Magnetic Field

Magnetism originates from electric charges in motion. A magnetic field is established by moving charges or permanent magnetic dipoles and exerts forces exclusively on moving charges.

1. The Lorentz Force Law

A particle carrying electric charge $q$ moving with velocity $\vec{v}$ in a region containing both an electric field $\vec{E}$ and a magnetic field $\vec{B}$ experiences the unified Lorentz Force: $$\vec{F} = q \left( \vec{E} + \vec{v} \times \vec{B} \right)$$ The magnetic force component is: $$\vec{F}_B = q (\vec{v} \times \vec{B})$$ Magnitude: $F_B = |q| v B \sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$. Direction: Governed by the vector cross product (Right-Hand Rule). SI Unit: **Tesla (T)** ($1 \text{ Tesla} = 1 \text{ N}/(\text{A}\cdot\text{m}) = 10^4 \text{ Gauss}$).

2. Fundamental Properties of the Magnetic Force

• Zero Work Property: Because $\vec{F}_B$ is everywhere perpendicular to velocity $\vec{v}$ ($\vec{F}_B \cdot \vec{v} = q (\vec{v} \times \vec{B}) \cdot \vec{v} \equiv 0$), the instantaneous power delivered by a magnetic field is identically zero: $$P = \vec{F}_B \cdot \vec{v} = 0 \implies dK = 0$$ A static magnetic field can NEVER change the kinetic energy or speed of a charged particle; it can only alter its direction of motion.

3. Cyclotron Motion and Helical Trajectories

Consider a particle of mass $m$ and charge $q$ injected into a uniform magnetic field $\vec{B} = B \hat{k}$ with initial velocity $\vec{v}_\perp = v_x \hat{i} + v_y \hat{j}$. The magnetic force acts as a pure centripetal force: $$q v_\perp B = \frac{m v_\perp^2}{r} \implies r_c = \frac{m v_\perp}{q B}$$ $r_c$ is the cyclotron (Larmor) radius. The period of circular revolution and the cyclotron angular frequency $\omega_c$ are: $$T = \frac{2\pi r_c}{v_\perp} = \frac{2\pi m}{q B}, \quad \omega_c = \frac{q B}{m}$$ Notice that $\omega_c$ and $T$ are **completely independent of the particle's speed or orbital radius** (isochronism of the cyclotron). If the particle possesses a parallel velocity component $v_\parallel = v_z \hat{k}$, it executes a **helical path** with pitch $p = v_\parallel T = \frac{2\pi m v_\parallel}{q B}$.

Because electric current consists of an ensemble of moving charges, magnetic forces manifest macroscopically on current-carrying wires.

1. Magnetic Force on a Wire Element

Consider a differential segment of wire of cross-section $A$ and length $d\vec{l}$ carrying current $I = n q A v_d$. The number of charge carriers in the segment is $dN = n A dl$. The total magnetic force on the element is: $$d\vec{F} = dN \cdot q (\vec{v}_d \times \vec{B}) = (n A dl) q (\vec{v}_d \times \vec{B}) = (n q A v_d) (d\vec{l} \times \vec{B})$$ $$d\vec{F} = I (d\vec{l} \times \vec{B})$$ For a straight wire of finite length $\vec{L}$ in a uniform magnetic field: $$\vec{F} = I (\vec{L} \times \vec{B})$$

2. Torque on a Planar Current Loop and Magnetic Dipole Moment

Consider a closed rectangular loop of dimensions $a \times b$ (area $A = ab$) carrying current $I$ in a uniform magnetic field $\vec{B}$. The net translational force vanishes ($\vec{F}_{\text{net}} = 0$). However, forces on opposite arms form a couple, generating net torque: $$\vec{\tau} = \vec{\mu} \times \vec{B}$$ where $\vec{\mu}$ is the magnetic dipole moment vector: $$\vec{\mu} = N I \vec{A} = N I A \hat{n}$$ for a coil of $N$ turns, where $\hat{n}$ is the unit normal given by the right-hand grip rule. SI Unit: $\text{A}\cdot\text{m}^2 = \text{J/T}$. The potential energy of the magnetic dipole in field $\vec{B}$ is: $$U = -\vec{\mu} \cdot \vec{B} = -\mu B \cos\theta$$

