AP Physics C Electricity and Magnetism Quick Review

Charge is quantized and conserved; the electric force between point charges obeys an inverse-square law identical in form to gravity but far stronger and able to attract or repel.

Coulomb's Law: $F = k/|q_1 q_2|r^2$, with $k = 8.99×10^9 N·m^2/C^2 = /14_0$; charge is quantized in units of $e = 1.60×10^-19 C$; total charge in an isolated system is conserved; force is a vector, so multiple-charge problems require superposition.

Coulomb's Law (force between two specific charges, can attract or repel) vs. Newton's Law of Gravitation (always attractive, uses masses) — same inverse-square form, different sign behavior.

Adding force magnitudes directly instead of adding force vectors component-by-component when charges are not collinear.

$F=kq_1q_2/r^2 arrow$ vector superposition, never scalar sum.

The electric field is force per unit positive test charge; it exists at every point in space regardless of whether a test charge is actually there.

$E = / Fq_0 = k/qr^2r$; for multiple sources, $E_net = _i k/q_ir_i^2r_i$ (vector sum); field points away from positive charges, toward negative charges.

Electric field (vector, N/C, defined everywhere) vs. electric potential (scalar, V) — field gives force direction, potential gives energy per charge.

Forgetting that the field of a negative charge points toward the charge, not away from it, when assigning $ r$.

$E=kq/r^2$, direction $arrow$ away from $+$, toward $-$.

For continuous charge, break the object into infinitesimal elements $dq$, find $d E$ from each, and integrate — calculus replaces the discrete sum.

$d E = k/dqr^2 r$, so $ E = k/dqr^2 r$; densities: linear $=dq/dl$, surface $=dq/dA$, volume $=dq/dV$; classic results include a ring on axis, $E=/kQx(x^2+R^2)^3/2$.

$$, $$, and $$ — linear, surface, and volume charge density — using the wrong density in $dq$ collapses the integral.

Integrating the vector magnitude of $d E$ directly instead of resolving it into components first and integrating only the components that survive by symmetry.

$dq →$ identify symmetry $arrow$ integrate surviving components.

A dipole is two equal and opposite charges separated by a small distance; in a uniform field it feels a torque but zero net force, since the forces on the two charges are equal and opposite.

Dipole moment $ p = q d$ (points $-$ to $+$); torque $ = p × E$; potential energy $U=- p · E$; field lines start on $+$ charges, end on $-$ charges, never cross, and their density indicates field strength.

Torque on a dipole ($=pE$, causes rotation) vs. net force on a dipole (zero in a uniform field, nonzero only in a nonuniform field).

Assuming a dipole accelerates translationally in a uniform external field — it only rotates; translation requires a field gradient.

Uniform field $arrow$ torque only, zero net force on dipole.

Electric flux counts field lines passing through a surface; Gauss's Law relates the total flux through any closed surface to the enclosed charge, regardless of the surface's shape.

Flux $_E= E· d A$; Gauss's Law: $ E · d A = /Q_enc_0$; only enclosed charge contributes net flux — charges outside the surface contribute zero net flux.

Zero flux through a closed surface (can happen even when $ E ≠ 0$ at points on it) vs. the field itself being zero.

Trying to use Gauss's Law to solve for $E$ on an asymmetric distribution — it only isolates $E$ algebraically when a Gaussian surface can be chosen where $E$ is constant and parallel or perpendicular to $d A$ everywhere.

$ E dA=Q_enc/_0 arrow$ needs symmetry to solve for $E$.

The three canonical symmetric geometries — spherical, cylindrical, and planar — each pair with a matching Gaussian surface that makes the flux integral trivial to evaluate.

Sphere/point charge, $r>R$: $E=/kQr^2$; uniformly charged solid insulating sphere, $r<R$: $E=/kQrR^3$; infinite line/cylinder: $E=/2_0 r$; infinite sheet: $E=/2_0$; just outside a conductor: $E=/_0$.

Field inside a uniformly charged insulating sphere ($E r$, grows linearly) vs. inside a conducting sphere (always zero — all charge sits on the surface).

Using the total charge $Q$ instead of only the charge enclosed within radius $r$ when the Gaussian surface sits inside a charged sphere.

Sphere $arrow r^2$ falloff outside, $r$ growth inside (insulator); sheet $arrow$ constant $E$.

