Understand Electric & Magnetic Fields — Understand D.2 Electric and magnetic fields - IB Physics

SECTION 1

Historical Context & Motivation

Humans have known about static electricity since the ancient Greeks rubbed amber against fur and watched lightweight objects leap toward it. Yet for centuries, no one could explain why forces appeared between objects that were not touching. Similarly, lodestones — naturally magnetized rocks — mystified navigators who relied on compass needles long before anyone understood the underlying physics. The breakthroughs that eventually explained these phenomena came in waves, with each generation of scientists building on what came before.

1785 • Coulomb's Law

Charles-Augustin de Coulomb used a torsion balance to show that the electric force between two point charges follows an inverse-square law, much like gravity. This gave electricity its first precise mathematical description.

1820 • Ørsted's Discovery

Hans Christian Ørsted noticed that a compass needle deflected when placed near a current-carrying wire, proving that electricity and magnetism are fundamentally connected.

1831 • Faraday's Field Lines

Michael Faraday introduced the concept of lines of force to visualize electric and magnetic fields, and discovered electromagnetic induction — a changing magnetic field creates an electric effect.

1865 • Maxwell's Equations

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

1905 • Einstein & Relativity

Albert Einstein showed that electric and magnetic fields are two aspects of the same electromagnetic field, with their relative strengths depending on the observer's frame of reference.

The central question this topic addresses is: how do we describe the invisible influence that a charge or a current exerts on the space around it? The answer lies in the concept of a field — a physical quantity that has a value at every point in space. Instead of thinking about forces acting magically across a distance, fields allow us to say that a charge creates something real in the space around it, and any other charge placed in that region feels a force because of that field.

SECTION 2

Core Principles & Definitions

Electric and magnetic fields share a common foundation: both are vector fields, meaning they have a magnitude (size) and a direction at every point in space. Understanding the core principles below gives you the tools to analyze almost any situation involving charges and currents.

Electric Field (E)

An electric field is the force per unit positive test charge at a point. It points away from positive source charges and toward negative ones. Units: N C⁻¹ (or equivalently V m⁻¹).

Magnetic Field (B)

A magnetic field is produced by moving charges (currents) or changing electric fields. It exerts a force on other moving charges. The SI unit is the tesla (T). Unlike electric field lines, magnetic field lines always form closed loops.

Field Lines

Field lines are visual tools: their direction shows the field direction, and their density (how close together they are) indicates the field strength. Lines never cross, because the field can only point in one direction at any given point.

Superposition

When multiple sources are present, the total field at any point is the vector sum of the individual fields. This principle of superposition lets us handle complex charge or current arrangements by adding contributions one at a time.

Force on Charges

An electric field exerts a force F = qE on any charge q placed in it. A magnetic field exerts a force F = qv × B on a charge moving with velocity v. Crucially, the magnetic force is always perpendicular to the velocity, so it changes direction but not speed.

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KEY TAKEAWAY

Think of a field like the wind around a fan. You can't see the wind itself, but you can feel it and see its effects on objects. The fan (a charge or current) creates the wind (the field), and anything that enters that region (another charge) feels a push or pull. The field exists whether or not a second charge is there to feel it — just like the wind blows even when nothing is in its path.

SECTION 3

Visualizing Electric Field Patterns

Electric field line diagrams are your most powerful tool for quickly understanding what happens in the space around charges. The diagram below shows the three most common configurations you will encounter in IB Physics: a single positive point charge, a single negative point charge, and a pair of equal and opposite charges (a dipole). Pay close attention to the direction of the arrows and the spacing of the lines.

Left: field lines radiate outward from a positive charge. Centre: field lines point inward toward a negative charge. Right: in a dipole, lines leave the positive charge, curve through space, and terminate on the negative charge. The density of lines is greatest near the charges, indicating a stronger field.

Notice three crucial features in the diagram. First, the arrows always show the direction a positive test charge would move — away from positive sources and toward negative ones. Second, the lines are closer together near each charge, reflecting the inverse-square relationship: the field gets much stronger as you approach the source. Third, in the dipole configuration the lines curve smoothly from the positive charge to the negative charge, never crossing each other. These visual rules apply whether you're sketching fields on an exam or analyzing a complex charge arrangement.

