checkpoint 1

units/em/unit-10.tex

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+\section{Capacitance, Dielectrics, and Energy Storage}
+
+This unit shifts the focus from electric fields and potentials produced by isolated charge distributions to systems designed to store electric energy: capacitors. The central idea is that a conductor (or pair of conductors) can hold charge and associated electric potential energy in a controlled geometry, and that the ability to store charge at a given voltage is quantified by the capacitance $C$.
+
+The flow begins with conductors in electrostatic equilibrium, where charge resides entirely on surfaces and the interior field vanishes. It then covers charge redistribution through contact, induction, and grounding. Next, capacitance and the standard capacitor geometries (parallel-plate, spherical, cylindrical) are introduced, followed by the energy stored in capacitor electric fields. The unit concludes with dielectrics, showing how inserting an insulating material increases capacitance through polarization.
+
+\input{concepts/em/u10/e10-1-conductors.tex}
+
+\input{concepts/em/u10/e10-2-charge-redistribution.tex}
+
+\input{concepts/em/u10/e10-3-capacitance.tex}
+
+\input{concepts/em/u10/e10-4-capacitor-energy.tex}
+
+\input{concepts/em/u10/e10-5-dielectrics.tex}

units/em/unit-11.tex

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+\section{Direct-Current Circuits}
+
+This unit develops the circuit viewpoint for electric charge motion. In AP Physics C: Electricity and Magnetism, the central idea is that steady currents through resistive elements obey precise relationships between current, voltage, and resistance. The treatment is restricted to direct-current (DC) circuits — configurations where sources are ideal constant-voltage batteries and fields have settled so that charge flows at a constant rate.
+
+The flow begins with the microscopic description of current, including drift velocity and current density, then introduces resistance, resistivity, and Ohm's law at both macroscopic and microscopic levels. It continues with electric power dissipation in resistors, then builds up to multi-resistor circuits through equivalent resistance for series and parallel combinations. Kirchhoff's junction and loop rules are introduced as the systematic tool for analyzing complex networks. RC transients with a time constant and internal resistance, which affects real batteries and measurement devices, round out the unit.
+
+\input{concepts/em/u11/e11-1-current-density.tex}
+
+\input{concepts/em/u11/e11-2-resistance-ohm.tex}
+
+\input{concepts/em/u11/e11-3-power.tex}
+
+\input{concepts/em/u11/e11-4-equivalent-resistance.tex}
+
+\input{concepts/em/u11/e11-5-kirchhoff.tex}
+
+\input{concepts/em/u11/e11-6-rc-circuits.tex}
+
+\input{concepts/em/u11/e11-7-internal-resistance.tex}

units/em/unit-12.tex

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+\section{Magnetism: Forces, Fields, and Sources}
+
+This unit develops the magnetic interaction, beginning with the force a magnetic field exerts on a moving charge — the vector cross-product law $\vec{F} = q\vec{v}\times\vec{B}$ — and the circular or helical motion that follows. In AP Physics C: Electricity and Magnetism, magnetic force is introduced as a velocity-dependent, perpendicular force that changes the direction of motion but does no work.
+
+From particle motion the unit extends to macroscopic currents: the force and torque on current-carrying conductors and loops placed in a magnetic field. That current viewpoint then flips to its source side — the Biot–Savart law and Ampère's law — which let you calculate the magnetic field produced by steady currents using symmetry. The unit closes with solenoids, parallel currents, and magnetic dipoles as canonical configurations.
+
+\input{concepts/em/u12/e12-1-magnetic-force-charge.tex}
+
+\input{concepts/em/u12/e12-2-particle-motion-in-b.tex}
+
+\input{concepts/em/u12/e12-3-force-on-current.tex}
+
+\input{concepts/em/u12/e12-4-biot-savart.tex}
+
+\input{concepts/em/u12/e12-5-ampere.tex}
+
+\input{concepts/em/u12/e12-6-solenoids-dipoles.tex}

units/em/unit-13.tex

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+\section{Electromagnetic Induction}
+
+This unit introduces the phenomena where changing magnetic conditions produce electric effects. In AP Physics C: Electricity and Magnetism, the core ideas begin with magnetic flux — a scalar measure of how much magnetic field penetrates a surface — and then culminate in Faraday's law, which quantifies how a time-varying magnetic flux induces an electromotive force. These concepts are the gateway from electrostatics to the full unification of electricity and magnetism.
+
+The flow starts with magnetic flux and Faraday's law, then addresses the direction of induced currents through Lenz's law. From there, the unit covers motional emf (emf generated by conductors moving through a field), inductance and the magnetic energy stored in fields, and finally the transient behavior of LR circuits and the harmonic oscillations of LC circuits.
+
+\input{concepts/em/u13/e13-1-magnetic-flux.tex}
+
+\input{concepts/em/u13/e13-2-faraday.tex}
+
+\input{concepts/em/u13/e13-3-lenz.tex}
+
+\input{concepts/em/u13/e13-4-motional-emf.tex}
+
+\input{concepts/em/u13/e13-5-inductance.tex}
+
+\input{concepts/em/u13/e13-6-lr-circuits.tex}
+
+\input{concepts/em/u13/e13-7-lc-circuits.tex}