Electric potential energy is the work required to assemble a charge configuration; it is a property of the system, defined relative to a reference (usually infinity).

$U=k/q_1q_2r$ for a pair of point charges; work by an external agent moving a charge slowly: $W_ext= U$; work by the field: $W_E=- U$; for a system, sum $U$ over every unique pair of charges.

Potential energy $U$ (Joules, a property of a pair or system) vs. potential $V$ (Volts, a property of a point in space); $U=qV$ links them.

Forgetting to sum over every unique pair when computing total potential energy for three or more point charges (three charges have three pairs, not two or six).

$U=kq_1q_2/r arrow$ sum over pairs, $U=0$ at $r=∞$.

Electric potential is potential energy per unit charge; unlike the field, potential is a scalar, so contributions from multiple charges simply add algebraically with sign.

$V=k/qr$ (reference $V=0$ at infinity); for multiple charges, $V=_i k/q_ir_i$; $ U = q V$; work-energy relation $W=q(V_i-V_f)$.

Scalar addition of potential (just add signed numbers) vs. vector addition of field (must add components) — the main computational advantage of working with $V$.

Assuming $V$ is always positive, or forgetting a negative charge contributes negative potential that can make $V=0$ at a point even though $ E ≠ 0$ there.

$V=kq/r$, scalar sum $arrow$ can be zero where $E≠0$.

The electric field is the negative gradient of the potential; it points in the direction of steepest decrease of $V$, and one quantity can be recovered from the other by calculus.

$ E = - V$; in one dimension, $E_x=-/dVdx$; conversely $V(r)-V(ref)=-_ref^r E· d l$; field lines are always perpendicular to equipotentials, pointing from high to low $V$.

$E=-dV/dr$ (differentiate $V$ to get $E$) vs. $V=- E dr$ (integrate $E$ to get $V$) — know which direction the calculus runs.

Dropping the negative sign in $E=-dV/dx$, which flips the field's direction, or assuming $E=0$ only where $V=0$ instead of where $V$ is constant.

$E=-dV/dx arrow$ steepest descent; flat $V arrow E=0$.

Because potential is a scalar, integrating $dV$ over a continuous distribution is often algebraically easier than integrating the vector field directly.

$V= k/dqr$; ring on axis: $V=/kQsqrtx^2+R^2$; once $V(x)$ is known, $E_x=-dV/dx$ recovers the field without ever resolving vector components.

Finding $V$ then differentiating for $E$ (scalar integral + one derivative) vs. integrating $ E$ directly (must resolve components before integrating).

Leaving two different variables (e.g., both $x$ and $$) in the integrand instead of expressing everything in terms of one variable before integrating.

Integrate scalar $V$ first, differentiate for $E$ $arrow$ usually less work.

An equipotential surface connects points at equal potential; no work is done moving a charge along one, and a conductor in electrostatic equilibrium is always entirely equipotential.

On an equipotential, $ V=0$ and $W_by field=-q V=0$; $ E$ is always perpendicular to equipotential surfaces; a conductor in equilibrium has $ E=0$ inside and constant $V$ throughout, including its surface.

Equipotential surfaces ($E$ surface) vs. field lines ($E$ line) — the two families are always mutually perpendicular.

Assuming closely spaced equipotential lines indicate high or low potential — spacing indicates field strength (rate of change of $V$), not the sign or size of $V$ itself.

Equipotentials $$ field lines; conductor surface = one equipotential.

In electrostatic equilibrium, free charges in a conductor redistribute until the internal field is exactly zero everywhere inside the material.

$ E_inside=0$; excess charge resides entirely on the outer surface; just outside the surface, $E=/_0$ perpendicular to the surface; surface charge density is greatest at points of sharpest curvature.

Field inside a conducting shell's empty cavity (zero — Faraday cage shielding) vs. field inside a cavity containing a charge (nonzero inside the cavity, but still zero in the conductor material itself).

Forgetting that a charge inside a cavity induces a uniformly distributed $+q$ on the outer surface regardless of where the charge sits within the cavity.

$E_inside\ conductor=0$ always; excess charge $arrow$ outer surface.

Capacitance measures a device's ability to store charge per unit potential difference; it depends only on geometry (and any dielectric present), never on $Q$ or $V$ individually.

$C=Q/V$ (Farads, C/V); parallel plate: $C=/_0 Ad$, derived from $E=/_0$ and $V=Ed$.

