Edexcel A-Level Physics Master Revision Guide

Particle Physics & Subatomic Particles

• Particle Classifications:

  • All particles are broadly categorized into Hadrons and Leptons.

  • Leptons: Fundamental particles that cannot be broken down any further. Examples include the electron ($e^-$), the muon ($\mu^-$, essentially a heavy electron), and neutrinos ($\nu$, which possess no electric charge and essentially zero mass).

  • Lepton Numbers: All standard leptons have a lepton number of $+1$, while their antiparticle equivalents have a lepton number of $-1$. Electron neutrinos ($\nu_e$) and muon neutrinos ($\nu_mu$) have their electron lepton number ($L_e$) and muon lepton number ($L_mu$) tracked separately in particle interactions.

  • Hadrons: Non-fundamental particles composed of quarks. They are split into two sub-groups:

    • Baryons: Composed of three quarks ($qqq$) or three antiquarks ($\bar{q}\bar{q}\bar{q}$). Baryons have a non-zero baryon number ($B = +1$ for baryons, $B = -1$ for antibaryons).

    • Mesons: Composed of a quark-antiquark pair ($q\bar{q}$). Their baryon number is $0$

• Quark Flavors & Properties:

  • Up ($u$): Electric charge = $+\frac{2}{3}e$, Baryon number = $+\frac{1}{3}$, Strangeness = $0$

  • Down ($d$): Electric charge = $-\frac{1}{3}e$, Baryon number = $+\frac{1}{3}$, Strangeness = $0$

  • Strange ($s$): Electric charge = $-\frac{1}{3}e$, Baryon number = $+\frac{1}{3}$, Strangeness = $-1$

  • Antiquarks ($\bar{u}, \bar{d}, \bar{s}$) possess opposite electric charge, baryon number, and strangeness. For instance, an anti-strange quark ($\bar{s}$) has a strangeness of $+1$

• Hadron Compositions:

  • Neutron: Composition is $udd$ ("nud"). Charge = $+\frac{2}{3} - \frac{1}{3} - \frac{1}{3} = 0$

  • Proton: Composition is $uud$ ("poo"). Charge = $+\frac{2}{3} + \frac{2}{3} - \frac{1}{3} = +1e$

  • Pions ($\pi^+, \pi^0, \pi^-$): Mesons that do not contain strange quarks.

  • Kaons ($K^+, K^-, K^0$): Mesons that contain strange or anti-strange quarks.

• Fundamental Forces & Exchange Particles:

  • Electromagnetic Force: Affects any electrically charged particle. The exchange particle is the virtual photon ($\gamma$), which transfers momentum between charged particles.

  • Gravity: Affects all particles with mass. The theoretical exchange particle is the graviton.

  • Weak Nuclear Force: Affects virtually all particles. Exchange particles are the $W^+$, $W^-$, and $Z^0$ bosons. It is responsible for flavor changes in quarks during weak interactions.

  • Strong Nuclear Force: Affects hadrons only. The exchange particle is the pion (or gluon). It overcomes electrostatic repulsion between protons to hold atomic nuclei together.

    • Range: Effective between $3\text{ to }4\,fm$. It is attractive down to $0.5\,fm$ and becomes strongly repulsive below $0.5\,fm$ to prevent the nucleus from collapsing on itself.

• Conservation Laws & Weak Interactions:

  • In any particle interaction, Electric Charge, Baryon Number, Electron Lepton Number, and Muon Lepton Number must be strictly conserved.

  • Beta Minus Decay ($\beta^-$): A neutron decays into a proton, emitting an electron ($e^-$) and an anti-electron neutrino ($\bar{\nu}_e$) to balance the electron lepton number. At the quark level, a down quark decays into an up quark ($d \rightarrow u + e^- + \bar{\nu}_e$) mediated by a $W^-$ boson.

  • Beta Plus Decay ($\beta^+$): An up quark turns into a down quark, emitting a positron ($e^+$) and an electron neutrino ($\nu_e$) mediated by a $W^+$ boson (u \rightarrow d + e^+ + \n\nu_e).

  • Electron Capture: A proton captures an inner-shell electron, turning into a neutron and emitting an electron neutrino (p + e^- \rightarrow n + \n\nu_e) mediated by a $W^+$ boson.

Radiation, Nuclear Decay & Mass-Energy Equivalence

• Types of Nuclear Radiation:

  • Gamma Radiation ($\gamma$): High-energy electromagnetic waves emitted directly by an excited nucleus with excess energy. Highly ionizing upon absorption, knocking electrons off atoms, which damages living cells and can cause cancer.

  • Alpha Radiation ($\alpha$): Consists of an alpha particle, which is a helium nucleus composed of two protons and two neutrons ($_2^4\text{He}$). Emitted by heavy, unstable nuclei (e.g., Americium-241).

    • Alpha Decay Reaction: $_{95}^{241}\text{Am} \rightarrow _{93}^{237}\text{Np} + _{2}^{4}\text{He}$

    • The atomic number decreases by $2$ and the mass number decreases by $4$

  • Beta Minus Radiation ($\beta^-$): Consists of a fast-moving electron ejected from the nucleus when a neutron transforms into a proton.

