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Waves – Complete Formula Revision

Waves – Complete Formula Revision          (JEE Mains) Every Formula | Every Trap | PYQ Mapping 1️⃣ Basic Wave Parameters Wave equation: \[ y(x,t) = A \sin(kx - \omega t + \phi) \] Amplitude = \( A \) Angular frequency = \( \omega = 2\pi f \) Wave number = \( k = \frac{2\pi}{\lambda} \) Wave speed = \( v = \frac{\omega}{k} \) Also: \( v = f\lambda \) ⚠ Trap: Sign in wave equation decides direction. 2️⃣ Speed of Transverse Wave on String \[ v = \sqrt{\frac{T}{\mu}} \] T = tension \( \mu = \frac{m}{L} \) (linear density) ⚠ Increasing tension increases speed, but frequency remains constant. 3️⃣ Longitudinal Sound Waves Speed in medium: \[ v = \sqrt{\frac{B}{\rho}} \] B = Bulk modulus \( \rho \) = density For gas: \[ v = \sqrt{\frac{\gamma P}{\rho}} \] ⚠ Trap: Speed independent of frequency. 4️⃣ Principle of Superposition Resultant displacement: \[ y = y_1 + y_2 \] Resultant amplitude: ...
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GRAVITATION — Complete JEE Mains Revision

GRAVITATION — Complete JEE Mains Revision.  Every Formula | Every Trap | Every Mains Question Type 1. Universal Law of Gravitation Newton’s Law: $$ F = G \frac{m_1 m_2}{r^2} $$ G = $6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$ Key Results Force ∝ product of masses Force ∝ 1 / r² (inverse square) Force always attractive Vector Form: $$ \vec{F} = - G \frac{m_1 m_2}{r^2} \hat{r} $$ Mains Trap: Distance is center-to-center, NOT surface distance. 2. Gravitational Field & Acceleration Due to Gravity Field Definition: $$ g = \frac{F}{m} $$ For planet of mass M: $$ g = \frac{GM}{R^2} $$ Variation with height (h): $$ g_h = \frac{GM}{(R+h)^2} $$ For small h: $$ g_h = g \left(1 - \frac{2h}{R}\right) $$ Variation with depth (d): $$ g_d = g \left(1 - \frac{d}{R}\right) $$ Mains Trap: g decreases linearly with depth but inverse square with height. 3. Gravitational Potential & Potential Energy Potential: $$ V = -\frac{G...
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Moving Charges & Magnetism- Revision Capsule

Moving Charges & Magnetism — StudyBeacon 1️⃣ Magnetic Force on a Moving Charge Lorentz Force: $$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$ In pure magnetic field: $$F = qvB\sin\theta$$ Special Cases: θ = 0° → F = 0 θ = 90° → F = qvB (maximum) Circular Motion: $$qvB = \frac{mv^2}{r}$$ $$r = \frac{mv}{qB}$$ $$T = \frac{2\pi m}{qB}$$ $$f = \frac{qB}{2\pi m}$$ ⚠ Trap: Frequency is independent of velocity. --- 2️⃣ Helical Motion If velocity has components: $$v_\parallel = v\cos\theta$$ $$v_\perp = v\sin\theta$$ Radius: $$r = \frac{mv_\perp}{qB}$$ Pitch: $$p = v_\parallel T$$ --- 3️⃣ Magnetic Force on Current Carrying Conductor $$\vec{F} = I \vec{L} \times \vec{B}$$ Magnitude: $$F = ILB\sin\theta$$ For parallel wires: $$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$ ⚠ Same direction currents → Attraction ⚠ Opposite → Repulsion --- 4️⃣ Biot–Savart Law $$dB = \frac{\mu_0}{4\pi}\frac{Idl\sin\theta}{r^2}$$ Circular loo...
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Current Electricity – Complete Formula, Laws & PYQ Profiling

Current Electricity – Complete Formula, Laws & PYQ Profiling (JEE Main Only) 1. Electric Current & Drift Velocity Electric current: I = Q / t Current density: J = nqv d Drift velocity: v d = eEτ / m Relation: I = nAev d Mobility: μ = v d / E Conductivity: σ = nqμ Resistivity: ρ = 1 / σ JEE Trap: Drift velocity is extremely small, but electrical signal propagates almost at speed of light. 2. Ohm’s Law & Resistance Ohm’s law: V = IR Resistance: R = ρL / A Temperature dependence: R = R₀(1 + αΔT) Combination of resistors: Series: R eq = R₁ + R₂ + … Parallel: 1/R eq = 1/R₁ + 1/R₂ + … PYQ Trap: For same material & volume, R ∝ L² (since A ∝ 1/L). 3. Electric Power & Heating Effect P = VI = I²R = V² / R Electrical energy: E = Pt Joule’s law of heating: H = I²Rt 4. EMF & Internal Resistance Terminal voltage (discharging): V = E − Ir Terminal voltage (charging): V = E + Ir Current: I = E / (R + r) M...
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Capacitors – Complete Formula, Traps & Question Profiling

