Magnetism and Electromagnetic Induction Cheat Sheet

The core ideas of Magnetism and Electromagnetic Induction distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Magnetic Force on a Moving Charge

A charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ experiences a force $\vec{F} = q\vec{v} \times \vec{B}$. The magnitude is $F = qvB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$. The force is always perpendicular to both $\vec{v}$ and $\vec{B}$, so it changes direction but not speed.

Magnetic Force on a Current-Carrying Wire

A straight wire of length $L$ carrying current $I$ in a magnetic field $\vec{B}$ experiences a force $\vec{F} = I\vec{L} \times \vec{B}$, with magnitude $F = BIL\sin\theta$. This force is the basis for electric motors, where current loops rotate in magnetic fields to produce mechanical work.

Magnetic Field from a Long Straight Wire

A long straight wire carrying current $I$ produces a magnetic field that circles the wire concentrically: $B = \mu_0 I / (2\pi r)$, where $\mu_0 = 4\pi \times 10^{-7}$ T m/A is the permeability of free space and $r$ is the distance from the wire. The direction follows the right-hand rule.

Solenoid and Its Magnetic Field

A solenoid is a coil of wire wound in a helix. Inside a long solenoid, the field is nearly uniform: $B = \mu_0 n I$, where $n$ is the number of turns per unit length and $I$ is the current. The field outside is approximately zero. Solenoids are used in electromagnets, relays, and MRI machines.

Ampere's Law

The line integral of the magnetic field around any closed loop equals $\mu_0$ times the enclosed current: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$. This is the magnetic analog of Gauss's law and is most useful for calculating fields with high symmetry (infinite wires, solenoids, toroids).

Magnetic Flux

The magnetic flux through a surface is $\Phi_B = \int \vec{B} \cdot d\vec{A}$. For a uniform field through a flat loop: $\Phi_B = BA\cos\theta$, where $\theta$ is the angle between $\vec{B}$ and the surface normal. The SI unit is the weber (Wb = T m$^2$).

Faraday's Law of Induction

A changing magnetic flux through a loop induces an EMF: $\varepsilon = -N \, d\Phi_B/dt$, where $N$ is the number of turns. The EMF drives a current if the loop is part of a complete circuit. This is the operating principle of generators, transformers, and many sensors.

Lenz's Law

The direction of an induced current is always such that it opposes the change in magnetic flux that produced it. This is a consequence of conservation of energy: if the induced current aided the flux change, it would create a perpetual motion machine. The negative sign in Faraday's law encodes Lenz's law.

Electromagnetic Waves

Self-propagating oscillations of perpendicular electric and magnetic fields that travel at the speed of light: $c = 1/\sqrt{\mu_0 \epsilon_0} \approx 3 \times 10^8$ m/s. Predicted by Maxwell's equations, they span the spectrum from radio waves to gamma rays and require no medium for propagation.

Key Terms at a Glance

Magnetic Field:A vector field that exerts a force on moving charges and current-carrying conductors. Symbol: $\vec{B}$. SI unit: tesla (T).

Magnetic Flux:The integral of the magnetic field over a surface: $\Phi_B = \int \vec{B} \cdot d\vec{A}$. For uniform fields: $\Phi_B = BA\cos\theta$. SI unit: weber (Wb).

Faraday's Law:A changing magnetic flux induces an EMF: $\varepsilon = -N d\Phi_B/dt$. The basis of electric generators and transformers.

Lenz's Law:The direction of induced current opposes the change in flux that produced it. Encodes conservation of energy in electromagnetic induction.

Ampere's Law:The line integral of $\vec{B}$ around a closed loop equals $\mu_0$ times the enclosed current: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$.

Solenoid:A helical coil of wire that produces a nearly uniform magnetic field inside: $B = \mu_0 nI$.

Right-Hand Rule:A mnemonic for finding the direction of magnetic forces, fields, and induced currents. Point thumb along current or velocity, curl fingers in field direction.

Permeability of Free Space:The constant $\mu_0 = 4\pi \times 10^{-7}$ T m/A that appears in magnetic field formulas. It quantifies the strength of magnetic interactions in vacuum.

Electromagnetic Induction:The process by which a changing magnetic flux induces an EMF and current in a conductor. Discovered by Faraday and Henry independently.

Transformer:A device that changes AC voltage levels using electromagnetic induction between two coils (primary and secondary) sharing a magnetic core.

Displacement Current:Maxwell's addition to Ampere's law: a changing electric field acts as a source of magnetic field, even without physical current. Completes Maxwell's equations.

Maxwell's Equations:The four fundamental equations governing all classical electromagnetic phenomena: Gauss's laws for E and B, Faraday's law, and the Ampere-Maxwell law.

Electromagnetic Wave:A self-sustaining oscillation of electric and magnetic fields propagating at the speed of light. Includes radio, microwave, infrared, visible, UV, X-ray, and gamma radiation.

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