SECTION 4

Mathematical Framework

The qualitative pictures of the previous section become quantitative once you attach equations. IB Physics D.2 requires you to work with the key formulas for electric and magnetic fields. Let's build them up, starting with Coulomb's law.

Electric Field Equations

COULOMB'S LAW

F = k × (q₁ × q₂) / r²

F = electrostatic force (N), k = Coulomb's constant ≈ 8.99 × 10⁹ N m² C⁻², q₁ and q₂ = charges (C), r = separation distance (m). Like charges → F is repulsive; unlike charges → F is attractive.

ELECTRIC FIELD STRENGTH

E = F / q = k × Q / r²

E = electric field strength (N C⁻¹), Q = source charge creating the field, q = small positive test charge, r = distance from Q. This equation defines the field as force per unit charge, independent of the test charge.

Magnetic Field Equations

FORCE ON A MOVING CHARGE IN B

F = qvB sin θ

F = magnetic force (N), q = charge (C), v = speed (m s⁻¹), B = magnetic field strength (T), θ = angle between v and B. When θ = 90°, the force is maximum; when θ = 0° (charge moves parallel to B), the force is zero.

FORCE ON A CURRENT-CARRYING WIRE

F = BIL sin θ

B = magnetic field (T), I = current (A), L = length of wire in the field (m), θ = angle between the wire and the field. This is the macroscopic version of the force on moving charges, since current is just many charges flowing together.

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Direction Rule: Right-Hand Rule

To find the direction of the magnetic force, point your right-hand fingers in the direction of the velocity (or current), curl them toward the magnetic field B, and your thumb points in the direction of the force on a positive charge. For a negative charge, the force is in the opposite direction.

SECTION 5

Magnetic Field Patterns & the Right-Hand Rule

While electric field lines start and end on charges, magnetic field lines always form closed loops. There are no isolated magnetic "charges" (magnetic monopoles have never been found). The diagram below shows the magnetic field around a long straight current-carrying wire and around a solenoid (a coil of many loops). These two configurations appear repeatedly in IB problems.

Left: looking down a straight wire with current flowing out of the page, the magnetic field forms concentric circles that weaken with distance. Right: inside a solenoid, the field is nearly uniform and parallel, resembling a bar magnet externally.

For the straight wire, the right-hand grip rule tells you the direction: point your thumb along the current and your fingers naturally curl in the direction of B. For the solenoid, curl your right-hand fingers in the direction the current flows around the coils, and your thumb points toward the north pole — the end from which field lines emerge. Inside the solenoid the field is almost perfectly uniform, making solenoids extremely useful in engineering applications such as MRI machines and particle accelerators.

Feature	Electric Field	Magnetic Field
Source	Stationary or moving charges	Moving charges (currents) only
Acts on	Any charge (stationary or moving)	Moving charges only
Force direction	Parallel to field lines	Perpendicular to both v and B
Field lines	Begin on + charges, end on − charges	Always form closed loops
SI Unit	N C⁻¹ (or V m⁻¹)	Tesla (T)

Comparison of electric and magnetic field properties

SECTION 6

Worked Example: Force on a Moving Charge

Let's apply the equations to a typical IB problem combining both electric and magnetic fields.

Proton in Crossed E and B Fields

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Step 1 — Read the Problem

A proton (charge q = 1.6 × 10⁻¹⁹ C) enters a region where a uniform electric field E = 4.0 × 10⁴ N C⁻¹ acts downward and a uniform magnetic field B = 0.20 T acts into the page. The proton moves horizontally to the right at speed v. Find the speed v at which the proton travels in a straight line (undeflected).
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Step 2 — Identify the Forces

The electric force on the proton is F_E = qE, directed downward (same direction as E for a positive charge). The magnetic force on the proton is F_B = qvB (sin 90° = 1 because v ⊥ B). Using the right-hand rule with velocity to the right and B into the page, the magnetic force is directed upward.
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Step 3 — Set the Condition for No Deflection