Capacitance $C$ (fixed by geometry) vs. $Q$ or $V$ (can vary in a circuit) — $C$ stays constant while $Q=CV$ links the other two.

Assuming increasing plate separation $d$ increases stored charge — for a capacitor held at constant $V$ (battery connected), larger $d$ means smaller $C$ and thus smaller $Q$.

$C=_0A/d arrow$ geometry only; $Q=CV$ links the rest.

Combining capacitors in series or parallel changes the effective capacitance in the opposite pattern from resistors — series capacitors add like parallel resistors, and vice versa.

Parallel: $C_eq=C_1+C_2+·s$ (same $V$, charges add); Series: $/1C_eq=/1C_1+/1C_2+·s$ (same $Q$, voltages add).

Capacitors in series ($1/C$ adds, like resistors in parallel) vs. resistors in series ($R$ adds directly) — the reciprocal rule flips between the two components.

Applying resistor combination rules to capacitors under time pressure instead of re-deriving from $Q=CV$ and identifying what is shared (charge or voltage).

Capacitors: series $arrow$ reciprocals add; parallel $arrow$ direct add (opposite of resistors).

Inserting a dielectric between capacitor plates polarizes the material, reducing the internal field and increasing capacitance by the dielectric constant.

$C'= C_0$ ($>1$); $E'=E_0/$; if the battery stays connected (constant $V$), $Q$ increases by $$; if the capacitor is isolated (constant $Q$), $V$ decreases by $$.

Constant-$V$ case (battery connected: $Q$ increases when dielectric inserted) vs. constant-$Q$ case (battery removed: $V$ decreases when dielectric inserted) — same dielectric, opposite consequence.

Applying the constant-$V$ result when the capacitor has actually been disconnected from the battery — always check whether the battery remains in the circuit first.

Dielectric in $arrow C$ by $$; battery connected $⇒ Q$; battery removed $⇒ V$.

Charging a capacitor stores energy in its electric field; this energy can be written in terms of $Q$, $V$, or $C$, and also expressed as a field energy density that generalizes beyond capacitors.

$U=/12QV=/12CV^2=/Q^22C$; energy density $u=/12_0E^2$, valid anywhere an electric field exists.

Energy stored, $U=/12QV$ (factor of $/12$, from integrating as charge builds gradually) vs. work to move a single charge, $W=qV$ (no $/12$, since $V$ is already fixed there).

Forgetting the $/12$ factor by treating capacitor charging like moving a fixed charge through a fixed potential difference, rather than integrating $ V dq$ as $V$ itself grows.

$U=/12CV^2 arrow /12$ comes from integrating $V(q)=q/C$ while charging.

Current is the rate of charge flow; resistance opposes that flow and depends on a material's intrinsic resistivity as well as its geometry.

$I=/dQdt$; current density $J=I/A$; $R=/LA$; Ohm's Law: $V=IR$; resistivity rises with temperature for most conductors, $(T)≈_0[1+(T-T_0)]$.

Resistivity $$ (intrinsic material property, $·$m) vs. resistance $R$ (also depends on shape) — doubling a wire's length doubles $R$ but leaves $$ unchanged.

Assuming $V=IR$ applies to every device — Ohm's Law only holds for ohmic materials with constant resistance, not diodes or bulbs with varying resistance.

$R= L/A arrow$ long, thin wire = high resistance.

Series resistors share the same current and split voltage; parallel resistors share the same voltage and split current — the opposite combination pattern from capacitors.

Series: $R_eq=R_1+R_2+·s$; Parallel: $/1R_eq=/1R_1+/1R_2+·s$; a parallel combination's equivalent resistance is always less than the smallest individual resistor.

Resistors in parallel ($1/R$ adds, $R_eq$ decreases) vs. capacitors in parallel ($C$ adds directly, $C_eq$ increases) — memorize which quantity is shared to avoid flipping the rule.

Assuming adding a resistor in parallel increases total resistance — more parallel paths always decrease $R_eq$.

Resistors: series adds $R$; parallel adds $1/R$ (opposite of capacitors).

Kirchhoff's rules are statements of charge conservation (junction rule) and energy conservation (loop rule) that let you solve any circuit, however complex.

Junction rule: $ I_in= I_out$; Loop rule: $ V=0$ around any closed loop; crossing a resistor along the assumed current direction is a drop ($-IR$); crossing a battery from $-$ to $+$ is a gain ($+$).

