05Magnetism and Electromagnetic Induction
Analyze magnetic fields and forces, apply the right-hand rule, and use Faraday's and Lenz's laws to analyze electromagnetic induction and generators.
Magnetism and induction are the principles behind electric motors, generators, and transformers — the machines that power modern civilization.
How do moving charges and changing magnetic fields generate the forces and voltages that power electric motors, generators, and transformers?
Magnetism and electricity are two faces of the same phenomenon — electromagnetism. A magnetic field B (measured in Tesla, T) exerts a force on any moving charge or current-carrying conductor. Conversely, a changing magnetic flux through a loop induces an electromotive force (EMF) that drives current — the principle behind every generator and transformer on the grid. In this lesson you will master the quantitative relationships that connect fields, forces, flux, and induced voltages, and learn to apply the right-hand rule to determine directions without ambiguity.
Right-Hand Rule Summary
Rule 1 — Force on a Moving Charge
Point fingers in the direction of velocity v. Curl them toward B. Your thumb points in the direction of the magnetic force F on a positive charge (reverse for negative).
Rule 2 — Force on a Current-Carrying Wire
Point fingers in the direction of conventional current I. Curl them toward B. Your thumb points in the direction of the force F on the wire.
Rule 3 — Induced Current Direction (Lenz)
Curl the fingers of your right hand in the direction of the induced current; your thumb points in the direction of the induced magnetic field, which opposes the change in flux.
Key Equations
F = qvB sinθMagnetic force on a moving chargeF = BIL sinθMagnetic force on a current-carrying wireτ = NIAB sinθTorque on a current loop (N turns)B = μ₀I / 2πrMagnetic field of a long straight wire (Ampere's law)Φ = BA cosθMagnetic flux through a surfaceEMF = −ΔΦ / ΔtFaraday's law of inductionEMF = BLvMotional EMF (conductor of length L moving at v)V₁/V₂ = N₁/N₂Transformer voltage ratioI₁N₁ = I₂N₂Transformer current ratio (ideal)Worked Examples
A proton (q = 1.6 × 10⁻¹⁹ C) moves at v = 3 × 10⁶ m/s perpendicular to a uniform magnetic field B = 0.5 T. What is the magnitude of the magnetic force on the proton?
Identify knowns: q = 1.6 × 10⁻¹⁹ C, v = 3 × 10⁶ m/s, B = 0.5 T, θ = 90° so sinθ = 1.
Write the formula: F = qvB sinθ.
Substitute: F = (1.6 × 10⁻¹⁹)(3 × 10⁶)(0.5)(1).
Multiply step by step: 1.6 × 3 = 4.8; 4.8 × 10⁻¹⁹ × 10⁶ = 4.8 × 10⁻¹³; 4.8 × 10⁻¹³ × 0.5 = 2.4 × 10⁻¹³ N.
A straight wire of length L = 0.3 m carries a current I = 5 A at right angles (θ = 90°) to a magnetic field B = 0.8 T. Find the magnetic force on the wire.
Identify knowns: B = 0.8 T, I = 5 A, L = 0.3 m, θ = 90° so sinθ = 1.
Write the formula: F = BIL sinθ.
Substitute: F = (0.8)(5)(0.3)(1).
Calculate: 0.8 × 5 = 4.0; 4.0 × 0.3 = 1.2 N.
A long straight wire carries a current I = 10 A. Using Ampere's law, find the magnetic field B at a perpendicular distance r = 0.05 m from the wire. (μ₀ = 4π × 10⁻⁷ T·m/A)
Write Ampere's law for a long wire: B = μ₀I / (2πr).
Substitute: B = (4π × 10⁻⁷ × 10) / (2π × 0.05).
Simplify numerator: 4π × 10⁻⁶.
Simplify denominator: 2π × 0.05 = 0.1π.
Divide: B = (4π × 10⁻⁶) / (0.1π) = 4 × 10⁻⁵ T = 40 μT.
A coil of N = 200 turns experiences a change in magnetic flux from Φ₁ = 0.05 Wb to Φ₂ = 0.02 Wb in Δt = 0.1 s. Find the magnitude of the induced EMF.