For the proton to travel in a straight line, the net force must be zero. This means the electric force downward must equal the magnetic force upward: qE = qvB.
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Step 4 — Solve for v

Cancel q from both sides: E = vB, so v = E / B. Substituting: v = (4.0 × 10⁴) / (0.20) = 2.0 × 10⁵ m s⁻¹.

v = 2.0 × 10⁵ m s⁻¹
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Step 5 — Interpret the Result

This device is called a velocity selector. Only particles travelling at exactly this speed pass through undeflected; faster particles curve one way, slower particles curve the other. Note that the result is independent of the charge and mass of the particle — any positive ion at 2.0 × 10⁵ m s⁻¹ passes through.

SECTION 7

Strengths, Limitations & Common Misconceptions

The field concept is extraordinarily powerful, but students often trip over a few persistent misconceptions. The table below highlights what the field model does well and where common mistakes arise.

Strength	Common Misconception
Fields let us predict forces without knowing the details of every interacting particle — just know E or B at a point.	"Field lines are real things." They are a visualization tool; the field itself is the real quantity.
Superposition allows complex systems to be broken into simpler parts.	"A stationary charge can feel a magnetic force." No — magnetic force requires motion (v ≠ 0).
The inverse-square law for E is mathematically identical to gravity, so techniques transfer between topics.	"The magnetic force does work on a charge." Since F ⊥ v always, the magnetic force does zero work.
Field diagrams offer quick qualitative insight — you can sketch the answer before calculating.	"Inside a conductor the field is always zero." Only in electrostatic equilibrium (no current flowing).

Strengths of the field model vs. common student misconceptions

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KEY TAKEAWAY

The most important distinction to internalize is this: an electric field pushes charges along the field direction, while a magnetic field pushes moving charges sideways. Think of it like a river vs. a spinning merry-go-round. In a river (E field), you float straight downstream. On a merry-go-round (B field), you feel pushed to the side as you try to walk across it — this sideways deflection is exactly what happens to a charged particle in a magnetic field.

SECTION 8

Connection to Electromagnetic Induction & Beyond

Understanding static electric and magnetic fields is the foundation, but the real magic happens when these fields change with time. A changing magnetic field creates an electric field (Faraday's law), and a changing electric field creates a magnetic field (the Maxwell addition to Ampère's law). This mutual creation is the basis of all electromagnetic waves — radio, light, X-rays — and leads directly to IB Topic D.4 on induction.

D.2 — Static Fields	D.4 — Electromagnetic Induction
E and B fields are constant in time	Changing B induces an EMF (Faraday's law)
Force on a charge: F = qE + qvB sin θ	Induced EMF: ε = −ΔΦ / Δt
Sources: static charges and steady currents	Sources: time-varying fields create each other
Applications: velocity selectors, cathode ray tubes	Applications: generators, transformers, wireless charging

How D.2 static fields connect to D.4 electromagnetic induction

At the university level, all of this unifies into Maxwell's four equations, which elegantly show that electricity and magnetism are two faces of the same fundamental interaction. If you pursue physics beyond IB, you'll also encounter the idea that what one observer sees as a purely electric field, another observer moving relative to the first may see as a mix of electric and magnetic fields — a startling consequence of special relativity.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A proton is placed at rest in a uniform magnetic field. Describe the force the proton experiences and explain your reasoning.

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The proton experiences zero magnetic force. The magnetic force on a charge is given by F = qvB sin θ, and since the proton is at rest (v = 0), the force is zero regardless of the field strength. A magnetic field can only exert a force on a moving charge. If there were also an electric field present, the proton would feel an electric force F = qE even at rest, but the question specifies only a magnetic field.

Question 2: BASIC CALCULATION

Calculate the electric field strength at a distance of 0.30 m from a point charge of +5.0 × 10⁻⁶ C. Use k = 8.99 × 10⁹ N m² C⁻².