    • Beta Minus Decay Reaction: $_{6}^{14}\text{C} \rightarrow _{7}^{14}\text{N} + _{-1}^{0}e + \bar{\nu}_e$

    • The atomic number increases by $1$ (since $6 = 7 + (-1)$), while the mass number remains unchanged at $14$

• Annihilation & Pair Production:

  • Annihilation: Occurs when a particle and its corresponding antiparticle (e.g., an electron and a positron) collide. All of their mass is converted into energy, producing two photons of electromagnetic radiation traveling in opposite directions to conserve momentum.

    • Minimum Photon Energy Formula: $E_{min} = mc^2 = hf$

    • Where $m$ is the rest mass of one particle, $c$ is the speed of light ($3.00 \times 10^8\,m/s$), $h$ is Planck's constant, and $f$ is photon frequency.

  • Pair Production: Occurs when a single high-energy photon spontaneously converts into a particle-antiparticle pair.

    • Minimum Photon Energy Formula: $hf_{min} = 2mc^2$

    • Any excess energy possessed by the photon above the minimum rest energy threshold is converted into kinetic energy of the resulting particle pair.

Electricity & Circuit Analysis

• Electric Current, Charge, and Voltage:

  • Electric Current ($I$): The rate of flow of electric charge. Measured in Amperes ($A$) using an ammeter connected in series.

    • Formula: $I = \frac{Q}{t}$

    • Conventional current flows from the positive terminal of a power source to the negative terminal.

  • Potential Difference ($V$): The energy transferred per unit charge between two points. Measured in Volts ($V$) using a voltmeter connected in parallel across a component.

    • Formula: $V = \frac{E}{Q}$

    • $1\,V = 1\,J/C$ (One Joule of energy supplied per Coulomb of charge).

• Resistance & Ohm's Law:

  • Ohm's Law: $V = IR$

  • Ohmic Resistor: Resistance remains constant regardless of potential difference. Its $I\text{-}V$ characteristic graph is a straight line passing through the origin. Steeper gradient implies lower resistance ($R = \frac{1}{\text{gradient}}$).

  • Filament Lamp (Non-Ohmic Conductor): Resistance increases as current increases. Current causes delocalized electrons to collide more frequently with the metal lattice ions, causing the ions to vibrate with greater amplitude and heat up, making it harder for electrons to flow.

  • Diodes & Light Emitting Diodes (LEDs): Allow current to flow in only one direction (forward bias). Resistance is extremely high in reverse bias and drops drastically once the threshold voltage (around $1\,V$) is reached in forward bias.

  • Superconductors: Materials that exhibit precisely zero electrical resistance when cooled below a specific critical temperature ($T_c$).

• Resistivity:

  • Definition: The resistance of a cube of unit length sides ($1\,m^3$) of a material, measured in Ohm-meters ($\Omega\,m$).

  • Resistivity Formula: $R = \frac{\rho L}{A} \implies \rho = \frac{R A}{L}$

  • Experimental Determination:

    1. Measure wire diameter $d$ using a micrometer at multiple locations to calculate cross-sectional area $A = \frac{\pi d^2}{4}$

    2. Measure resistance $R$ for various lengths $L$ using a meter rule, voltmeter, and ammeter.

    3. Plot $R$ against $L$. The gradient equals $\frac{R}{L}$. Multiply the gradient by $A$ to calculate $\rho$

• Series & Parallel Circuits and Kirchhoff's Laws:

  • Kirchhoff's First Law: The total current entering a junction equals the total current leaving it (Conservation of Charge): $\sum I_{in} = \sum I_{out}$

  • Kirchhoff's Second Law: In any closed circuit loop, the sum of electromotive forces (EMFs) equals the sum of potential difference drops (Conservation of Energy): $\sum \mathcal{E} = \sum V$

  • Series Circuits:

    • Current is identical across all components ($I_{total} = I_1 = I_2$).

    • Total potential difference is shared ($V_{total} = V_1 + V_2$).

    • Total resistance is the sum of individual resistances ($R_{total} = R_1 + R_2 + \dots$).

  • Parallel Circuits:

    • Potential difference is equal across every parallel branch ($V_{total} = V_1 = V_2$).

    • Current is shared between branches ($I_{total} = I_1 + I_2$).

    • Total resistance decreases as more parallel branches are added: $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots$

• Sensing Circuits & Potential Dividers:

  • Potential Divider Equation: $V_{out} = V_{in} \times \left(\frac{R_x}{R_{total}}\right)$

  • Negative Temperature Coefficient (NTC) Thermistor: Resistance decreases as temperature increases. In a potential divider circuit, as temperature drops, thermistor resistance rises, increasing its share of the potential difference.

  • Light Dependent Resistor (LDR): Resistance decreases as light intensity increases. When dark, its resistance rises, yielding a higher voltage output across it.