Junction rule (charge conservation, applies at nodes) vs. loop rule (energy conservation, applies around closed paths) — a multiloop circuit needs both, applied enough times for $N$ independent equations in $N$ unknown currents.

Treating a negative solved current as an arithmetic error — it simply means the actual current flows opposite to the assumed direction.

Junctions $arrow I=0$; loops $arrow V=0$; negative $I$ = wrong assumed direction.

Electric power is the rate at which energy is delivered or dissipated; for resistors, this energy is irreversibly converted to heat.

$P=IV=I^2R=/V^2R$; energy dissipated $E=Pt$; a real battery has terminal voltage $V=-Ir$, where $r$ is internal resistance.

EMF $$ (ideal, constant source voltage) vs. terminal voltage $V$ (actual delivered voltage, reduced by the internal drop $Ir$).

Using EMF instead of terminal voltage to compute power delivered to the external circuit when internal resistance is given — power $I^2r$ is dissipated inside the battery itself.

$P=I^2R=V^2/R$; real battery $arrow V_terminal=-Ir$.

As a capacitor charges through a resistor, current decays exponentially because the growing capacitor voltage increasingly opposes the driving EMF — governed by a first-order linear differential equation.

Loop equation: $ = IR+/QC = R/dQdt+/QC$; solution: $Q(t)=C≤ft(1-e^-t/RC)$, $I(t)=/Re^-t/RC$; time constant $=RC$.

Charging ($Q$ grows toward $C$, $I$ decays from $/R$ toward 0) vs. discharging (both $Q$ and $I$ decay toward 0 from their initial values).

Forgetting that an initially uncharged capacitor acts like a short circuit at $t=0$ (maximum current $/R$) and like an open circuit as $t→∞$ (zero current).

Charging: $Q=C(1-e^-t/RC)$, $I=(/R)e^-t/RC$; $=RC$.

A charged capacitor discharging through a resistor releases its stored energy exponentially, governed by the same time constant $=RC$ that describes charging.

Loop equation: $R/dQdt=-/QC$; solution $Q(t)=Q_0e^-t/RC$, $I(t)=-/Q_0RCe^-t/RC$; after one time constant, charge falls to $1/e≈37%$ of its initial value.

Time constant $=RC$ (time to fall to $1/e$ of the initial value) vs. total discharge time (no finite value exists — exponential decay only approaches zero asymptotically).

Treating $$ as "the time to fully discharge" instead of expressing the remaining charge at $t=, 2, 3$ using $e^-n$.

$Q=Q_0e^-t/RC$; at $t=$, $37%$ remains; large $R$ or $C$ $arrow$ slower decay.

A magnetic field exerts a force on a moving charge only when velocity has a component perpendicular to the field; this force is always perpendicular to both $ v$ and $ B$, so it does no work.

$ F=q v× B$; magnitude $F=qvB$; direction from the right-hand rule for positive charges (reverse for negative); since $ F v$, magnetic force changes direction of motion but never speed or kinetic energy.

Magnetic force (always $$ velocity, does zero work, changes direction only) vs. electric force (can be parallel to velocity, does work, changes speed).

Applying the right-hand rule without reversing it for negative charges, giving the wrong force direction for electrons.

$F=qv× B arrow $ to both; magnetic force does NO work.

Because the magnetic force is always perpendicular to velocity, a charge moving perpendicular to a uniform field travels in a circle at constant speed, with the magnetic force acting as the centripetal force.

$qvB=/mv^2r⇒ r=/mvqB$; period $T=/2 mqB$, independent of speed; a velocity component parallel to $ B$ produces helical motion.

Circular motion in a magnetic field (period independent of speed, radius depends on speed) vs. orbital motion under gravity or an electric force (period generally tied to orbit radius and speed together).

Assuming the period changes when a problem changes the particle's speed — $T=2 m/(qB)$ is speed-independent.

$r=mv/(qB)$; $T=2 m/(qB) arrow$ speed-independent.

A current-carrying wire in a magnetic field feels a force because each moving charge composing the current experiences the Lorentz force; summing over all charges gives a simple formula in terms of current.

$ F=I L× B$, magnitude $F=BIL$; force per length between two parallel wires: $/FL=/_0 I_1I_22 d$ — attractive if currents are parallel, repulsive if antiparallel.