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Using E = kQ / r²: E = (8.99 × 10⁹)(5.0 × 10⁻⁶) / (0.30)² = (4.495 × 10⁴) / (0.09) = 5.0 × 10⁵ N C⁻¹. The field points radially outward from the positive charge. At 0.30 m, this is a strong field — about five times what you'd encounter near high-voltage power lines. The direction is important: since Q is positive, a positive test charge placed at this point would be pushed directly away from Q.

Question 3: INTERMEDIATE

An electron (charge −1.6 × 10⁻¹⁹ C, mass 9.11 × 10⁻³¹ kg) enters a uniform magnetic field of 0.050 T at right angles with a speed of 2.0 × 10⁶ m s⁻¹. (a) Calculate the magnitude of the magnetic force on the electron. (b) Calculate the radius of its circular path.

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(a) F = qvB sin 90° = (1.6 × 10⁻¹⁹)(2.0 × 10⁶)(0.050)(1) = 1.6 × 10⁻¹⁴ N. (b) The magnetic force provides the centripetal force: qvB = mv²/r, so r = mv / (qB) = (9.11 × 10⁻³¹ × 2.0 × 10⁶) / (1.6 × 10⁻¹⁹ × 0.050) = 1.822 × 10⁻²⁴ / 8.0 × 10⁻²¹ = 2.3 × 10⁻⁴ m ≈ 0.23 mm. The electron follows a tiny circle because its mass is so small compared to the force acting on it. This principle is used in mass spectrometers to separate ions by mass.

Question 4: APPLIED

A velocity selector uses an electric field of 3.0 × 10⁴ V m⁻¹ and a magnetic field of 0.15 T, arranged perpendicular to each other and to the beam of incoming ions. (a) What speed are the ions that pass through undeflected? (b) If the magnetic field strength is doubled but the electric field stays the same, what happens to the selected speed?

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(a) For zero net force: qE = qvB → v = E / B = (3.0 × 10⁴) / (0.15) = 2.0 × 10⁵ m s⁻¹. (b) If B is doubled to 0.30 T, then v = E / B = (3.0 × 10⁴) / (0.30) = 1.0 × 10⁵ m s⁻¹. The selected speed is halved. This makes sense physically: a stronger magnetic field exerts more force at the same speed, so only slower ions (which experience less magnetic force) will balance the unchanged electric force. This demonstrates how engineers can tune a velocity selector by adjusting either field.

Question 5: CRITICAL THINKING

A charged particle moves through a region where both E and B fields are present. The particle travels in a straight line at constant speed. A student claims: "The electric and magnetic fields must be perpendicular to each other." Evaluate this claim. Under what conditions is it correct, and can you describe a situation where it is not necessary?

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The student's claim is correct for the standard velocity-selector configuration, where E and B are perpendicular to each other and to the particle's velocity, allowing their forces to cancel. However, the claim is not universally true. Consider a particle moving parallel to both E and B. The magnetic force would be zero (sin 0° = 0), and the electric force would accelerate the particle along the field — so the particle could not move at constant speed unless E = 0. Alternatively, if E and B are parallel and the particle moves perpendicular to both, the electric force would accelerate the particle, changing its speed, so straight-line constant-speed motion is impossible. Therefore, the perpendicular arrangement with E ⊥ B ⊥ v is indeed the specific configuration that allows undeflected constant-speed motion, and the student's claim holds in this standard context but misses the nuance that the velocity must also be perpendicular to both fields.

SUMMARY

Lesson Summary

Summary

Electric and magnetic fields are vector quantities that exist at every point in space around their sources. An electric field E is created by charges (stationary or moving) and exerts a force F = qE on any charge placed in it. The field of a point charge follows Coulomb's inverse-square law, E = kQ / r². Field lines begin on positive charges and end on negative charges, and their density indicates field strength.

A magnetic field B is produced only by moving charges or currents and exerts a force F = qvB sin θ that is always perpendicular to the velocity, meaning it changes direction but does no work. Magnetic field lines form closed loops. The right-hand rule determines force and field directions. When E and B are arranged perpendicular to each other and to a beam of charged particles, a velocity selector passes only particles with speed v = E / B, an elegant application that connects both fields in a single device.