Calculate the change in flux: ΔΦ = Φ₂ − Φ₁ = 0.02 − 0.05 = −0.03 Wb.
Apply Faraday's law for N turns: EMF = −N × (ΔΦ / Δt).
Substitute: EMF = −200 × (−0.03 / 0.1).
Simplify: −0.03 / 0.1 = −0.3; EMF = −200 × (−0.3) = +60 V.
The magnitude is 60 V; the positive sign confirms the induced EMF opposes the decrease in flux (Lenz's law).
An ideal transformer has a primary coil of N₁ = 500 turns connected to V₁ = 120 V. The secondary coil has N₂ = 2000 turns. (a) Find the secondary voltage V₂. (b) If the secondary current I₂ = 2 A, find the primary current I₁.
Part (a) — Voltage ratio: V₁/V₂ = N₁/N₂ → V₂ = V₁ × (N₂/N₁).
Substitute: V₂ = 120 × (2000/500) = 120 × 4 = 480 V.
Part (b) — Current ratio (ideal transformer, power conserved): I₁N₁ = I₂N₂.
Solve for I₁: I₁ = I₂ × (N₂/N₁) = 2 × (2000/500) = 2 × 4 = 8 A.
Guided Practice
An electron (q = 1.6 × 10⁻¹⁹ C) moves at v = 2 × 10⁷ m/s at an angle of 30° to a magnetic field B = 0.4 T. Find the magnitude of the magnetic force on the electron.
Hint: Use F = qvB sinθ with θ = 30°. Remember sin 30° = 0.5. The charge magnitude is the same as a proton's.
A horizontal wire of length L = 0.5 m carries I = 8 A directed east in a magnetic field B = 0.6 T directed vertically upward. Find the force on the wire and its direction.
Hint: Use F = BIL sinθ. The angle between east (current) and up (B) is 90°. Apply the right-hand rule: fingers east, curl up → thumb points north.
A rectangular coil (N = 50 turns, area A = 0.04 m²) is placed in a B = 0.3 T field. The flux changes from maximum to zero in Δt = 0.02 s. Calculate the induced EMF.
Hint: Maximum flux Φ_max = BA. Then ΔΦ = 0 − BA = −BA. Apply EMF = −N(ΔΦ/Δt) and take the magnitude.
A conducting rod of length L = 0.25 m moves at v = 4 m/s perpendicular to a magnetic field B = 1.2 T. Calculate the motional EMF.
Hint: Use the motional EMF formula: EMF = BLv. All three quantities are perpendicular to each other, so no sinθ correction is needed here.
An ideal step-down transformer converts 240 V to 12 V. If the primary has N₁ = 1200 turns, how many turns does the secondary have? If the primary current is I₁ = 0.5 A, what is the secondary current?
Hint: Use V₁/V₂ = N₁/N₂ to find N₂. Then use I₁N₁ = I₂N₂ to find I₂. A step-down transformer reduces voltage but increases current.
Key Vocabulary
Magnetic Field (B)
A vector field (measured in Tesla, T) that exerts forces on moving charges and current-carrying conductors. Field lines run from north to south poles outside a magnet.
Example: Earth's magnetic field is approximately 25–65 μT at the surface.
Magnetic Force
The force exerted on a moving charge or current-carrying conductor by a magnetic field. It is always perpendicular to both the velocity (or current) and the field: F = qvB sinθ.
Example: A proton moving horizontally through a vertical magnetic field is deflected sideways.
Right-Hand Rule
A mnemonic for determining the direction of the magnetic force, magnetic field, or induced current. Point fingers in the direction of v (or I), curl toward B; the thumb indicates the force direction.
Example: Current flowing north in a field pointing east produces a force directed upward.
Magnetic Flux (Φ)
The total magnetic field passing through a surface area A at angle θ to the field: Φ = BA cosθ. Measured in Webers (Wb = T·m²).
Example: A 0.5 T field through a 2 m² loop at 0° gives Φ = 1 Wb.
Faraday's Law of Induction
The induced EMF in a loop equals the negative rate of change of magnetic flux through it: EMF = −ΔΦ/Δt. For N turns: EMF = −N(ΔΦ/Δt).
