Comprehensive Study Guide for Magnetism and Electromagnetism

Magnetic Torque on a Current Loop

When a loop of wire carrying a current is placed within a magnetic field, it may experience a magnetic torque that tends to cause the loop to rotate around a specific axis, typically passing through the center of mass of the loop. The plane of the loop is mathematically defined by a unit vector $\hat{n}$ that is oriented perpendicular to the loop's surface. The magnetic force acting on a segment of length $L$ is given by the vector product $F = iL \times B$, or in differential form as $dF = i ds \times B$. The magnitude of this force is calculated as $F = i L B \sin(\theta)$.

The magnetic dipole moment, denoted as $\mu$, is a vector quantity defined by the number of turns in the loop $N$, the current $i$, the area of the loop $A$, and the unit vector perpendicular to the plane $\hat{n}$. The formula is expressed as $\mu = NiA\hat{n}$. The resulting torque $\tau$ experienced by the loop is the cross product of the magnetic dipole moment and the external magnetic field: $\tau = \mu \times B$. The magnitude of this torque is $\tau = \mu B \sin(\theta)$, or specifically $\tau = NiAB \sin(\theta)$. This torque reaches its maximum value when the vectors are perpendicular ($\theta = 90^{\circ}$) and is at its minimum (zero) when the vectors are parallel ($\theta = 0^{\circ}$) or antiparalel ($\theta = 180^{\circ}$).

The magnetic potential energy $U$ of the system is given by the dot product of the magnetic dipole moment and the field, with a negative sign: $U = -\mu \cdot B$, which expands to $U = -\mu B \cos(\theta)$ or $U = -NiAB \cos(\theta)$. The energy is at its maximum when the dipole and field are antiparallel ($\theta = 180^{\circ}$), at its minimum when they are parallel ($\theta = 0^{\circ}$), and is null (zero) when they are perpendicular ($\theta = 90^{\circ}$ or $270^{\circ}$). Stable equilibrium occurs at $\theta = 0^{\circ}$ (aligned with the field), while unstable equilibrium occurs at $\theta = 180^{\circ}$ (opposite to the field).

Fundamental Laws of Electromagnetism

The Biot-Savart Law provides a mathematical model for the differential magnetic field $dB$ generated by a current element $i ds$. It states that the magnetic field is directly proportional to the cross product of the current element and the position vector, and inversely proportional to the cube of the distance to the point of observation: $dB = \frac{\mu_0}{4\pi} \frac{i ds \times \mathbf{r}}{r^3}$. In terms of magnitude and angles, this is often written as $dB = \frac{\mu_0}{4\pi} \frac{i ds \sin(\theta)}{r^2}$. The constant $\mu_0$ is the permeability of free space, valued at $4\pi \times 10^{-7}\,T \cdot m/A$.

Ampere's Law describes the magnetic circulation around a closed path, known as an Amperian loop. It states that the line integral of the magnetic field around any closed path is directly proportional to the net current enclosed by that path: $\oint \mathbf{B} \cdot d\mathbf{s} = \mu_0 i_{\text{enclosed}}$. Current density is defined as $J = \frac{i}{A}$.

Faraday's Law of Induction states that the induced electromotive force (EMF), denoted as $\mathcal{E}$, is directly proportional to the negative rate of change over time of the magnetic flux $\Phi_B$ passing through a loop: $\mathcal{E} = -\frac{d\Phi_B}{dt}$. Magnetic flux is the net quantity of magnetic field lines passing perpendicularly through a surface area, calculated as $\Phi_B = B \cdot A$. The unit of magnetic flux in the International System is the Weber ($Wb$).

Lenz's Law provides the direction of the induced current, stating that the induced current will flow in a direction such that the magnetic field it creates opposes the change in the magnetic flux that produced it. This is the physical reason for the negative sign in Faraday's Law.

Magnetic Field Calculations for Various Geometries

For a long, straight, infinite wire, the magnetic field at a distance $r$ is given by $B = \frac{\mu_0 i}{2\pi r}$. For a semi-infinite wire at a point perpendicular to one end, the field is $B = \frac{\mu_0 i}{4\pi r}$. For a finite segment of length $L$ at a perpendicular distance from its midpoint, the formula involves specific trigonometric or geometric considerations involving the angles to the wire ends.

At the center of a circular ring or coil of radius $R$ with $N$ turns, the field is $B = \frac{N μ_0 i}{2R}$. If calculating the field on the axis of a circular coil at a distance $z$ from the center, the formula is $B = \frac{N μ_0 i R^2}{2(R^2 + z^2)^{3/2}}$.

A solenoid is described as a long wire wound into a tightly packed helix, used to create a uniform magnetic field. Inside the core of a solenoid, the field is $B = \mu_0 ni$, where $n = N/L$ is the number of turns per unit length. A toroid is a solenoid bent into a donut or ring shape (rosica); its internal field is $B = \frac{\mu_0 Ni}{2\pi r}$.