Force on a moving point charge ($F=qvB$) vs. force on a current-carrying wire ($F=BIL$) — the wire formula sums the point-charge formula over all charge carriers.

Reversing the attract/repel rule for parallel wires — same-direction currents attract, opposite of like electric charges, which repel.

$F=BIL$; same-direction currents $arrow$ attract.

Just as continuous charge distributions require integrating $d E$, a current-carrying wire's magnetic field is found by integrating the field contribution from each infinitesimal current element.

$d B=/_04/I d l× rr^2$, $_0=4×10^-7 T·m/A$; infinite straight wire: $B=/_0 I2 r$; on-axis field of a current loop: $B=/_0IR^22(x^2+R^2)^3/2$, and at the center $B=/_0I2R$.

Biot-Savart Law (general, works for any current geometry, always requires integration) vs. Ampere's Law (faster, but only when symmetry allows $B$ to be pulled out of the integral).

Applying the right-hand rule to the whole wire at once instead of checking $d l× r$ at each current element, which can give the wrong direction for non-straight geometries.

$dB=/_04/I dl× rr^2 arrow$ integrate; straight wire $B=_0I/(2 r)$.

Ampere's Law relates the circulation of the magnetic field around a closed loop to the current piercing that loop, providing a shortcut to Biot-Savart integration whenever symmetry permits.

$ B· d l=_0 I_enc$; works cleanly for infinite wires, solenoids, and toroids, where $B$ can be pulled outside the integral along a well-chosen Amperian loop; the sign of $I_enc$ follows the right-hand rule relative to the loop's traversal direction.

Ampere's Law's $I_enc$ (net current through the loop, sign depends on traversal direction) vs. Gauss's Law's $Q_enc$ (simple algebraic sum of enclosed charge).

Applying Ampere's Law to a non-symmetric current distribution where $B$ cannot be factored out of the line integral — it only isolates $B$ algebraically with sufficient symmetry.

$ B dl=_0I_enc arrow$ needs symmetry (wire, solenoid, toroid).

A tightly wound solenoid produces a strong, nearly uniform field inside and negligible field outside — the magnetic analog of a parallel-plate capacitor's uniform field.

Ideal solenoid interior: $B=_0nI$ ($n=N/L$, turns per length); toroid: $B=/_0NI2 r$ inside the core; single loop at center: $B=/_0I2R$.

Solenoid field $B=_0nI$ (uniform inside, uses turn density $n$) vs. toroid field $B=_0NI/(2 r)$ (varies with radial position, uses total turns $N$).

Using total turns $N$ in place of turns per length $n=N/L$ in the solenoid formula — a longer solenoid with the same $N$ has a weaker field.

Solenoid: $B=_0nI$, uniform inside; toroid: $B1/r$ inside the core.

Magnetic flux measures how much magnetic field passes through a surface; it is the magnetic analog of electric flux and is the quantity whose change induces an EMF.

$_B= B· d A$; for uniform $B$ and a flat area, $_B=BA$; units: Weber (Wb) $=T·m^2$; Gauss's Law for magnetism: $ B· d A=0$ always, since magnetic monopoles do not exist.

Magnetic flux (changes due to changing $B$, changing area, or changing orientation — three independent mechanisms) vs. electric flux (analogous formula, but $ E· dA≠0$ in general because electric monopoles exist).

Forgetting that flux depends on $$ between $ B$ and the area's normal vector — a loop whose plane is parallel to $ B$ has zero flux, not maximum flux.

$_B=BA arrow$ zero when $ B$ lies in the plane of the loop.

A changing magnetic flux through a loop induces an EMF that drives a current; Lenz's Law fixes the direction of that current to oppose the change that caused it, a direct statement of energy conservation.

Faraday's Law: $=-/d_Bdt$; for $N$ loops, $=-N/d_Bdt$; the negative sign encodes Lenz's Law — the induced current's magnetic field opposes the change in flux.

Lenz's Law opposing the change in flux (not the flux itself) — if flux is already decreasing, the induced current reinforces the existing flux direction rather than canceling it.

Applying Lenz's Law to oppose the current direction of the existing flux instead of opposing its rate of change.

$=-d_B/dt arrow$ induced current opposes the CHANGE, not the flux itself.