• Electrical Power & Alternating Current (AC):

  • Power Equations: $P = VI = I^2 R = \frac{V^2}{R}$

  • Alternating Current: UK mains supplies AC where the neutral wire stays at $0\,V$ and the live wire fluctuates between $+325\,V$ and $-325\,V$ (Peak voltage $V_{peak} = 325\,V$, Peak-to-Peak $V_{p\text{-}p} = 650\,V$).

  • Root Mean Square (RMS) Values: Used to find the DC equivalent values for AC circuits:

    • $V_{RMS} = \frac{V_{peak}}{\sqrt{2}} = \frac{325}{\sqrt{2}} \approx 230\,V$

    • $I_{RMS} = \frac{I_{peak}}{\sqrt{2}}$

    • Mean Power $P_{mean} = V_{RMS} \times I_{RMS} = \frac{P_{peak}}{2}$

• Electromotive Force (EMF) & Internal Resistance:

  • Electromotive Force ($\mathcal{E}$): Total energy converted per unit charge by a source from chemical to electrical energy.

  • Terminal Potential Difference ($V$): Energy available to the external load circuit per unit charge.

  • Internal Resistance ($r$): The inherent resistance inside a cell or battery that causes voltage to be lost internally ("lost volts" = $Ir$).

    • Formula: $\mathcal{E} = V + Ir = I(R + r)$

  • Graphical Determination: Plot Terminal PD ($V$) against Current ($I$):

    • $V = -rI + \mathcal{E}$

    • Y-intercept = $\mathcal{E}$

    • Magnitude of gradient ($|m|$) = internal resistance $r$

Mechanics, Motion & Vectors

• Vectors & Scalars:

  • Scalar: Has magnitude only (e.g., distance, speed, mass, time, energy).

  • Vector: Has magnitude and direction (e.g., displacement, velocity, acceleration, force, momentum, weight).

  • Vector Addition: Perpendicular vectors are combined using Pythagoras' theorem ($R = \sqrt{A^2 + B^2}$) and trigonometry ($\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$).

  • Vector Resolution: For a force $F$ at angle $\theta$ to the horizontal:

    • Horizontal Component: $F_x = F \cos(\theta)$

    • Vertical Component: $F_y = F \sin(\theta)$

• Work, Energy, and Power:

  • Work Done ($W$): Energy transferred by a force acting over a distance. Force and displacement must be parallel.

    • Formula: $W = F d \cos(\theta)$

  • Gravitational Potential Energy: $E_p = mgh$

  • Kinetic Energy: $E_k = \frac{1}{2}mv^2$

  • Power Developed ($P$): The rate of work done.

    • Formula: $P = \frac{W}{t} = F v$

• Newton's Laws of Motion:

  • First Law: An object remains at rest or continues at a constant velocity unless acted upon by a non-zero resultant force.

  • Second Law: Resultant force is proportional to the rate of change of momentum, simplified to $F_{net} = ma$

  • Third Law: For every action force, there is an equal and opposite reaction force acting on a different body.

• Mass on an Inclined Plane:

  • Weight ($mg$) acts vertically downward.

  • Component perpendicular to ramp = $mg \cos(\theta)$ (balanced by normal contact force $N$).

  • Component parallel down the ramp = $mg \sin(\theta)$

  • Resultant force down ramp = $mg \sin(\theta) - F_{friction} = ma$

• Equations of Motion (SUVAT):

  • Valid only for constant acceleration:

    1. $v = u + at$

    2. $s = ut + \frac{1}{2}at^2$

    3. $s = \frac{(u + v)t}{2}$

    4. $v^2 = u^2 + 2as$

  • Projectiles: Motion is resolved into independent horizontal and vertical components:

    • Horizontal: Zero acceleration ($a_x = 0$), constant velocity ($v_x = u_x$), distance $s_x = v_x t$

    • Vertical: Accelerated by gravity ($a_y = -g = -9.81\,m/s^2$), initial velocity $u_y = u \sin(\theta)$. SUVAT equations apply.

• Momentum & Impulse:

  • Linear Momentum ($p$): $p = mv$ (Units: $kg\,m/s$ or $N\,s$). Vector quantity.

  • Conservation of Momentum: Total momentum before an interaction equals total momentum after, provided no external forces act: $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$

  • Rebound Dynamics: An object hitting a wall at speed $u$ and rebounding at speed $u$ in the opposite direction experiences a change in momentum (Impulse) of $\Delta p = -2mu$

  • Elastic vs. Inelastic Collisions:

    • Elastic Collision: Total kinetic energy is conserved.

    • Inelastic Collision: Total kinetic energy is not conserved (lost as heat/sound/deformation).

  • Force-Momentum Relation: $F = \frac{\Delta p}{\Delta t}$. Area under a Force-time graph equals Impulse (change in momentum).

  • Fluid Flow Force: For a fluid of density $\rho$ flowing through cross-sectional area $A$ at speed $v$: $F = \rho A v^2$

• Moments, Couples & Equilibrium:

  • Moment (Torque): Turning effect of a force around a pivot: $\text{Moment} = F \times d_{\perp}$ (Units: $N\,m$).

  • Principle of Moments: For an object in equilibrium, the sum of clockwise moments about any pivot point equals the sum of anticlockwise moments about the same point.

