All Physics Formulas for JEE Advanced 2026: Complete List for Easy Reference

UNITS & DIMENSIONS

Fundamental SI Units:

• Length → Metre (m)

• Mass → Kilogram (kg)

• Time → Second (s)

• Electric Current → Ampere (A)

• Temperature → Kelvin (K)

• Luminous Intensity → Candela (cd)

• Amount of Substance → Mole (mol)

Supplementary Units:

• Plane Angle → Radian (rad)

• Solid Angle → Steradian (sr)

Metric Prefixes:

• Centi (c) = 10⁻²

• Milli (m) = 10⁻³

• Micro (μ) = 10⁻⁶

• Nano (n) = 10⁻⁹

• Pico (p) = 10⁻¹²

• Kilo (K) = 10³

• Mega (M) = 10⁶

RECTILINEAR MOTION

Velocity & Acceleration Definitions:

v_avg = Δr / Δt = (r_f − r_i) / t

Average Speed = Total distance / Total time

v_inst = lim(Δt→0) Δr/Δt = dr/dt

a_avg = Δv / Δt = (v_f − v_i) / Δt

a_inst = dv/dt = d²x/dt²

Equations of Motion (Constant Acceleration):

(1) v = u + at (2) s = ut + ½at² (3) v² = u² + 2as (4) s = [(u + v)/2] × t (5) S_n = u + (a/2)(2n − 1) [displacement in nth second]

Also: s = vt − ½at² Also: x = x₀ + ut + ½at²

Free Fall (u = 0, taking downward as positive):

v = gt s = ½gt² v² = 2gs S_n = (g/2)(2n − 1) h = h₀ − ½gt² (if upward is positive)

Graphs in Uniform Acceleration:

• x–t graph: Parabola (x is quadratic in t)

• v–t graph: Straight line with slope = a

• a–t graph: Horizontal line (a = constant)

Maxima & Minima:

dy/dx = 0 and d²y/dx² < 0 → Maximum dy/dx = 0 and d²y/dx² > 0 → Minimum

PROJECTILE MOTION & VECTORS

Standard Projectile (from ground, angle θ):

Time of flight: T = 2u sinθ / g Horizontal range: R = u² sin2θ / g Maximum height: H = u² sin²θ / 2g

Trajectory Equation:

y = x tanθ − gx² / (2u² cos²θ) y = x tanθ (1 − x/R)

Projection on an Inclined Plane (inclination β, α = angle w.r.t. incline):

Up the incline: Range = 2u² sin(α) cos(α+β) / [g cos²β] Time of flight = 2u sinα / [g cosβ] Angle for max range = π/4 − β/2 Max range = u² / [g(1 + sinβ)]

Down the incline: Range = 2u² sin(α) cos(α−β) / [g cos²β] Time of flight = 2u sinα / [g cosβ] Angle for max range = π/4 + β/2 Max range = u² / [g(1 − sinβ)]

RELATIVE MOTION

River Crossing — Shortest Time:

V_x = V_R (along river) V_y = V_mR (perpendicular, man's velocity w.r.t. river) Net speed: V_m = √(V_mR² + V_R²) Time = d / V_mR

River Crossing — Shortest Path (Zero Drift):

V_x = 0 → sinθ = V_R / V_mR V_y = √(V_mR² − V_R²) Time = d / √(V_mR² − V_R²) Condition: V_mR > V_R

NEWTON'S LAWS OF MOTION

Weighing Machine: Reads the normal force on its surface, NOT weight.

Spring Force:

F = −kx (x = deformation from natural length) Spring property: kℓ = constant

If spring is cut in ratio m : n: ℓ₁ = mℓ/(m+n), ℓ₂ = nℓ/(m+n) k₁ℓ₁ = k₂ℓ₂ = kℓ

Spring Combinations:

Series: 1/k_eq = 1/k₁ + 1/k₂ + ... Parallel: k_eq = k₁ + k₂ + k₃ + …

Spring Balance: Reads the force exerted by the object at the hook.

Atwood Machine:

a = (m₁ − m₂)g / (m₁ + m₂) T = 2m₁m₂g / (m₁ + m₂)

Wedge Constraint: Components of velocity along the perpendicular to the contact surface are equal if bodies remain in contact (no deformation).

FRICTION

Kinetic Friction:

f_k = μ_k × N μ_k depends on the nature of the two surfaces in contact.

Static Friction:

0 ≤ f_s ≤ f_s,max f_s,max = μ_s × N

Static friction is variable and self-adjusting. It exists when there is a tendency of relative motion but no actual relative motion.