A conducting rod moving through a magnetic field has its free charges pushed by the magnetic force, separating charge and creating an EMF — a special case of Faraday's Law viewed from the force perspective.

$=BLv$ (rod, field, and velocity mutually perpendicular); equivalently, charge separates until $qE_induced=qvB$; induced current $I=/R=BLv/R$; the induced current creates a retarding force $F=BIL$ opposing the rod's motion.

Motional EMF ($=BLv$, force-based, moving conductor in a static field) vs. transformer EMF ($=-d/dt$, changing field with stationary conductor) — both are applications of Faraday's Law, just different physical origins of the changing flux.

Forgetting that the retarding force on a sliding rod does negative work equal to the electrical energy dissipated, which is what keeps total energy conserved as an external force maintains constant velocity.

$=BLv arrow$ induced current opposes motion, converting mechanical work to electrical energy.

An inductor opposes changes in the current through it by inducing a back-EMF proportional to the rate of change of current, storing energy in its magnetic field.

Self-inductance: $_L=-L/dIdt$; solenoid inductance $L=/_0N^2Al$; energy stored $U_L=/12LI^2$; units: Henry (H) $=V·s/A$.

Inductor voltage $_L=-L dI/dt$ (depends on the rate of change of current) vs. resistor voltage $V=IR$ (depends on instantaneous current) vs. capacitor voltage $V=Q/C$ (depends on accumulated charge).

Assuming an inductor's voltage is nonzero whenever current is large — it is zero whenever $dI/dt=0$, no matter how large the steady current is.

$_L=-L dI/dt arrow$ opposes CHANGE in current; steady current $⇒$ inductor acts like a wire.

In a resistor-inductor circuit, current cannot change instantaneously; it approaches its steady-state value exponentially, governed by a first-order differential equation analogous to the RC circuit.

Loop equation: $=IR+L/dIdt$; growth solution: $I(t)=/R≤ft(1-e^-tR/L)$, time constant $=L/R$; decay after the source is removed: $I(t)=I_0e^-tR/L$.

RL time constant $=L/R$ (larger $L$ or smaller $R$ slows the change) vs. RC time constant $=RC$ (larger $R$ or $C$ slows the change) — $R$'s effect is opposite between the two circuits.

Forgetting that an inductor with no initial current acts like an open circuit at $t=0$ and like an ideal wire as $t→∞$ — the exact opposite of a capacitor's initial/final behavior.

$I(t)=(/R)(1-e^-tR/L)$, $=L/R$; inductor at $t=0$: open; at $t=∞$: wire.

An inductor and capacitor connected together exchange energy back and forth with no resistance to dissipate it, producing sustained electromagnetic oscillations analogous to a mass-spring system.

Loop equation: $L/d^2Qdt^2+/QC=0$; solution $Q(t)=Q_0( t+)$, angular frequency $=/1sqrtLC$; total energy is conserved: $U=/Q^22C+/12LI^2=/Q_0^22C$.

LC oscillation (undamped, energy sloshes forever between $C$ and $L$) vs. RC/RL decay (energy dissipates monotonically in a resistor, no oscillation) — the presence of $R$ distinguishes decay from oscillation.

Forgetting that maximum current occurs when the capacitor is fully discharged ($Q=0$) and maximum charge occurs when current is zero ($I=0$) — exactly out of phase, mirroring position and velocity in SHM.

$L Q+Q/C=0 arrow =1/sqrtLC$; energy sloshes between $C$ and $L$, like SHM.

Maxwell's four equations unify all of electricity and magnetism studied in this course, including the crucial insight that a changing electric field itself acts as a source of magnetic field.

Gauss's Law (E): $ E· d A=Q_enc/_0$; Gauss's Law (B): $ B· d A=0$; Faraday's Law: $ E· d l=-/d_Bdt$; Ampere-Maxwell Law: $ B· d l=_0I_enc+_0_0/d_Edt$.

Plain Ampere's Law (current-only source of $B$) vs. the full Ampere-Maxwell Law (adds the displacement current term $_0_0 d_E/dt$, needed wherever conduction current is absent but flux is changing).

Applying plain Ampere's Law inside a charging capacitor's gap and concluding $B=0$ there because no conduction current flows — the changing electric flux sustains a real magnetic field via the displacement current term.

4 equations $arrow$ Gauss (E), Gauss (B, always 0), Faraday, Ampere-Maxwell (adds displacement current).