  • Conditions for Equilibrium:

    1. Resultant force is zero ($\sum F = 0$).

    2. Resultant moment about any point is zero ($\sum \text{Moments} = 0$).

  • Couple: Two equal and opposite parallel forces that do not act along the same line, producing rotation without translation: $\text{Torque of a couple} = F \times d$ (where $d$ is the perpendicular distance between forces).

Circular Motion & Field Theory

• Uniform Circular Motion Mechanics:

  • An object traveling at constant speed $v$ in a circle of radius $r$ is continuously accelerating because its direction of velocity changes.

  • Angular Velocity ($\omega$): $\omega = \frac{2\pi}{T} = 2\pi f = \frac{v}{r}$ (Units: $rad/s$).

  • Centripetal Acceleration ($a$): $a = \frac{v^2}{r} = \omega^2 r$

  • Centripetal Force ($F$): Directed toward the center of the circle: $F = \frac{mv^2}{r} = m \omega^2 r$

  • Vertical Loops:

    • Bottom of Loop: Support Force $S - mg = \frac{mv^2}{r} \implies S = mg + \frac{mv^2}{r}$

    • Top of Loop: Support Force $S + mg = \frac{mv^2}{r} \implies S = \frac{mv^2}{r} - mg$

  • Banked Tracks: Horizontal component of normal reaction force provides centripetal force: $N \sin(\theta) = \frac{mv^2}{r}$ and $N \cos(\theta) = mg \implies \tan(\theta) = \frac{v^2}{gr}$

• Magnetic Fields & Force on Charges:

  • Motor Effect: A current-carrying wire in a magnetic field experiences a force: $F = BIL \sin(\theta)$ (where $B$ is magnetic flux density in Tesla, $T$).

  • Fleming's Left-Hand Rule: Thumb = Force ($F$), First Finger = Magnetic Field ($B$), Second Finger = Conventional Current ($I$) or Positive Particle Velocity ($v$).

  • Moving Particle in B-Field: $F = BQv$. Since this force is perpendicular to velocity, it acts as a centripetal force: $BQv = \frac{mv^2}{r} \implies r = \frac{mv}{BQ}$

  • Cyclotron Acceleration: High-energy particle accelerator utilizing two hollow metal D-shaped electrodes inside a perpendicular magnetic field. An alternating potential difference across the gap accelerates particles every time they cross.

    • Orbital Frequency: $f = \frac{BQ}{2\pi m}$ (Frequency is independent of orbital radius $r$).

  • Velocity Selector & Mass Spectrometer:

    • Velocity Selector: Uses perpendicular electric ($E$) and magnetic ($B$) fields. Particles experience balanced forces when $EQ = BQv \implies v = \frac{E}{B}$

    • Mass Spectrometer: Selected velocity particles enter a uniform B-field and undergo deflection into a semicircle. Radius $r = \frac{mv}{BQ}$ is directly proportional to particle mass $m$

Electromagnetic Induction & Transformers

• Magnetic Flux & Flux Linkage:

  • Magnetic Flux ($\Phi$): Measure of magnetism passing through an area: $\Phi = BA$ (Units: Weber, $Wb = V\,s$ or Tesla-meter squared, $T\,m^2$).

  • Magnetic Flux Linkage: Total flux passing through a coil of $N$ turns: $\text{Flux Linkage} = BAN$

• Laws of Induction:

  • Faraday's Law: The magnitude of induced EMF is proportional to the rate of change of magnetic flux linkage: $\mathcal{E} = \frac{\Delta(BAN)}{\Delta t}$

  • Lenz's Law: The direction of induced EMF/current always opposes the magnetic change that created it: $\mathcal{E} = -\frac{\Delta(BAN)}{\Delta t}$

  • Moving Conductor: A straight wire of length $L$ cutting flux at velocity $v$ induces an EMF: $\mathcal{E} = BLv$

  • Fleming's Right-Hand Rule: Used to find the direction of induced current in a generator (Thumb = Motion, First Finger = Field, Middle Finger = Induced Current).

• AC Generators & Transformers:

  • AC Generator Peak EMF: $\mathcal{E}_{peak} = BAN\omega$. Instantaneous EMF is $\mathcal{E} = BAN\omega \sin(\omega t)$

  • Transformer Equation: $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$ (for an ideal $100\%$ efficient transformer where $P_{in} = P_{out}$).

  • Transformer Efficiency: $\text{Efficiency} = \frac{V_s I_s}{V_p I_p}$

  • National Grid Transmission: Voltage is stepped up prior to long-distance transmission to reduce current, minimising heat power loss ($P_{loss} = I^2 R$).

  • Back EMF in Electric Motors: As a motor turns, it acts as a generator creating a back-EMF that opposes the supply voltage: $\mathcal{E}_{net} = V_{supply} - \mathcal{E}_{back}$

Gravitational & Electric Fields Detailed

• Inverse-Square Field Laws:

  • Newton's Law of Universal Gravitation: $F_g = \frac{G M m}{r^2}$ (where $G = 6.67 \times 10^{-11}\,N\,m^2/kg^2$).