WORK, POWER & ENERGY

Kinetic Energy & Momentum Relation:

K = p²/(2m) p = √(2mK) p = linear momentum

Conservative Force:

F = −dU/dr

Work-Energy Theorem:

W_C + W_NC + W_PS = ΔK

Since W_C = −ΔU: W_NC + W_PS = ΔK + ΔU = ΔE

(W_C = conservative, W_NC = non-conservative, W_PS = pseudo)

CIRCULAR MOTION

Angular Kinematics:

ω_avg = Δθ / Δt ω_inst = dθ/dt α_avg = Δω / Δt α_inst = dω/dt = ω(dω/dθ)

For constant α: ω = ω₀ + αt θ = ω₀t + ½αt² ω² = ω₀² + 2αθ

Radius of Curvature:

R = v² / a_⊥ = mv² / F_⊥

If y = f(x): R = [1 + (dy/dx)²]^(3/2) / |d²y/dx²|

Normal Reaction on Bridges:

Concave bridge: N = mg cosθ + mv²/r Convex bridge: N = mg cosθ − mv²/r

Skidding on Level Road:

v_safe = √(μgr)

Skidding on Rotating Platform:

ω_max = √(μg/r)

Bending of Cyclist:

tanθ = v²/(rg)

Banking of Road (without friction):

tanθ = v²/(rg)

Banking of Road (with friction):

v² = rg(μ + tanθ)/(1 − μ tanθ)

Maximum & Minimum Safe Speeds on Banked Road:

v_max = √[rg(tanθ + μ)/(1 − μ tanθ)] v_min = √[rg(tanθ − μ)/(1 + μ tanθ)]

Centrifugal Force (Pseudo):

f = mω²r (acts outward in rotating frame)

Effect of Earth's Rotation on Apparent Weight:

N = mg − mRω² cos²λ (λ = latitude)

Vertical Circular Motion (String/Smooth Track): Critical conditions apply — speed and tension vary with position. At the top of the loop: minimum speed v² = gR for the string.

Conical Pendulum:

T cosθ = mg T sinθ = mω²r Time period = 2π√(L cosθ / g)

CENTRE OF MASS & COLLISIONS

Mass Moment:

M = mr

Centre of Mass — Continuous Distribution:

x_cm = ∫x dm / M y_cm = ∫y dm / M z_cm = ∫z dm / M

COM of Common Bodies:

Two point masses: m₁r₁ = m₂r₂ (closer to heavier mass) Rectangular plate: x_c = L/2, y_c = b/2 Triangular plate: y_c = h/3 (at centroid) Semi-circular ring: y_c = 2R/π, x_c = 0 Semi-circular disc: y_c = 4R/(3π), x_c = 0 Hemispherical shell: y_c = R/2, x_c = 0 Solid hemisphere: y_c = 3R/8, x_c = 0 Solid cone: y_c = h/4 (from base) Hollow cone: y_c = h/3 (from base)

Important points:

• Gravitational force and spring force are always non-impulsive.

• An impulsive force can only be balanced by another impulsive force.

Coefficient of Restitution (e):

e = (Velocity of separation along line of impact) / (Velocity of approach along line of impact) e = (Impulse of reformation) / (Impulse of deformation)

e = 1 → Perfectly elastic collision (KE may be conserved) e = 0 → Perfectly inelastic collision (KE not conserved, bodies stick) 0 < e < 1 → Inelastic collision (KE not conserved)

Variable Mass System:

Thrust force: F_t = v_rel × (dm/dt)

Rocket Propulsion (gravity ignored, u = 0):

v = v_r ln(m₀/m)

RIGID BODY DYNAMICS

Moment of Inertia:

Single particle: I = mr² System of particles: I = Σm_i r_i² Continuous body: I = ∫r² dm Composite body: I = ∫dI_element

SI unit: kg·m²

Perpendicular Axis Theorem (2D lamina only):

I_z = I_x + I_y (object in x-y plane)

Parallel Axis Theorem (any object):

I_AB = I_cm + Md²

Radius of Gyration:

I = MK²

Rotational Kinetic Energy:

KE_rot = ½Iω²

Combined Translation + Rotation:

Total KE = ½Mv_cm² + ½I_cm ω² F⃗_ext = Ma⃗_cm τ_cm = I_cm × α P⃗ = Mv⃗_cm

Net external force has two parts: Tangential: F_t = ma_t = mαr Centripetal: F_c = mv²/r = mω²r

Rotational Equilibrium:

ΣF_x = 0 ΣF_y = 0 Στ_z = 0

Angular Momentum:

For a particle about a point: L = rp sinφ = r_⊥ p = r p_⊥ L⃗ = r⃗ × p⃗

Rigid body about fixed axis H: L_H = I_H × ω

Conservation of Angular Momentum:

If τ_ext = 0 → L = constant

Relation between Torque and Angular Momentum:

τ⃗ = dL⃗/dt

Impulse of Torque:

J = ∫τ dt = ΔL (change in angular momentum)

Angular Momentum about any axis AB:

L_AB = I_cm ω + r_cm × Mv_cm

For rigid body velocities:

V_P² = V_Q² + (ωr)² + 2V_Q(ωr)cosθ

α, ω are the same at every point of the body (or any rigidly attached point outside).