3. The Moving-Coil Galvanometer

In a d'Arsonval galvanometer, a rectangular coil of $N$ turns is suspended in a cylindrical soft iron core that produces a radial magnetic field ($\vec{B} \parallel$ plane of coil always, $\sin\theta = 1$). Deflecting magnetic torque: $\tau_d = N I A B$. Restoring torsional torque of phosphor-bronze suspension fiber: $\tau_r = C \theta$. In equilibrium ($\tau_d = \tau_r$): $$\theta = \left( \frac{N A B}{C} \right) I$$ The angular deflection is strictly linear with current.

• Current Sensitivity ($S_I$): $S_I = \frac{\theta}{I} = \frac{N A B}{C}$ (rad/A or div/$\mu$A).
• Voltage Sensitivity ($S_V$): $S_V = \frac{\theta}{V} = \frac{N A B}{C R_g}$ (rad/V).

Edwin Herbert Hall (1879) discovered that when a magnetic field is applied perpendicular to a current-carrying conducting strip, a transverse potential difference develops across the strip.

1. Physical Mechanism of the Hall Effect

Consider a flat conducting slab of width $w$ and thickness $t$ carrying current $I$ along $+x$. A uniform magnetic field $\vec{B} = B \hat{k}$ is applied along $+z$.

1. Charge carriers moving with drift velocity $\vec{v}_d$ experience transverse Lorentz magnetic force: $$\vec{F}_B = q (\vec{v}_d \times \vec{B})$$
2. If charge carriers are **negative electrons** ($q = -e, \vec{v}_d = -v_d \hat{i}$): $$\vec{F}_B = (-e) [(-v_d \hat{i}) \times (B \hat{k})] = -e v_d B \hat{j}$$ Electrons are deflected toward the bottom edge, charging it negative and leaving the top edge positive.
3. If charge carriers are **positive holes** ($q = +e, \vec{v}_d = +v_d \hat{i}$): $$\vec{F}_B = (+e) [(+v_d \hat{i}) \times (B \hat{k})] = -e v_d B \hat{j}$$ Positive charges also deflect downward, charging the bottom edge positive!

Sign of Carriers: The polarity of the transverse **Hall Voltage ($V_H$)** immediately reveals the sign of the charge carriers (confirming that metals conduct via negative electrons, while p-type semiconductors conduct via positive holes).

2. Derivation of the Hall Voltage and Hall Coefficient

Accumulating transverse charge creates a transverse Hall electric field $\vec{E}_H$ pointing toward the negative edge. At steady state, the electrostatic force balances the magnetic force: $$q E_H = q v_d B \implies E_H = v_d B$$ The measured transverse potential difference is: $$V_H = E_H w = v_d B w$$ Using $I = n q A v_d = n q (w t) v_d \implies v_d = \frac{I}{n q w t}$: $$V_H = \left( \frac{I}{n q w t} \right) B w = \frac{I B}{n q t}$$ We define the Hall Coefficient ($R_H$): $$R_H = \frac{E_H}{J B} = \frac{1}{n q}$$ $$V_H = R_H \frac{I B}{t}$$ Applications: Hall effect sensors measure magnetic fields non-invasively (Gaussmeters), sense motor rotor position in brushless DC motors, and quantify carrier density $n$ in semiconductor wafer fabrication.

Jean-Baptiste Biot and Félix Savart (1820) established the differential law governing the magnetic field generated by an infinitesimal current element.

