Model Electric and Magnetic Fields — Model interactions through electric and magnetic fields - Grade 11 Physics

SECTION 1

Historical Context & Motivation

For most of human history, the forces of lightning, static cling, and compass needles seemed like separate mysteries. Ancient Greeks noticed that rubbing amber attracted straw, and Chinese sailors used lodestones to navigate, but nobody could explain why these effects occurred. It took centuries of experiment and brilliant reasoning before scientists realized that electricity and magnetism are two sides of the same coin—governed by invisible fields that fill the space around charged and magnetic objects.

1600 • Gilbert's De Magnete

William Gilbert publishes De Magnete, the first systematic study of magnetism. He shows Earth itself is a giant magnet and distinguishes magnetic attraction from the static-electric "amber effect."

1785 • Coulomb's Law

Charles-Augustin de Coulomb measures the force between charged spheres using a torsion balance and establishes that the electric force follows an inverse-square law, much like gravity.

1820 • Ørsted's Discovery

Hans Christian Ørsted accidentally discovers that an electric current deflects a nearby compass needle, revealing the first link between electricity and magnetism.

1831 • Faraday's Field Lines

Michael Faraday introduces the concept of "lines of force" to visualize electric and magnetic fields. He also discovers electromagnetic induction—changing magnetic fields produce electric currents.

1865 • Maxwell's Equations

James Clerk Maxwell unifies electricity, magnetism, and light into four elegant equations, predicting electromagnetic waves that travel at the speed of light.

This historical arc raises a central question: How do electric charges and magnets influence objects at a distance, even through empty space? The answer lies in the field model—an idea so powerful that it underpins everything from smartphone screens to MRI machines. In this lesson, you will learn to describe, draw, and calculate with both electric and magnetic fields.

SECTION 2

Core Principles & Definitions

A field is a region of space where a force acts on an appropriate object—charges feel electric fields, and magnets (or moving charges) feel magnetic fields. Rather than thinking of forces that magically leap across empty space, we picture each charge or magnet as creating a field that extends outward in all directions. When another charge or magnet enters that field, it experiences a force.

Electric Field (E)

The electric field at a point is the force per unit positive test charge placed there. It points away from positive sources and toward negative sources. Its SI unit is newtons per coulomb (N/C).

Magnetic Field (B)

The magnetic field is produced by moving charges or magnetic materials. It is measured in teslas (T). Magnetic field lines form closed loops—they exit the north pole of a magnet and enter the south pole.

Field Lines

Field lines are imaginary curves that show the direction a positive test charge (or north pole) would move. Where lines are closely spaced the field is strong; where they spread apart the field is weak.

Superposition

When multiple sources create fields in the same region, the net field at any point is the vector sum of all individual fields. This is the principle of superposition.

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KEY TAKEAWAY

Think of a field like the smell of a campfire. The fire (source charge) creates a scent (field) that fills the space around it. You don't need to touch the fire to smell it—the scent is already in the air wherever you stand. Similarly, a charge creates an electric field everywhere around it, and another charge "feels" that field as a force, even without direct contact.

SECTION 3

Visualizing Electric and Magnetic Fields

Electric Field Lines Around Point Charges

Left: field lines radiate symmetrically outward from a single positive charge. Right: in an electric dipole, lines emerge from the positive charge and curve toward the negative charge. The density of lines indicates field strength—notice they are most concentrated between the two charges.

The diagram above illustrates two fundamental patterns. A single positive charge pushes field lines outward uniformly—like a sprinkler spraying water in all directions. An electric dipole (one positive and one negative charge) bends those lines so they start on the positive charge and terminate on the negative charge. The closer the lines are together, the stronger the field at that point. If you placed a small positive test charge anywhere in either diagram, it would accelerate along the direction of the nearest arrow.

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Drawing Rule of Thumb

Electric field lines never cross each other. If they did, a test charge at the crossing point would have two conflicting directions of force, which is physically impossible. Also, lines always begin on positive charges and end on negative charges (or extend to infinity if no opposite charge is nearby).