units/em/unit-8.tex

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+\section{Electrostatics: Charge, Field, Flux}
+
+This unit develops the core ideas of electrostatics by starting with charge as a conserved quantity and then building the interaction model for stationary charges. In AP Physics C: Electricity and Magnetism, the emphasis is on careful charge bookkeeping, Coulomb's law, superposition, and the interpretation of the electric field vector $\vec{E}$ as a map of the force per unit charge that space assigns to a test charge.
+
+From there, the unit extends point-charge reasoning to standard continuous charge distributions, then introduces electric flux as a measure of how much electric field passes through a surface. That flux viewpoint leads naturally to Gauss's law, especially for highly symmetric charge distributions and Gaussian surfaces that make the field easy to determine.
+
+\input{concepts/em/u8/e8-1-charge-conservation.tex}
+
+\input{concepts/em/u8/e8-2-coulomb-superposition.tex}
+
+\input{concepts/em/u8/e8-3-electric-field.tex}
+
+\input{concepts/em/u8/e8-4-continuous-distributions.tex}
+
+\input{concepts/em/u8/e8-5-electric-flux.tex}
+
+\input{concepts/em/u8/e8-6-gauss-law.tex}

units/em/unit-9.tex

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+\section{Electric Potential and Energy}
+
+This unit develops the energy viewpoint for electrostatics. In AP Physics C: Electricity and Magnetism, the central idea is that electric interactions can be described not only with the vector field $\vec{E}$ but also with the scalar quantities electric potential energy $U$ and electric potential $V$. That scalar viewpoint often makes multi-charge systems and energy changes easier to analyze.
+
+The flow begins with electric potential energy for charge configurations, then defines electric potential and voltage, connects potential to the electric field, and finishes with equipotentials and the energy changes of moving charges. The scope here is electrostatics only, so the field remains conservative and no induction-related electric fields are considered yet.
+
+\input{concepts/em/u9/e9-1-electric-potential-energy.tex}
+
+\input{concepts/em/u9/e9-2-potential-voltage.tex}
+
+\input{concepts/em/u9/e9-3-field-potential.tex}
+
+\input{concepts/em/u9/e9-4-equipotentials.tex}

units/mechanics/unit-1.tex

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+\section{Kinematics}
+
+Kinematics describes motion by first choosing a reference frame and then tracking how an object's position vector $\vec{r}(t)$ changes with time. In AP Physics C: Mechanics, the focus is on motion in inertial frames, with 1D and 2D models that connect physical situations to equations, graphs, and vector components.
+
+This unit develops the core chain of ideas for motion: define position and displacement, differentiate locally to obtain $\vec{v}$ and $\vec{a}$, interpret motion graphs, use constant-acceleration relationships when applicable, and extend the same framework to relative motion and projectile motion with negligible air resistance.
+
+\input{concepts/mechanics/u1/m1-1-scalars-vectors}
+
+\input{concepts/mechanics/u1/m1-2-position-displacement}
+
+\input{concepts/mechanics/u1/m1-3-velocity-acceleration}
+
+\input{concepts/mechanics/u1/m1-4-motion-graphs}
+
+\input{concepts/mechanics/u1/m1-5-constant-acceleration}
+
+\input{concepts/mechanics/u1/m1-6-relative-projectile}

units/mechanics/unit-2.tex

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+\section{Force and Translational Dynamics}
+
+Dynamics explains changes in motion by relating the net external force on a chosen system to its acceleration through Newton's laws. In AP Physics C: Mechanics, this unit stays within inertial frames and emphasizes careful system definition, free-body reasoning, and consistent vector notation for quantities such as $\vec{F}$, $\vec{a}$, $\vec{g}$, $\vec{T}$, and $\vec{N}$.
+
+The flow of the unit begins with Newton's laws and center-of-mass ideas, then develops common force models such as gravitation, normal force, tension, friction, and springs. It closes with velocity-dependent forces and radial dynamics, including circular motion and circular orbits, with idealized strings, pulleys, and springs used throughout where appropriate.
+
+\input{concepts/mechanics/u2/m2-1-newton-laws}
+
+\input{concepts/mechanics/u2/m2-2-center-of-mass}
+
+\input{concepts/mechanics/u2/m2-3-gravitation}
+
+\input{concepts/mechanics/u2/m2-4-normal-tension}
+
+\input{concepts/mechanics/u2/m2-5-friction}
+
+\input{concepts/mechanics/u2/m2-6-springs}
+
+\input{concepts/mechanics/u2/m2-7-drag-terminal}
+
+\input{concepts/mechanics/u2/m2-8-circular-orbital}