  • Coulomb's Law: $F_e = \frac{k Q q}{r^2} = \frac{Q q}{4\pi \varepsilon_0 r^2}$ (where $k = 9 \times 10^9\,N\,m^2/C^2$ and $\varepsilon_0$ is permittivity of free space).

• Field Strengths ($g$ and $E$):

  • Gravitational Field Strength ($g$): Force per unit mass: $g = \frac{G M}{r^2}$ (Units: $N/kg$ or $m/s^2$).

  • Electric Field Strength ($E$): Force per unit positive charge: $E = \frac{k Q}{r^2} = \frac{Q}{4\pi \varepsilon_0 r^2}$ (Units: $N/C$ or $V/m$).

  • Inside a Planet: Gravitational field strength decreases linearly from the surface to zero at the center ($g \propto r$ assuming constant density $\rho$).

• Potential ($V$) & Potential Energy ($E_p$):

  • Gravitational Potential ($V_g$): Work done per unit mass in bringing a mass from infinity to a point: $V_g = -\frac{G M}{r}$ (Units: $J/kg$). Always negative.

  • Electric Potential ($V_e$): Work done per unit positive charge in bringing a charge from infinity to a point: $V_e = \frac{k Q}{r}$ (Units: $J/C$ or $V$).

  • Potential Energy: $E_{p,g} = -\frac{G M m}{r}$ and $E_{p,e} = \frac{k Q q}{r}$

  • Potential Gradient: Field strength is equal to the negative potential gradient ($g = -\frac{\Delta V_g}{\Delta r}$ and $E = -\frac{\Delta V_e}{\Delta r}$).

• Uniform Electric Fields (Parallel Plates):

  • Electric field strength between parallel plates separated by distance $d$ with potential difference $V$: $E = \frac{V}{d}$

  • Acceleration of a particle with charge $Q$ and mass $m$: $a = \frac{EQ}{m} = \frac{VQ}{md}$

  • Millikan Oil Drop Experiment: Droplet levitates when electric force equals weight ($EQ = mg \implies \frac{VQ}{d} = mg$).

• Orbital Mechanics & Escape Velocity:

  • Orbital Speed: Setting gravity equal to centripetal force ($\frac{G M m}{r^2} = \frac{m v^2}{r}$) yields $v = \sqrt{\frac{GM}{r}}$

  • Kepler's Third Law: $T^2 = \left(\frac{4\pi^2}{GM}\right) r^3 \implies T^2 \propto r^3$

  • Geostationary Orbit: Has an orbital period of exactly $24\text{ hours}$ ($86,400\text{ seconds}$), orbits horizontally above the Earth's equator in the direction of Earth's rotation.

  • Escape Velocity ($v_{esc}$): Speed required at the surface for kinetic energy to equal potential energy to reach infinity:

    • $\frac{1}{2} m v_{esc}^2 = \frac{G M m}{r} \implies v_{esc} = \sqrt{\frac{2GM}{r}}$

  • Distance of Closest Approach: For a positive particle fired at a nucleus, kinetic energy converts fully to potential energy at closest approach $r$:

    • $\frac{1}{2} m v^2 = \frac{k Q q}{r}$

Capacitance & Dielectrics

• Capacitor Fundamentals:

  • Capacitance ($C$): Charge stored per unit potential difference: $C = \frac{Q}{V}$ (Units: Farad, $F = C/V$).

  • Energy Stored ($E$): Equal to the area under a Charge-Voltage ($Q\text{-}V$) graph:

    • $E = \frac{1}{2} Q V = \frac{1}{2} C V^2 = \frac{1}{2} \frac{Q^2}{C}$

• Discharging & Charging Equations:

  • Discharging: Exponential decay for voltage, current, and charge:

    • $V = V_0 e^{-\frac{t}{RC}}$, $I = I_0 e^{-\frac{t}{RC}}$, $Q = Q_0 e^{-\frac{t}{RC}}$

  • Charging: Voltage and charge build up exponentially, while current decays:

    • $V = V_0 \left(1 - e^{-\frac{t}{RC}}\right)$, $Q = Q_0 \left(1 - e^{-\frac{t}{RC}}\right)$, $I = I_0 e^{-\frac{t}{RC}}$

  • Time Constant ($\tau$): $\tau = RC$ (Units: seconds). The time taken for potential difference, current, or charge to decay to $\frac{1}{e}$ ($\approx 37\%$) of its initial value.

  • Linearized Logarithmic Discharge Graph: Plotting $\ln(V)$ against $t$ yields a straight line with gradient $m = -\frac{1}{RC}$ and y-intercept $\ln(V_0)$.

• Parallel Plate Capacitance & Dielectrics:

  • Formula: $C = \frac{\varepsilon A}{d} = \frac{\varepsilon_0 \varepsilon_r A}{d}$

  • Dielectric Materials: Insulating materials placed between plates containing polar molecules. The molecules align with the electric field, creating an opposing counter-electric field. This reduces net potential difference across the plates for a given charge, thereby increasing capacitance.

  • Circuit State Rules: If connected to a battery, voltage $V$ remains constant. If disconnected, total charge $Q$ remains constant.