Example: Spinning a coil in a magnetic field continuously changes flux, producing AC voltage.
Lenz's Law
The induced current flows in a direction such that its own magnetic field opposes the change in flux that caused it. This is the physical meaning of the minus sign in Faraday's law.
Example: Pushing a north pole into a coil induces a current that creates a north pole facing the approaching magnet, repelling it.
Electromagnetic Induction
The process by which a changing magnetic flux through a conductor induces an EMF (and thus a current if the circuit is closed). The foundation of generators and transformers.
Example: Moving a bar magnet in and out of a solenoid lights a connected LED.
Transformer
A device that uses mutual induction between two coils to step AC voltage up or down. Voltage ratio equals turns ratio: V₁/V₂ = N₁/N₂. An ideal transformer conserves power: P₁ = P₂.
Example: Power-line transformers step 345 kV down to 120 V for household use.
Check Your Understanding
Interactive Practice — 5 Questions
A positive charge moves parallel to a magnetic field. The magnetic force on the charge is:
Which of the following correctly states Lenz's law?
A transformer has N₁ = 100 turns and N₂ = 400 turns. If V₁ = 50 V, what is V₂?
The SI unit of magnetic flux is the:
A conducting rod of length 0.4 m moves at 5 m/s perpendicular to B = 2 T. The motional EMF is:
Independent Practice
A proton moves at 3.0 × 10⁶ m/s perpendicular to a magnetic field of 0.50 T. Calculate the magnetic force on the proton. (q = 1.6 × 10⁻¹⁹ C)
A straight wire carries a current of 5.0 A in a magnetic field of 0.80 T. The wire is 0.30 m long and perpendicular to the field. Find the force on the wire.
A rectangular coil (0.10 m × 0.15 m, 200 turns) rotates in a 0.40 T field. The flux changes from maximum to zero in 0.025 s. Calculate the average induced EMF.
A transformer has 500 primary turns and 2,000 secondary turns. If the primary voltage is 120 V, find (a) the secondary voltage and (b) the secondary current if the primary current is 8.0 A (assume 100% efficiency).
★ A conducting rod of length 0.50 m slides along rails separated by 0.50 m in a 0.30 T magnetic field (perpendicular to the circuit plane) at a speed of 4.0 m/s. The circuit has total resistance 2.0 Ω. (a) Calculate the induced EMF. (b) Find the induced current. (c) Find the force needed to maintain constant speed. (d) Calculate the power input and verify it equals the electrical power dissipated.
ChallengeCommon Mistakes
Forgetting that the magnetic force on a charge is zero when v is parallel to B
F = qvB sinθ. When v ∥ B, θ = 0° and F = 0. Maximum force occurs when v ⊥ B (θ = 90°)
Confusing the direction of the induced current with the direction of the changing flux
Lenz's law: the induced current creates a magnetic field that OPPOSES the change in flux — not the flux itself. If flux is increasing, induced B opposes the increase
Using the transformer equation V₁/V₂ = N₁/N₂ without checking energy conservation (I₁N₁ = I₂N₂)
An ideal transformer conserves power: P₁ = P₂ → V₁I₁ = V₂I₂. Step-up voltage means step-down current by the same ratio
Applying the right-hand rule for force on a wire without distinguishing current direction from charge velocity
For conventional current (positive charge flow), use the right-hand rule: fingers point in current direction, curl toward B, thumb points in force direction. For electrons, reverse the result
Math Tips
Right-hand rule for force: point fingers in the direction of v (or I), curl toward B — thumb points in the direction of F on a positive charge. For negative charges (electrons), the force is opposite
Faraday's law: |EMF| = N|ΔΦ/Δt| = N|ΔBAcosθ/Δt|. For a coil rotating in a uniform field: EMF_max = NBAω (peak EMF of an AC generator)
Magnetic field of a long straight wire: B = μ₀I/(2πr). Field circles around the wire (right-hand rule: thumb in current direction, fingers curl in B direction)
Transformer efficiency: ideal P_primary = P_secondary. Real transformers are 95–99% efficient. High-voltage transmission reduces I²R losses in power lines — that's why transformers are essential