SECTION 4

Mathematical Framework

Now that you can visualize fields, let's quantify them. The equations below let you calculate the strength of electric and magnetic fields and the forces they exert on charges.

COULOMB'S LAW

F = k × (q₁ × q₂) / r²

F = electric force (N), k = Coulomb's constant ≈ 8.99 × 10⁹ N·m²/C², q₁ and q₂ = charges (C), r = distance between charges (m). Positive F means repulsion; negative F means attraction.

ELECTRIC FIELD STRENGTH

E = F / q = k × Q / r²

E = electric field strength (N/C), F = force on test charge (N), q = test charge (C), Q = source charge (C). The field exists whether or not a test charge is present.

MAGNETIC FORCE ON A MOVING CHARGE

F = q × v × B × sin(θ)

F = magnetic force (N), q = charge (C), v = velocity (m/s), B = magnetic field strength (T), θ = angle between velocity and field. The force is maximum when the charge moves perpendicular (θ = 90°) and zero when moving parallel (θ = 0°).

MAGNETIC FORCE ON A CURRENT-CARRYING WIRE

F = I × L × B × sin(θ)

I = current (A), L = length of wire in the field (m), B = magnetic field strength (T), θ = angle between wire and field direction. This is the principle behind electric motors.

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Right-Hand Rule

To find the direction of the magnetic force on a positive charge, point the fingers of your right hand in the direction of the velocity (v), then curl them toward the magnetic field (B). Your thumb points in the direction of the force (F). For negative charges, the force is in the opposite direction.

SECTION 5

Comparing Electric and Magnetic Fields

Electric and magnetic fields are related but behave differently. Understanding their similarities and differences is essential for mastering electromagnetism. The following diagram shows magnetic field lines around a bar magnet alongside field lines between parallel plates, and the comparison table below highlights the key distinctions.

Left: magnetic field lines form closed loops around a bar magnet, exiting the north pole and entering the south pole. Right: between parallel charged plates, the electric field is nearly uniform—lines are straight, equally spaced, and point from the positive plate toward the negative plate.

Feature	Electric Field (E)	Magnetic Field (B)
Source	Electric charges (positive or negative), whether stationary or moving	Moving charges (electric currents) or intrinsic magnetic moments of particles
Acts on	Any electric charge (stationary or moving)	Only moving charges or other magnets
Line behavior	Begin on + charges, end on − charges (or extend to infinity)	Always form closed loops (no magnetic monopoles)
Force direction	Parallel to the field lines	Perpendicular to both v and B (right-hand rule)
SI unit	N/C (or V/m)	Tesla (T)
Does work?	Yes — it can accelerate or decelerate charges along its direction	No — it only changes direction of motion, not speed (force is always perpendicular to velocity)

Key differences between electric and magnetic fields

One of the most important distinctions is that magnetic fields never do work on charged particles. Because the magnetic force is always perpendicular to a particle's velocity, it changes the direction of motion but cannot speed the particle up or slow it down. This is why a magnetic field can bend the path of a charged particle into a circle without adding kinetic energy.

SECTION 6

Worked Example — Calculating Electric Field and Force

Let's walk through a complete problem that uses Coulomb's law and the electric field equation.

Electric Field and Force Between Two Charges

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Step 1 — Read the Problem

A charge Q₁ = +3.0 × 10⁻⁶ C is placed 0.40 m from a second charge Q₂ = −2.0 × 10⁻⁶ C. (a) Find the electric field at the location of Q₂ due to Q₁. (b) Find the magnitude and direction of the force on Q₂.
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Step 2 — Identify Given Values

Q₁ = +3.0 × 10⁻⁶ C, Q₂ = −2.0 × 10⁻⁶ C, r = 0.40 m, k = 8.99 × 10⁹ N·m²/C².
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Step 3 — Calculate the Electric Field due to Q₁

Using E = k × Q / r²: E = (8.99 × 10⁹) × (3.0 × 10⁻⁶) / (0.40)² E = (8.99 × 10⁹) × (3.0 × 10⁻⁶) / 0.16 E = 26,970 / 0.16