units/mechanics/unit-3.tex

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+\section{Work, Energy, and Power}
+
+In this unit, we develop the energy viewpoint for mechanics. We begin with work as the accumulated effect of a force acting through a displacement, using the line-integral form $W=\int \vec{F}\cdot d\vec{r}$ as the calculus backbone and then connecting net work to changes in kinetic energy $K$.
+
+We then shift to conservative forces and potential energy $U$, which lets us track mechanical energy with $E_{\mathrm{mech}}=K+U$. With that accounting framework in place, we finish by defining power $P$ as the rate at which energy is transferred or transformed. The emphasis throughout stays on AP-level mechanical energy, including one-dimensional potential-energy graphs and clear bookkeeping of energy changes.
+
+\input{concepts/mechanics/u3/m3-1-work.tex}
+
+\input{concepts/mechanics/u3/m3-2-work-energy.tex}
+
+\input{concepts/mechanics/u3/m3-3-potential-energy.tex}
+
+\input{concepts/mechanics/u3/m3-4-energy-conservation.tex}
+
+\input{concepts/mechanics/u3/m3-5-power.tex}

units/mechanics/unit-4.tex

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+\section{Linear Momentum and Collisions}
+
+This unit develops the momentum viewpoint for interactions between objects and systems. We begin by defining linear momentum $\vec{p}$ for a particle and total momentum $\vec{P}$ for a system, using consistent vector notation and emphasizing that momentum conservation is applied componentwise in both one and two dimensions.
+
+We then connect external forces to momentum transfer through impulse $\vec{J}$, use isolated-system reasoning to establish conservation of momentum, and apply those ideas to collisions, recoil, and explosions. Throughout, the AP focus is on classifying collisions correctly, recognizing that kinetic energy is not always conserved, and relating system motion to the center-of-mass velocity $\vec{v}_{\mathrm{cm}}$ when helpful.
+
+\input{concepts/mechanics/u4/m4-1-linear-momentum.tex}
+
+\input{concepts/mechanics/u4/m4-2-impulse.tex}
+
+\input{concepts/mechanics/u4/m4-3-momentum-conservation.tex}
+
+\input{concepts/mechanics/u4/m4-4-collisions.tex}
+
+\input{concepts/mechanics/u4/m4-5-recoil-explosions.tex}

units/mechanics/unit-5.tex

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+\section{Torque and Rotational Dynamics}
+
+This unit extends translational mechanics to fixed-axis rotational motion. We begin with angular position $\theta$, angular velocity $\omega$, and angular acceleration $\alpha$, then connect those quantities to the corresponding linear quantities for points on a rigid body.
+
+We next develop the rotational analogs of force and mass through torque $\vec{\tau}$ and moment of inertia $I$, apply equilibrium ideas to situations with no angular acceleration, and conclude with the rotational dynamics relation $\sum \tau = I\alpha$ for planar rigid-body rotation in AP Physics scope.
+
+\input{concepts/mechanics/u5/m5-1-angular-kinematics.tex}
+
+\input{concepts/mechanics/u5/m5-2-linear-rotational-link.tex}
+
+\input{concepts/mechanics/u5/m5-3-torque.tex}
+
+\input{concepts/mechanics/u5/m5-4-moment-of-inertia.tex}
+
+\input{concepts/mechanics/u5/m5-5-rotational-equilibrium.tex}
+
+\input{concepts/mechanics/u5/m5-6-rotation-dynamics.tex}

units/mechanics/unit-6.tex

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+\section{Angular Momentum and Rolling Motion}
+
+This unit continues fixed-axis rotational mechanics by extending the work-energy ideas of Unit 5 to rotational kinetic energy and the rotational work-energy theorem. We then introduce angular momentum $\vec{L}$ and angular impulse, emphasizing how torque $\vec{\tau}$ changes rotational motion in the same way that force changes linear momentum.
+
+We next focus on conservation of angular momentum for isolated systems, apply that reasoning to rolling without slipping, and conclude with the AP Physics treatment of circular Newtonian orbits, including orbit speed and energy. Throughout, the emphasis is on fixed-axis rotation, rolling motion, and standard circular orbit results within AP scope.
+
+\input{concepts/mechanics/u6/m6-1-rotational-energy.tex}
+
+\input{concepts/mechanics/u6/m6-2-angular-momentum.tex}
+
+\input{concepts/mechanics/u6/m6-3-angular-momentum-conservation.tex}
+
+\input{concepts/mechanics/u6/m6-4-rolling.tex}
+
+\input{concepts/mechanics/u6/m6-5-orbits.tex}

units/mechanics/unit-7.tex

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+\section{Oscillations}
+
+This unit develops the mechanics of oscillatory motion about stable equilibrium. We begin with the defining linear restoring-force model for simple harmonic motion (SHM), write its governing differential equation, and connect the motion to sinusoidal solutions, angular frequency, period, and frequency.
+
+We then apply that framework to spring-mass oscillators, interpret SHM through energy, and finish with pendulum models. In AP Physics C: Mechanics, the pendulum results here are used in the small-angle regime, and the focus remains on undamped, unforced oscillations rather than broader circuit or driven-oscillation extensions.
+
+\input{concepts/mechanics/u7/m7-1-shm.tex}
+
+\input{concepts/mechanics/u7/m7-2-spring-oscillator.tex}
+
+\input{concepts/mechanics/u7/m7-3-shm-energy.tex}
+
+\input{concepts/mechanics/u7/m7-4-simple-pendulum.tex}
+
+\input{concepts/mechanics/u7/m7-5-physical-pendulum.tex}