Capacitors – Complete Formula, Traps & Question Profiling (JEE Main Only) 1. Basic Concept of Capacitor Capacitor: Device to store electric charge Consists of two conductors separated by dielectric C = Q / V SI unit → Farad (F) 1 μF = 10⁻⁶ F 2. Parallel Plate Capacitor C = ε₀ A / d A → area of plates d → separation With dielectric (k): C = k ε₀ A / d 3. Effect of Dielectric Capacitance increases k times Electric field reduces: E = E₀ / k Potential reduces: V = V₀ / k Dielectric constant: k = ε / ε₀ 4. Series & Parallel Combination Series: 1/C eq = 1/C₁ + 1/C₂ + ... Charge same on each capacitor V distributes Parallel: C eq = C₁ + C₂ + ... Potential same Charge distributes 5. Energy Stored in Capacitor U = ½ C V² U = Q² / (2C) U = ½ QV Energy density: u = ½ ε E² 6. Capacitor with Battery Connected / Disconnected Battery connected: V constant Q increases with dielectric Energy increas...
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Oscillations – Complete Formula & Question Profiling

Oscillations – Complete Formula & Question Profiling (JEE Main Only) 1. Periodic & Oscillatory Motion Periodic motion: Repeats itself after equal intervals of time Oscillatory motion: To-and-fro about mean position All oscillatory motions are periodic (not vice-versa) 2. Simple Harmonic Motion (SHM) Defining condition: a ∝ −x a = −ω²x Restoring force always towards mean position 3. SHM Kinematics Displacement: x = A sin(ωt + φ) Velocity: v = ω√(A² − x²) Acceleration: a = −ω²x Maximum values: v max = ωA a max = ω²A 4. Time Period & Frequency T = 2π / ω f = 1 / T Independent of amplitude (for SHM) 5. Mass–Spring System ω = √(k / m) T = 2π√(m / k) Horizontal and vertical springs → same T Extension due to gravity does not affect T 6. Simple Pendulum T = 2π√(l / g) Valid for small angular displacement Independent of mass Effective length change: In lift accelerating upward → g' = g + ...
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Mechanical Properties of Fluids – Complete Formula Sheet physics

Mechanical Properties of Fluids – Complete Formula Sheet (JEE Main Only) 1. Basic Definitions Fluid: Substance that can flow (liquids + gases) Density (ρ): Mass / Volume Relative density: Density of substance / Density of water ρ = m / V 2. Pressure in Fluids Pressure: P = F / A Pressure at depth h: P = P₀ + ρgh Independent of container shape Same at same depth 3. Pascal’s Law Pressure applied to confined fluid is transmitted equally Hydraulic lift: F₂ / F₁ = A₂ / A₁ 4. Buoyancy & Archimedes’ Principle Buoyant force: F B = ρ fluid g V displaced Acts vertically upward Independent of depth 5. Floatation Conditions Floating body: Weight = Buoyant force Fraction submerged = ρ body / ρ fluid Apparent weight: W apparent = mg − F B 6. Continuity Equation A₁v₁ = A₂v₂ Valid for incompressible fluids Velocity ↑ when area ↓ 7. Bernoulli’s Theorem P + ½ρv² + ρgh = constant Valid for steady, incompressi...
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Thermal Properties of Matter – Complete Formula Sheet

Thermal Properties of Matter – Complete Formula Sheet (JEE Main Only) 1. Temperature & Heat Temperature: Measure of degree of hotness Heat: Energy transferred due to temperature difference Heat always flows: Higher T → Lower T SI unit of heat: Joule (J) 2. Temperature Scales K = °C + 273 °C = (5/9)(°F − 32) Zero points: 0 K → Absolute zero Triple point of water = 273.16 K 3. Thermal Expansion (a) Linear Expansion ΔL = α L ΔT (b) Areal Expansion ΔA = β A ΔT    (β = 2α) (c) Volume Expansion ΔV = γ V ΔT    (γ = 3α) 4. Expansion of Solids – PYQ Traps Hole expands as if material absent α is same in all directions (isotropic solid) If expansion prevented → thermal stress develops Thermal stress: σ = Y α ΔT 5. Calorimetry Heat absorbed/released: Q = m c ΔT c → specific heat capacity Water has maximum specific heat Principle: Heat lost = Heat gained 6. Latent Heat Q = m L L → latent heat Temper...
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Mechanical Properties of Solids – Complete Formula Sheet

Mechanical Properties of Solids – Complete Formula Sheet (JEE Main Only) 1. Elasticity – Basic Terms Stress (σ) = Force / Area Strain (ε) = Change / Original dimension Elastic limit → Max stress up to which Hooke’s law holds Plastic region → Permanent deformation σ = F / A ε = ΔL / L 2. Types of Stress & Strain Longitudinal stress (Tensile / Compressive) Shearing stress Bulk stress Longitudinal strain = ΔL / L Shear strain = θ (radian) Volumetric strain = ΔV / V 3. Hooke’s Law Stress ∝ Strain σ = Yε τ = Gθ P = K(ΔV / V) 4. Elastic Moduli Young’s Modulus: Y = (FL) / (AΔL) Bulk Modulus: K = −P / (ΔV / V) Shear Modulus: G = τ / θ 5. Poisson’s Ratio μ = Lateral strain / Longitudinal strain −1 ≤ μ ≤ 0.5 Rubber → μ ≈ 0.5 6. Relations between Elastic Constants Y = 2G(1 + μ) Y = 3K(1 − 2μ) G = Y / [2(1 + μ)] K = Y / [3(1 − 2μ)] 7. Stress–Strain Curve Proportional limit Elastic limit Yield point Ultimate tensile stress ...
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