E ≈ 1.69 × 10⁵ N/C, directed from Q₁ toward Q₂ (since Q₁ is positive, the field points away from it).
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Step 4 — Calculate the Force on Q₂

Using F = q × E (or equivalently Coulomb's law): F = |Q₂| × E = (2.0 × 10⁻⁶) × (1.69 × 10⁵)

F ≈ 0.337 N. Since Q₁ is positive and Q₂ is negative, the force is attractive — Q₂ is pulled toward Q₁.
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Step 5 — Check Your Answer

The force is less than 1 N, which is reasonable for microcoulomb-level charges at 0.40 m separation. The direction makes sense: opposite charges attract. If we doubled the distance to 0.80 m, the force would drop to one-quarter of 0.337 N (about 0.084 N), confirming the inverse-square relationship.

SECTION 7

Real-World Applications & Limitations

Electric and magnetic field models are not just abstract physics—they are the foundation of modern technology. Here are some of the most impactful applications, along with the limitations of simplified field models.

Application	Field Type Used	How It Works
Electric motors	Magnetic (B)	Current-carrying coils in a magnetic field experience a torque (F = ILB sin θ), causing rotation that drives machines.
Touchscreens	Electric (E)	Your finger changes the local electric field on a capacitive screen, which the device detects as a touch location.
MRI scanners	Magnetic (B)	Strong magnetic fields align hydrogen atoms in the body. Radio pulses disturb this alignment, and the resulting signals create detailed images.
Power generators	Both (E and B)	A changing magnetic field (from spinning turbines) induces an electric field in wire coils, creating the current that powers your home.
Lightning rods	Electric (E)	The pointed tip concentrates the electric field, providing a preferred path for lightning to safely reach the ground.

Everyday technologies that rely on electric and magnetic fields

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LIMITATIONS TO KEEP IN MIND

The field-line model is a powerful visualization tool, but it has limits. Field lines are not physical objects—they are a human-made map, not the territory itself. The simple equations in this lesson assume point charges and uniform fields; real-world situations (like irregular conductors or time-varying fields) require more advanced math such as Maxwell's equations and vector calculus. Additionally, at very small scales, quantum electrodynamics replaces the classical field picture.

SECTION 8

Connection to Electromagnetic Theory

The electric and magnetic fields you have studied in this lesson are actually two aspects of a single entity called the electromagnetic field. James Clerk Maxwell showed that a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field. This mutual creation is what allows electromagnetic waves—including visible light, radio waves, and X-rays—to propagate through empty space at 3.0 × 10⁸ m/s.

Concept	This Lesson (Classical Fields)	Advanced Theory
Field description	Separate E and B fields with arrows and field lines	Unified electromagnetic field tensor in special relativity
Governing equations	Coulomb's law, F = qvB sin θ, simple superposition	Maxwell's four equations using vector calculus (divergence, curl)
Waves	Not covered in this lesson	E and B fields oscillate perpendicular to each other, traveling as electromagnetic waves
Quantum scale	Continuous fields	Fields are mediated by virtual photons in quantum electrodynamics (QED)

Classical fields vs. advanced electromagnetic theory

In future courses such as AP Physics or college electromagnetism, you will learn how Maxwell's equations unify everything you've studied here into a complete, elegant framework. For now, the key insight is that electricity and magnetism are inseparable partners. Every time you charge your phone, listen to a speaker, or see a rainbow, you are witnessing the electromagnetic field in action.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A stationary positive charge is placed in a uniform magnetic field. Will it experience a magnetic force? Explain your reasoning using the magnetic force equation.

Show answer

No, the stationary charge will not experience a magnetic force. According to the equation F = qvB sin(θ), the velocity v = 0 for a stationary charge, which makes the entire product zero regardless of the charge or field strength. Magnetic fields only exert forces on moving charges. However, the charge would experience a force if an electric field were also present, since electric forces act on charges whether they are moving or not.