Mechanics of Materials & Fluids

• Upthrust & Archimedes' Principle:

  • Upthrust ($U$): Upward force exerted by a fluid on an object equal to the weight of the fluid displaced:

    • $U = m_{fluid} g = \rho_{fluid} V_{object} g$

• Hooke's Law & Spring Combinations:

  • Hooke's Law: $F = k \Delta x$ (where $k$ is stiffness/spring constant in $N/m$).

  • Elastic Potential Energy: $E_p = \frac{1}{2} F \Delta x = \frac{1}{2} k (\Delta x)^2$

  • Springs in Series: Equivalent stiffness halves: $\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2}$

  • Springs in Parallel: Equivalent stiffness doubles: $k_{eq} = k_1 + k_2$

• Stress, Strain, and Young Modulus:

  • Tensile Stress ($\sigma$): Force per unit cross-sectional area: $\sigma = \frac{F}{A}$ (Units: Pascals, $Pa = N/m^2$).

  • Tensile Strain ($\epsilon$): Ratio of extension to original length: $\epsilon = \frac{\Delta L}{L}$ (Dimensionless).

  • Young Modulus ($E$): Ratio of stress to strain within the limit of proportionality:

    • $E = \frac{\sigma}{\epsilon} = \frac{F L}{A \Delta L}$

    • Given $A = \frac{\pi d^2}{4}$, doubling wire diameter quarters the extension for a given force.

• Deformation & Stress-Strain Graphs:

  • Limit of Proportionality: Point beyond which stress is no longer proportional to strain.

  • Elastic Limit: Point beyond which the material will deform plastically (permanently extended) and will not return to its original length when unloaded.

  • Ultimate Tensile Strength (UTS): Maximum stress a material can withstand before necking and breaking.

  • Hysteresis Loop: Area under the loading curve represents work done stretching the material. Area under unloading curve represents energy returned. Enclosed area represents energy lost as thermal energy during cyclic loading.

Simple Harmonic Motion & Waves

• Simple Harmonic Motion (SHM) Conditions:

  1. Acceleration is directly proportional to displacement from equilibrium ($a \propto x$).

  2. Acceleration acts in the opposite direction to displacement ($a = -\omega^2 x$).

• SHM Kinematic Equations:

  • Displacement: $x = A \sin(\omega t)$ (starting at equilibrium) or $x = A \cos(\omega t)$ (starting at maximum amplitude).

  • Maximum Acceleration: $a_{max} = \omega^2 A$

  • Velocity: $v = \omega \sqrt{A^2 - x^2}$

  • Maximum Velocity: $v_{max} = \omega A = 2\pi f A$ (occurs at equilibrium $x = 0$).

• Oscillator Periods:

  • Simple Pendulum: $T = 2\pi \sqrt{\frac{L}{g}}$ (valid for small angle approximations \theta < 10^\circ; independent of mass).

  • Mass-Spring System: $T = 2\pi \sqrt{\frac{m}{k}}$

• Damping & Resonance:

  • Damping: Resistive forces remove mechanical energy from the system, decreasing amplitude over time. Energy is proportional to amplitude squared ($E \propto A^2$).

    • Light Damping: Amplitude decays gradually over many cycles.

    • Heavy/Over Damping: No oscillation occurs; system returns to equilibrium very slowly.

    • Critical Damping: System returns to equilibrium in the minimum possible time without overshooting (e.g., car suspension shock absorbers).

  • Resonance: Occurs when driving frequency matches the natural resonant frequency ($f_0$) of a lightly damped system, resulting in maximum energy transfer and maximum amplitude. Driving force is $90^\circ$ ($\frac{\pi}{2}\text{ radians}$) out of phase with displacement.

• Wave Properties & Equations:

  • Longitudinal Wave: Particle oscillations are parallel to the direction of energy transfer (consists of compressions and rarefactions).

  • Transverse Wave: Particle oscillations are perpendicular to the direction of energy transfer.

  • Wave Equation: $v = f \lambda$ and $f = \frac{1}{T}$

  • Intensity ($I$): Proportional to amplitude squared ($I \propto A^2$).

• Refraction & Total Internal Reflection (TIR):

  • Refractive Index ($n$): $n = \frac{c}{v}$ (where $c = 3.00 \times 10^8\,m/s$).

  • Snell's Law: $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$

  • Critical Angle ($\theta_c$): Angle of incidence above which total internal reflection occurs when traveling from higher to lower refractive index (n_1 > n_2):

    • $\sin(\theta_c) = \frac{n_2}{n_1}$

  • Optical Fibers: Composed of a core surrounded by cladding with a lower refractive index (n_{core} > n_{cladding}). Modal/multipath dispersion causes pulse broadening; mitigated by using thin single-mode fibers, repeaters, or graded-index fibers.

• Thin Lenses:

  • Lens Power ($P$): $P = \frac{1}{f}$ (Units: Diopters, $D$; focal length $f$ in meters).

  • Lens Formula: $\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$ (where $u$ is object distance, $v$ is image distance).

  • Convex Lens: Converging lens; forms real inverted images or magnified virtual upright images.

