content(warnings): Add W17-W28, X8-X12, N7 — E&M misconceptions, cross-refs, notes

concepts/em/u12/e12-1-magnetic-force-charge.tex

@@ -24,6 +24,8 @@ where $\theta$ is the angle between the vectors $\vec{v}$ and $\vec{B}$ measured
 \item \textbf{SI unit of $B$:} The tesla, $\mathrm{T} = \dfrac{\mathrm{N}}{\mathrm{C}\cdot\mathrm{m/s}} = \dfrac{\mathrm{N}}{\mathrm{A}\cdot\mathrm{m}} = \dfrac{\mathrm{kg}}{\mathrm{C}\cdot\mathrm{s}}$.
 \end{itemize}}
 
+The magnetic force on a single charge (this section) generalizes to the magnetic force on a current-carrying wire (Section 12.3) by summing over all charge carriers. The Biot-Savart law (Section 12.4) gives the reverse: how currents produce the $\vec{B}$ field that exerts these forces.
+
 \pf{Lorentz magnetic force law from cross-product geometry}{The vector cross product $\vec{v}\times\vec{B}$ is defined to have magnitude $vB\sin\theta$ and direction given by the right-hand rule. Multiplying by $q$ scales the magnitude by $|q|$ and reverses direction if $q<0$. Thus
 \[
 \vec{F}_B = q\,(\vec{v}\times\vec{B})
@@ -32,6 +34,8 @@ has magnitude $|q|vB\sin\theta$ and the correct directional behaviour. This is t
 
 \cor{Charge at rest or parallel to field}{When $\vec{v}=\vec{0}$ or when $\vec{v}\parallel\vec{B}$, we have $\sin\theta=0$ and therefore $F_B=0$. The magnetic field exerts no force on a stationary charge or on a charge moving exactly along the field lines.}
 
+\wc{A stationary charge produces zero magnetic field}{A point charge at rest produces only an \emph{electric} field. A magnetic field is produced only by \emph{moving} charges (currents) or changing electric fields (displacement current). A single stationary charge $q$ has $\vec{B}=\vec{0}$ everywhere.}
+
 \mprop{Magnetic force vs.\ electric force}{For the same charge $q$ placed in both an electric field $\vec{E}$ and a magnetic field $\vec{B}$, the total Lorentz force is
 \[
 \vec{F} = q\,\vec{E} + q\,\vec{v}\times\vec{B}.
@@ -49,6 +53,8 @@ P = \vec{F}_B\cdot\vec{v} = q\,(\vec{v}\times\vec{B})\cdot\vec{v} = 0.
 \]
 The magnetic force can change the direction of a particle's velocity but never its kinetic energy. This is the mathematical expression of the scalar triple-product identity $(\vec{a}\times\vec{b})\cdot\vec{a}=0$.}
 
+\wc{Magnetic force does zero work --- always}{Because $\vec{F}_B=q\vec{v}\times\vec{B}$ is always perpendicular to $\vec{v}$, the power $P=\vec{F}_B\cdot\vec{v}=0$. The magnetic force can change a particle's direction but never its speed or kinetic energy. Even in complex field configurations, the magnetic force contributes zero to the work integral $\int\vec{F}\cdot d\vec{r}$.}
+
 \ex{Illustrative example}{When a charged particle enters a uniform magnetic field perpendicularly, it follows a circular path. The magnetic force provides the centripetal force:
 \[
 |q|\,v\,B = \frac{m\,v^2}{R} \quad\Rightarrow\quad R = \frac{m\,v}{|q|\,B},

concepts/em/u12/e12-2-particle-motion-in-b.tex

@@ -72,6 +72,8 @@ The sense of the circular rotation follows the same rule as cyclotron motion: co
 
 \nt{If $\alpha=0^\circ$, then $v_{\perp}=0$ and the particle travels in a straight line along $\vec{B}$ (no magnetic force). If $\alpha=90^\circ$, then $v_{\parallel}=0$ and the particle undergoes pure circular motion (no drift along $\vec{B}$). Helical motion interpolates between these two extremes. The pitch increases as $\alpha\to 0^\circ$ and approaches zero as $\alpha\to 90^\circ$.}
 
+\wc{Field lines are a visualization tool, not physical threads}{Magnetic field lines are a mathematical construct to visualize $\vec{B}$. The field is continuous and exists at every point --- field lines are just a convenient drawing. Charges do not follow field lines (except when $\vec{v}\parallel\vec{B}$). The density of lines indicates field strength, but the lines themselves have no physical substance.}
+
 \mprop{Cyclotron and helical motion parameters}{For a particle of mass $m$ and charge $q$ in a uniform magnetic field $\vec{B}$, with velocity $\vec{v}$ at angle $\alpha$ to $\vec{B}$:
 \begin{align}
 R&=\frac{m\,v\,\sin\alpha}{|q|\,B} && \text{(helix radius)} \\

concepts/em/u12/e12-5-ampere.tex

@@ -12,6 +12,8 @@ This subsection states Amp\`ere's law and shows how symmetry can reduce a diffic
 \]
 This law is always true. It becomes a practical method for solving for the magnetic field when the current distribution has enough symmetry that one can choose an Amperian loop for which the magnitude $B=|\vec{B}|$ is constant on each field-contributing part of the loop and the angle between $\vec{B}$ and $d\vec{\ell}$ is everywhere $0^\circ$, $180^\circ$, or $90^\circ$. Then the line integral reduces to algebraic terms such as $B\ell$, $-B\ell$, or $0$. Common useful cases are cylindrical symmetry (long straight wires), planar symmetry (infinite current sheets), and solenoidal symmetry (ideal solenoids). The direction of $\vec{B}$ follows the right-hand rule relative to the enclosed current: if the thumb of your right hand points in the direction of the current, your fingers curl in the direction of the magnetic field circulation.}
 
+\wc{Ampere's law is always true for steady currents}{Like Gauss's law, $\oint\vec{B}\cdot d\vec{\ell}=\mu_0 I_{\mathrm{enc}}$ holds for \emph{any} steady current distribution and \emph{any} closed path. Symmetry only makes it \emph{useful} for calculating $\vec{B}$. Without symmetry, you know the line integral but cannot extract $\vec{B}$ at each point.}
+
 \nt{Amp\`ere's law is the magnetic analogue of Gauss's law. Gauss's law relates the electric field flux through a closed surface to the enclosed charge, $\oint\vec{E}\cdot d\vec{A}=q_{\mathrm{enc}}/\varepsilon_0$. Amp\`ere's law relates the magnetic field circulation around a closed loop to the enclosed current, $\oint\vec{B}\cdot d\vec{\ell}=\mu_0 I_{\mathrm{enc}}$. Both are universally valid but are practically useful for finding fields only when the source distribution has high symmetry. The matching of symmetry to geometry is parallel: spherical symmetry $\to$ spherical Gaussian surface, cylindrical symmetry $\to$ circular Amperian loop, planar symmetry $\to$ rectangular Amperian loop.}
 
 \pf{How symmetry reduces the line integral}{Let a long straight wire carry current $I$ along the $+z$ axis. By cylindrical symmetry, the magnetic field circulates around the wire in concentric circles in planes perpendicular to the wire, and its magnitude $B(r)$ depends only on the radial distance $r$ from the wire axis. Choose a circular Amperian loop of radius $r$ centred on the wire. Along this loop, $\vec{B}$ is everywhere tangent to $d\vec{\ell}$, so $\vec{B}\cdot d\vec{\ell}=B(r)\,d\ell$, and $B(r)$ is constant everywhere on the loop. Therefore,