1. The Biot-Savart Law

The magnetic induction $d\vec{B}$ at field point $P$ due to a differential current element $I d\vec{l}$ at source position $\vec{r}'$ is: $$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times (\vec{r} - \vec{r}')}{|\vec{r} - \vec{r}'|^3}$$ where $\mu_0$ is the permeability of free space: $$\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A (exact by historical definition)} \approx 1.2566 \times 10^{-6} \text{ H/m}$$ For any closed circuit loop $C$: $$\vec{B}(\vec{r}) = \frac{\mu_0 I}{4\pi} \oint_C \frac{d\vec{l}' \times (\vec{r} - \vec{r}')}{|\vec{r} - \vec{r}'|^3}$$

2. Magnetic Field of a Long Straight Conductor

Integrating along an infinite straight wire carrying current $I$: At perpendicular distance $R$: $$B = \frac{\mu_0 I}{4\pi} \int_{-\infty}^\infty \frac{dx \sin\theta}{r^2} = \frac{\mu_0 I}{2\pi R}$$ Field lines form concentric circles centered on the wire.

3. Magnetic Field on the Axis of a Circular Current Loop

Consider a circular wire loop of radius $R$ carrying current $I$ lying in the yz-plane. At axial distance $x$ along the symmetry axis: By symmetry, components perpendicular to the axis cancel. The axial component is: $$B_x = \int dB \sin\alpha = \frac{\mu_0 I}{4\pi (x^2 + R^2)} (2\pi R) \left( \frac{R}{\sqrt{x^2 + R^2}} \right)$$ $$B(x) = \frac{\mu_0 I R^2}{2(x^2 + R^2)^{3/2}}$$

• At the center of the loop ($x = 0$): $$B(0) = \frac{\mu_0 I}{2R}$$
• Far from the loop ($x \gg R$): using dipole moment $\mu = I A = I (\pi R^2)$: $$B(x) \approx \frac{\mu_0 \mu}{2\pi x^3}$$ Decays as $1/x^3$, identical to an electric dipole.

4. Helmholtz Coils

A pair of identical coaxial coils of radius $R$, separated by distance equal to their radius ($d = R$), carrying identical current $I$ in the same direction. At the midpoint $x = R/2$: $$\frac{dB}{dx} = 0, \quad \frac{d^2 B}{dx^2} = 0$$ The second derivative vanishes, producing an exceptionally uniform magnetic field over a wide central volume.

André-Marie Ampère (1826) formulated Ampere's circuital law, the magnetic analog of Gauss's law for high-symmetry current distributions.

1. Ampere's Circuital Law

The line integral of magnetic field $\vec{B}$ around any closed Amperian loop $C$ equals $\mu_0$ times the total net electric current enclosed by the loop: $$\oint_C \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}}$$ In differential form (applying Stokes' Theorem): $$\nabla \times \vec{B} = \mu_0 \vec{J}$$

2. The Ideal Long Solenoid

A helical coil of length $L$ and $N$ closely spaced turns carrying current $I$ ($n = N/L$ turns per unit meter). Inside an infinitely long solenoid, the magnetic field is uniform and parallel to the axis; outside, it is zero. Construct a rectangular Amperian loop of length $h$ with one side inside and one outside: $$\oint_C \vec{B} \cdot d\vec{l} = B h + 0 + 0 + 0 = B h$$ Enclosed current: $I_{\text{encl}} = n h I$. $$B h = \mu_0 (n h I) \implies B = \mu_0 n I$$ The field depends exclusively on turn density $n$ and current $I$, independent of solenoid cross-sectional diameter or position.

3. The Toroid (Toroidal Solenoid)

A solenoid bent into a closed donut-shaped ring of inner radius $a$ and outer radius $b$ with $N$ total turns. Construct a circular Amperian loop of radius $r$ inside the core ($a < r < b$): $$\oint \vec{B} \cdot d\vec{l} = B (2\pi r) = \mu_0 (N I) \implies B(r) = \frac{\mu_0 N I}{2\pi r}$$ Outside the toroid ($r < a$ or $r > b$), enclosed current is zero, so $\vec{B} = 0$ everywhere. Toroids have zero external magnetic leakage.