  • Concave Lens: Diverging lens; always forms virtual, upright, diminished images.

• Polarization:

  • Transverse waves oscillating in a single plane perpendicular to energy travel. Unpolarized light passing through a linear polarizer loses $50\%$ of its intensity. A second filter placed at $90^\circ$ absorbs all remaining light.

• Superposition, Phase & Stationary Waves:

  • Phase Difference: $\Delta \phi = 2\pi \times \left(\frac{\Delta x}{\lambda}\right)$ or $2\pi \times \left(\frac{\Delta t}{T}\right)$ (measured in radians or degrees).

  • Stationary Waves: Formed when two progressive waves of identical frequency and amplitude traveling in opposite directions superpose.

    • Nodes: Points of zero amplitude (destructive interference).

    • Antinodes: Points of maximum amplitude (constructive interference).

  • Fundamental Frequency (First Harmonic) on a String: $f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$ (where $T$ is tension and $\mu$ is mass per unit length in $kg/m$).

• Diffraction & Interference:

  • Young's Double Slit Equation: $w = \frac{\lambda D}{s}$ (where $w$ is fringe spacing, $D$ is slit-to-screen distance, $s$ is slit separation).

    • Constructive interference (bright fringe): Path difference = $n \lambda$

    • Destructive interference (dark fringe): Path difference = $\left(n + \frac{1}{2}\right) \lambda$

  • Single Slit Diffraction: Central maximum is double the width of secondary maxima, with intensity dropping rapidly.

  • Diffraction Grating Equation: $d \sin(\theta) = n \lambda$ (where $d = \frac{1}{\text{lines per meter}}$). Maximum order occurs when $\theta = 90^\circ$ ($\sin(\theta) = 1$).

Quantum Physics, Photons & Thermodynamics

• Atomic Energy Levels & Spectra:

  • Electrons occupy discrete energy levels ($N_1, N_2, \dots$). Excitation occurs when an electron absorbs precise photon energy $\Delta E = hf$ or absorbs energy from colliding free electrons.

  • De-excitation releases photons of energy: $\Delta E = E_2 - E_1 = hf = \frac{hc}{\lambda}$

  • Electron-Volt Conversion: $1\,eV = 1.60 \times 10^{-19}\,J$. To convert Joules to $MeV$, divide by $1.60 \times 10^{-13}$

  • Fluorescent Tubes: Free electrons accelerate through mercury vapor, exciting mercury atoms via collisions. Mercury de-excites emitting UV photons. The phosphor coating absorbs UV photons and de-excites via smaller energy steps, emitting visible light photons.

• Photoelectric Effect:

  • Provides evidence for the particle model of light (one-to-one photon-electron interaction).

  • Einstein's Photoelectric Equation: $E_{k,max} = hf - \phi$

  • Work Function ($\phi$): Minimum energy required to liberate an electron from the metal surface.

  • Threshold Frequency ($f_0$): Minimum frequency required for photoemission: $\phi = h f_0$

  • Stopping Potential ($V_s$): $e V_s = E_{k,max}$

• De Broglie Wavelength & Wave-Particle Duality:

  • Particles display wave properties (evidenced by electron diffraction rings through graphite target).

  • De Broglie Equation: $\lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2m E_k}}$

• Thermal Physics & Specific Heat/Latent Heat:

  • Temperature measures mean kinetic energy of particles. During a phase change, temperature remains constant as energy alters particle potential energy.

  • Specific Heat Capacity: $Q = m c \Delta T$

  • Specific Latent Heat: $Q = m L$

  • Thermal Equilibrium: Total energy gained by colder object = Total energy lost by warmer object ($m_1 c_1 (T - T_1) = m_2 c_2 (T_2 - T)$).

  • Absolute Zero: $0\text{ Kelvin} = -273.15^\circ C$. Temperature conversion: $T(K) = \theta(^\circ C) + 273$

• Ideal Gas Laws & Kinetic Theory:

  • Boyle's Law: $P \propto \frac{1}{V}$ (constant $T$).

  • Charles's Law: $V \propto T$ (constant $P$).

  • Pressure Law: $P \propto T$ (constant $V$).

  • Ideal Gas Equation: $PV = nRT = NkT$ (where $n$ = moles, $R = 8.31\,J\,mol^{-1}\,K^{-1}$, $N$ = molecule count, $k$ = Boltzmann constant).

  • Assumptions of Kinetic Theory (RAVED):

    • Random motion of particles.

    • Attraction forces between molecules are zero.

    • Volume of gas molecules is negligible compared to container volume.

    • Elastic collisions.

    • Duration of collisions is negligible compared to time between collisions.

  • Kinetic Theory Pressure Formula: $PV = \frac{1}{3} N m c_{rms}^2 \implies P = \frac{1}{3} \rho c_{rms}^2$

  • Molecular Kinetic Energy: $\frac{1}{2} m c_{rms}^2 = \frac{3}{2} k T$ (Mean kinetic energy of a gas particle depends solely on absolute temperature).

  • Work Done by Expanding Gas: $W = P \Delta V$ (Area under $P\text{-}V$ graph).