Question 2: BASIC CALCULATION

Calculate the electric field strength at a distance of 0.25 m from a point charge of +5.0 × 10⁻⁶ C. Use k = 8.99 × 10⁹ N·m²/C².

Show answer

Using E = k × Q / r²: E = (8.99 × 10⁹) × (5.0 × 10⁻⁶) / (0.25)² E = 44,950 / 0.0625 E ≈ 7.19 × 10⁵ N/C, directed radially outward from the positive charge. The field is quite strong because the charge is relatively large (microcoulombs) and the distance is small.

Question 3: INTERMEDIATE

A proton (q = 1.6 × 10⁻¹⁹ C) moves at 4.0 × 10⁶ m/s perpendicular to a magnetic field of 0.50 T. (a) Calculate the magnetic force on the proton. (b) Describe the resulting motion of the proton.

Show answer

(a) Since the proton moves perpendicular to B, θ = 90° and sin(90°) = 1: F = qvB sin(θ) = (1.6 × 10⁻¹⁹)(4.0 × 10⁶)(0.50)(1) F = 3.2 × 10⁻¹³ N (b) Because the magnetic force is always perpendicular to the velocity, it acts as a centripetal force. The proton follows a circular path. The speed remains constant (the magnetic field does no work), but the direction continuously changes. The right-hand rule determines which way the circle curves.

Question 4: APPLIED

A straight wire carrying 12 A of current passes through a 0.30 T magnetic field. The wire is 0.50 m long and oriented at 60° to the field. (a) Calculate the force on the wire. (b) If this wire is one side of a rectangular coil in an electric motor, explain how the motor produces rotation.

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(a) Using F = ILB sin(θ): F = (12)(0.50)(0.30) sin(60°) = (12)(0.50)(0.30)(0.866) F ≈ 1.56 N (b) In a motor, opposite sides of the rectangular coil carry current in opposite directions. This means the magnetic force pushes one side up and the other side down, creating a torque that spins the coil. A device called a commutator reverses the current direction every half-turn so the torque always acts in the same rotational direction, producing continuous spinning.

Question 5: CRITICAL THINKING

Two identical positive charges are fixed 1.0 m apart. A small positive test charge is placed exactly at the midpoint between them. (a) What is the net electric field at the midpoint? (b) Is the midpoint a stable or unstable equilibrium position if the test charge is nudged along the line connecting the two charges? What about if it is nudged perpendicular to that line? Justify your answers.

Show answer

(a) By symmetry, each fixed charge creates an electric field of equal magnitude at the midpoint, but these two fields point in opposite directions (each points away from its respective source charge). They cancel exactly, so the net electric field at the midpoint is zero. (b) Along the line connecting the charges: if the test charge is nudged toward one source charge, the force from that nearer charge grows (inverse-square law) while the force from the farther charge shrinks. The net force pushes the test charge further away from the midpoint—this is unstable equilibrium along this axis. Perpendicular to the line: both source charges now push the test charge back toward the midpoint (their force components along the perpendicular direction both point inward), so it is stable in the perpendicular direction. This illustrates that in electrostatics, you cannot have stable equilibrium in all directions simultaneously (a result known as Earnshaw's theorem).

SUMMARY

Lesson Summary

Summary

Electric and magnetic fields are the invisible agents that transmit forces between charges and magnets across space. An electric field (E) is created by any electric charge and exerts a force on other charges, described quantitatively by Coulomb's law (F = kq₁q₂/r²) and the field equation E = kQ/r². A magnetic field (B) is produced by moving charges or magnets and exerts a force only on moving charges, given by F = qvB sin(θ). Field lines let us visualize these fields: they start on positive charges and end on negative charges for E-fields, and they form closed loops for B-fields.

Key differences to remember: electric fields do work on charges (changing their speed), while magnetic fields cannot do work (they only change direction of motion). The right-hand rule determines the direction of the magnetic force. When multiple sources are present, use the principle of superposition to find the net field by vector addition. These concepts form the foundation of electromagnetism, which powers motors, generators, medical imaging, and nearly all modern technology.