cyclotron, a particle accelerator.

{{ZOOM: title=The Cyclotron Principle | text=In a cyclotron, charged particles spiral outward as they gain energy, but their revolution frequency remains constant. This allows a fixed-frequency alternating voltage to accelerate them repeatedly at the same phase — a cornerstone of high-energy physics.}}

Helical Motion: When v Has a Component Along B

What if the velocity has a component parallel to B?

Let v = v_⊥ + v_∥, where v_⊥ is perpendicular to B and v_∥ is parallel to B.

{{VISUAL: diagram: 3D perspective showing a charged particle moving in a helical path, with velocity decomposed into perpendicular component v_perp causing circular motion and parallel component v_parallel causing linear drift along the field}}

• The perpendicular component v_⊥ causes circular motion in the plane perpendicular to B, exactly as before, with radius r = (m v_⊥) / (q B).
• The parallel component v_∥ is unaffected by the magnetic field (since F = q v_∥ × B = 0), so the particle drifts at constant speed along the field lines.

The result: the particle traces a helix — a spiral that advances along the field direction.

{{VISUAL: chart: side view of helical motion showing one complete turn, with pitch p labeled as the distance moved along the field in one revolution}}

The pitch of the helix is the distance travelled parallel to B in one revolution:

p = v_∥ T = v_∥ × (2π m) / (q B)

{{KEY: type=concept | title=Helical Motion in a Magnetic Field | text=When velocity has a component parallel to B, the particle moves in a helix. The perpendicular component causes circular motion; the parallel component causes linear drift. The pitch p = 2π m v_parallel / (q B) is the advance per revolution.}}

Worked Example from NCERT

Let's apply these ideas to a concrete problem.

Example 4.3 from NCERT:

What is the radius of the path of an electron (mass 9 × 10⁻³¹ kg, charge 1.6 × 10⁻¹⁹ C) moving at speed 3 × 10⁷ m/s in a magnetic field of 6 × 10⁻⁴ T perpendicular to it? What is its frequency? Calculate its energy in keV.

Solution:

Using the formula for radius:

r = (m v) / (q B)

r = (9 × 10⁻³¹ kg × 3 × 10⁷ m/s) / (1.6 × 10⁻¹⁹ C × 6 × 10⁻⁴ T)

r = (27 × 10⁻²⁴) / (9.6 × 10⁻²³) = 0.28 m = 28 cm

For frequency:

ν = v / (2π r) = (3 × 10⁷) / (2π × 0.28) ≈ 1.7 × 10⁷ Hz = 17 MHz

Kinetic energy:

E = ½ m v² = ½ × 9 × 10⁻³¹ × (3 × 10⁷)²

E = ½ × 9 × 10⁻³¹ × 9 × 10¹⁴ = 40.5 × 10⁻¹⁷ J ≈ 4 × 10⁻¹⁶ J

Converting to eV (since 1 eV = 1.6 × 10⁻¹⁹ J):

E = (4 × 10⁻¹⁶) / (1.6 × 10⁻¹⁹) = 2500 eV = 2.5 keV

{{VISUAL: diagram: vector diagram showing electron velocity v perpendicular to magnetic field B, with resulting circular path of radius r = 28 cm and force F pointing toward center}}

{{KEY: type=exam | title=Common Exam Question | text=CBSE often asks you to calculate radius, frequency, or pitch given particle mass, charge, speed, and field. Remember to decompose velocity into perpendicular and parallel components if the angle is not 90°. Show all substitutions clearly for full marks.}}

Comparison with Electric Field

Solved NCERT Exercises

Q4.11 A charged particle is moving in a circular orbit in a uniform magnetic field. Obtain the frequency of revolution of the electron in its circular orbit. Does the answer depend on the speed of the electron? Explain.

Solution:

Step 1: For circular motion in a magnetic field, the magnetic force provides the centripetal force:

q v B = (m v²) / r

Step 2: Solving for radius:

r = (m v) / (q B)

Step 3: The angular frequency is:

ω = v / r = (q B) / m

Step 4: The frequency of revolution is:

ν = ω / (2π) = (q B) / (2π m)

Step 5: For an electron, q = e = 1.6 × 10⁻¹⁹ C and m = 9.1 × 10⁻³¹ kg.

Does the answer depend on speed?

No. The frequency ν = (q B) / (2π m) depends only on the charge q, magnetic field B, and mass m. It is independent of the speed of the electron. This is because a faster electron traces a larger circle, but completes it in the same time.

Final Answer: ν = (q B) / (2π m), which is independent of speed.

Q4.13(a) A circular coil of 30 turns and radius 8.0 cm carrying a current of 6.0 A is suspended vertically in a uniform horizontal magnetic field of magnitude 1.0 T. The field lines make an angle of 60° with the normal of the coil. Calculate the magnitude of the counter torque that must be applied to prevent the coil from turning.

Solution:

Step 1: The torque on a current-carrying coil in a magnetic field is:

τ = n I A B sin θ

where n = number of turns, I = current, A = area of coil, B = magnetic field, θ = angle between normal to the coil and the field.

Step 2: Given data:

• n = 30 turns
• r = 8.0 cm = 0.08 m
• I = 6.0 A
• B = 1.0 T
• θ = 60°

Step 3: Area of the coil:

A = π r² = π × (0.08)² = π × 0.0064 = 0.0201 m²

Step 4: Calculate torque:

τ = 30 × 6.0 × 0.0201 × 1.0 × sin 60°

τ = 30 × 6.0 × 0.0201 × 1.0 × 0.866

τ = 30 × 6.0 × 0.0201 × 0.866 = 3.14 N·m

Final Answer: The counter torque required is 3.14 N·m.

Q4.13(b) Would your answer change, if the circular coil in (a) were replaced by a planar coil of some irregular shape that encloses the same area? (All other particulars are also unaltered.)

Solution:

Step 1: The torque on a current loop depends on the magnetic moment M = n I A, where A is the area enclosed by the loop, regardless of its shape.

Step 2: The formula τ = M B sin θ = n I A B sin θ does not depend on the shape of the loop, only on the area it encloses.

Step 3: Since the irregular coil encloses the same area, has the same number of turns, carries the same current, and is in the same magnetic field at the same angle, the torque remains the same.

Final Answer: No, the answer would not change. The torque depends only on the area enclosed, not the shape of the coil.