For more complex shapes, such as a regular polygon with $n$ sides at a distance $a$ from the center, the field is $B = \frac{n μ_0 i}{2π a} \tan(\frac{\pi}{n})$. At the center of a rectangular loop of width $w$ and length $L$, the field is expressed as $B = \frac{2 μ_0 i \sqrt{w^2 + L^2}}{\pi w L}$.

Motion of Charged Particles and Lorentz Force

The Lorentz Force ($F_L$) represents the total force experienced by a charged particle $q$ moving through both electric ($E$) and magnetic ($B$) fields: $F_L = q(E + v \times B)$. When a particle moves perpendicular through a uniform, constant magnetic field, it follows a circular path. The radius of this circle is calculated as $R = \frac{mv}{qB}$. The period of the motion is the time required to complete one full cycle, given by $T = \frac{2π m}{qB}$. The angular frequency or velocity is $w = \frac{qB}{m}$.

In a mass spectrometer, a substance is ionized, and the ions are accelerated by an electric field before entering a magnetic field. This instrument measures the molecular mass of chemical compounds by separating ions based on their specific mass-to-charge ($m/q$) ratios. The magnetic force remains perpendicular to the motion, meaning it does no work and therefore does not change the speed (kinetic energy) of the charged particle.

Instruments and Practical Applications

A galvanometer is an instrument used to detect and measure very small electric currents through the movement of a needle in a magnetic field. It can be converted into an ammeter by connecting a very low resistance (shunt) in parallel, allowing it to measure higher currents. Conversely, it becomes a voltmeter by connecting a very high resistance (multiplier) in series with the coil to limit current and measure voltage across a component without damage.

Transformers utilize electromagnetic induction to transfer energy between two coils (primary and secondary) to increase or decrease alternating current (AC) voltage. In a step-down (bajada) transformer, the primary current is lower than the secondary current (I_p < I_s) as the voltage is reduced. The relationship between the number of turns ($N$) and voltage ($V$) is $\frac{V_p}{V_s} = \frac{N_p}{N_s}$.

The Hall Effect is a phenomenon occurring when a conductor or semiconductor carrying a current is placed in a magnetic field perpendicular to the current. A transversal potential difference (Hall voltage) is created because the Lorentz force pushes charge carriers toward one side of the material. This allows for the determination of the type and density of charge carriers.

Geophysics and Material Properties

Geomagnetism is the study of the Earth's magnetic field. The Magnetic Declination is the horizontal angle between the true Geographic North and the Magnetic North indicated by a compass. Magnetic Inclination is the vertical angle of the total magnetic field vector relative to the horizontal plane; it varies from $0^{\circ}$ at the magnetic equator to $90^{\circ}$ at the magnetic poles.

Materials are categorized by their magnetic response: Ferromagnetic materials (e.g., iron) are strongly attracted to magnetic fields and can retain magnetism. Paramagnetic materials are weakly attracted and do not retain magnetism. Diamagnetic materials are weakly repelled by magnetic fields. The Curie Temperature is the specific temperature threshold above which a ferromagnetic material loses its permanent magnetic properties and becomes paramagnetic.

The Aurora Borealis is a natural phenomenon formed when charged particles from the sun collide with the Earth's atmosphere, guided by the magnetic field toward the poles. The Oersted Experiment (1820) demonstrated that a current-carrying wire deflects a nearby compass needle, proving that electric currents generate circular magnetic fields around them, serving as the foundation of electromagnetism.

Circuit Definitions and Constants

Basic circuit elements include nodes (where three or more elements meet), branches (elements in series between two nodes), and meshes (any closed path within a circuit). Kirchhoff's First Law (Junction Rule) is based on the conservation of charge, stating that the algebraic sum of currents at a node is zero ($\Sigma I = 0$). Kirchhoff's Second Law (Loop Rule) is based on the conservation of energy, stating that the sum of voltage rises equals the sum of voltage drops in any closed loop.

The Ampere is defined as the current which, if maintained in two straight parallel conductors of infinite length and negligible circular cross-section, placed $1\,m$ apart in a vacuum, would produce a force between them of $2 \times 10^{-7}\,N/m$. A Coulomb is the amount of charge that passes through a cross-section of a circuit in one second when a current of one Ampere flows.

Useful physical constants include:

• Permeability of free space: $\mu_0 = 4π \times 10^{-7}\,T \cdot m/A$

• Elementary charge: $e = 1.6 \times 10^{-19}\,C$

• Mass of an electron: $m_e = 9.11 \times 10^{-31}\,kg$

• Mass of a proton: $m_p = 1.67 \times 10^{-27}\,kg$

• Speed of light: $c = 3 \times 10^8\,m/s$

• Conversion: $1\,Tesla = 10^4\,gauss$