Nuclear Physics, Astrophysics & Cosmology

• Nuclear Stability & Binding Energy:

  • Plotting Neutrons ($N$) vs Protons ($Z$) shows stability line curving above $N = Z$ for heavy nuclei. Unstable lighter nuclei decay via $\beta^-$, heavy nuclei via $\alpha$

  • Atomic Mass Unit ($u$): $1\,u = \frac{1}{12}\text{ mass of Carbon-12} = 1.66 \times 10^{-27}\,kg \equiv 931.5\,MeV$

  • Mass Defect ($\Delta m$): Difference between constituent nucleons mass total and bound nucleus mass.

  • Binding Energy: Energy needed to separate a nucleus into constituent nucleons: $E = \Delta m c^2 = \Delta m \times 931.5\,MeV$

  • Binding Energy Curve: Peak average binding energy per nucleon is Iron-56 ($^{56}\text{Fe}$, $\approx 8.8\,MeV/\text{nucleon}$), making it the most stable isotope. Fusion occurs left of Iron-56; Fission occurs to the right.

• Radioactive Decay Kinetics:

  • Activity ($A$): Rate of nuclear decay: $A = \lambda N$ (Units: Becquerel, $Bq = s^{-1}$).

  • Exponential Decay: $A = A_0 e^{-\lambda t}$ and $N = N_0 e^{-\lambda t}$

  • Half-Life ($t_{1/2}$): Time taken for activity or active nuclei count to half:

    • $t_{1/2} = \frac{\ln(2)}{\lambda} \approx \frac{0.693}{\lambda}$

  • Gamma Radiation Inverse-Square Law: Intensity drops with distance $d$: $I = \frac{k}{d^2} \implies I_1 d_1^2 = I_2 d_2^2$

• Stellar Physics & Radiation Laws:

  • Stefan-Boltzmann Law: Total stellar luminosity $L = 4\pi r^2 \sigma T^4$ (where $\sigma = 5.67 \times 10^{-8}\,W\,m^{-2}\,K^{-4}$).

  • Wien's Displacement Law: $\lambda_{peak} T = 2.898 \times 10^{-3}\,m\,K$

  • Hertzsprung-Russell (H-R) Diagram: Plot of Luminosity vs Temperature (decreasing along x-axis). Features Main Sequence, Red Giants (top right), White Dwarfs (bottom left).

• Cosmology & Astronomical Distances:

  • Parsec ($pc$): Distance at which $1\text{ Astronomical Unit (AU)}$ subtends an angle of $1\text{ arcsecond}$ ($\frac{1}{3600}\text{ degree}$).

  • Doppler Redshift: Shift in spectral lines from receding galaxies:

    • $z = \frac{\Delta \lambda}{\lambda} = \frac{\Delta f}{f} = \frac{v}{c}$

  • Hubble's Law: Recessional velocity $v$ is proportional to galaxy distance $d$:

    • $v = H_0 d$

    • Age of Universe $\approx \frac{1}{H_0}$ (when $H_0$ converted to $s^{-1}$).

  • Big Bang Evidence: Cosmic Microwave Background Radiation (CMBR) and galactic redshift.

Experimental Methods, Measurement & Data Analysis

• Instrument Resolution & Techniques:

  • Meter Rule: Resolution = $1\,mm$. Minimize parallax error by positioning eye perpendicular to scale.

  • Vernier Caliper: Resolution = $0.05\,mm$

  • Micrometer Screw Gauge: Resolution = $0.01\,mm$. Calibrate for zero error by closing jaws using ratchet prior to measurement.

• Uncertainty Calculations:

  • Absolute Uncertainty: Equal to resolution of instrument or half range of repeated values ($\frac{\text{Max} - \text{Min}}{2}$).

  • Percentage Uncertainty: $\text{Percentage Uncertainty} = \left(\frac{\text{Absolute Uncertainty}}{\text{Mean Value}}\right) \times 100\%$

  • Combining Uncertainties:

    • Adding/Subtracting Values ($y = a \pm b$): Add absolute uncertainties ($\Delta y = \Delta a + \Delta b$).

    • Multiplying/Dividing Values ($y = a \times b$ or $y = \frac{a}{b}$): Add percentage uncertainties (\%\Delta y = \%\Delta a + \%\Delta b).

    • Power Functions ($y = a^n$): Multiply percentage uncertainty by power $n$ (\%\Delta y = n \times \%\Delta a).

• Graphical Uncertainty & Logarithmic Transformations:

  • Uncertainty in Gradient: $\text{Uncertainty} = |\text{Gradient of Best Fit} - \text{Gradient of Worst Fit}|$

  • Power Law Transformation: If $y = k x^n$, taking logs yields $\ln(y) = n \ln(x) + \ln(k)$. Plotting $\ln(y)$ against $\ln(x)$ gives a straight line with gradient $n$ and y-intercept $\ln(k)$.

  • Exponential Transformation: If $y = k e^{cx}$, taking natural logs yields $\ln(y) = cx + \ln(k)$. Plotting $\ln(y)$ against $x$ gives a straight line with gradient $c$ and y-intercept $\ln(k)$.