SIMPLE HARMONIC MOTION

SHM Basics:

Restoring force: F = −kx General equation: x = A sin(ωt + φ) (ωt + φ) is the phase, φ is the initial phase

Angular Frequency & Time Period:

ω = 2π/T = 2πf T = 2π/ω = 2π√(m/k)

Velocity:

v = ω√(A² − x²)

Acceleration:

a = −ω²x

Kinetic Energy:

KE = ½mω²(A² − x²) = ½k(A² − x²)

Potential Energy:

PE = ½kx²

Total Mechanical Energy:

TME = KE + PE = ½kA² = constant

Spring-Mass Systems:

(i) Single spring: T = 2π√(m/k) (ii) Two-body oscillator: T = 2π√(μ/k) where μ = m₁m₂/(m₁ + m₂) = reduced mass

Superposition of Two SHMs (same direction, same frequency):

x₁ = A₁ sin(ωt), x₂ = A₂ sin(ωt + δ) Resultant: x = A sin(ωt + φ)

A = √(A₁² + A₂² + 2A₁A₂ cosδ) tanφ = A₂ sinδ / (A₁ + A₂ cosδ)

WAVES ON STRINGS

General Wave Equation:

∂²y/∂t² = v² × ∂²y/∂x²

y(x,t) = f(t − x/v) → wave in +x direction y(x,t) = f(t + x/v) → wave in −x direction

Sinusoidal: y = A sin(ωt − kx) [+x direction] y = A sin(ωt + kx) [−x direction]

Wave Parameters:

Wave number: k = 2π/λ = ω/v (rad/m) Phase: (ωt − kx ± φ) Phase difference: Δφ = (2π/λ)Δx = (2π/T)Δt

Speed on String:

v = √(T/μ) (T = tension, μ = mass per unit length)

Power & Intensity:

Average power: P = 2π²f²A²μv Intensity: I = P/S = 2π²f²A²ρv

Reflection & Transmission of Waves:

Incident: y_i = A sin(ωt − k₁x)

Rarer → Denser (v₂ < v₁): y_r = A_r sin(ωt + k₁x) [reflected, phase change] y_t = A_t sin(ωt − k₂x) [transmitted]

Denser → Rarer (v₂ > v₁): y_r = A_r sin(ωt + k₁x) [reflected, no phase change] y_t = A_t sin(ωt − k₂x)

Amplitude relations: A_r = [(k₁ − k₂)/(k₁ + k₂)] × A_i A_t = [2k₁/(k₁ + k₂)] × A_i

Standing / Stationary Waves:

y₁ = A sin(ωt − kx), y₂ = A sin(ωt + kx) y₁ + y₂ = 2A cos(kx) sin(ωt)

Resultant amplitude at position x = 2A|cos(kx)|

Nodes: positions where amplitude = 0 Antinodes: positions where amplitude = 2A

Distance between successive nodes = λ/2 Distance between successive antinodes = λ/2 Distance between adjacent node and antinode = λ/4

All particles in the same segment (between 2 nodes) vibrate in the same phase. Particles in 2 consecutive segments vibrate in opposite phase. Energy cannot be transmitted across nodes.

Vibrations of Strings — Fixed at Both Ends:

λ = 2L/n (n = 1, 2, 3, ...) f_n = (n/2L)√(T/μ) n = number of loops (harmonics)

String Fixed at One End, Free at Other (Odd Harmonics Only):

Fundamental: λ = 4L, f₁ = (1/4L)√(T/μ) nth overtone: f = [(2n+1)/4L]√(T/μ) (n = 0, 1, 2, ...)

General: f_n = [(2n+1)/(4L)]√(T/μ)

SOUND WAVES

Displacement & Pressure Waves:

Displacement: ξ = A sin(ωt − kx) Pressure excess: P_ex = −B(∂ξ/∂x) = BAk cos(ωt − kx) Pressure amplitude = BAk

Speed of Sound:

General: C = √(E/ρ) (E = elastic modulus) Solid: C = √(Y/ρ) Liquid: C = √(B/ρ) Gas (Laplace correction): C = √(γP/ρ) = √(γRT/M₀)

Intensity & Loudness:

Intensity: I = 2π²f²A²ρv = P_m²/(2ρv) Loudness: L = 10 log₁₀(I/I₀) dB (I₀ = 10⁻¹² W/m²) Point source: I = P/(4πr²)

Interference of Sound:

Resultant pressure amplitude: p₀² = p_m1² + p_m2² + 2p_m1 p_m2 cos(Δφ)

Resultant intensity: I = I₁ + I₂ + 2√(I₁I₂) cos(Δφ)

Phase difference from path difference: Δφ = (2π/λ)Δx

Constructive: Δx = nλ, Δφ = 2nπ Destructive: Δx = (2n+1)λ/2, Δφ = (2n+1)π

If p_m1 = p_m2 and Δφ = 0: p₀ = 2p_m, I = 4I₁ If p_m1 = p_m2 and Δφ = π: p₀ = 0, I = 0 (no sound)

Closed Organ Pipe (one end closed):

f = (2n+1)v/(4L) (only odd harmonics: fundamental, 3rd, 5th, ...)

Open Organ Pipe (both ends open):

f = nv/(2L) (all harmonics)

Beats:

Beat frequency = |f₁ − f₂|

Doppler Effect:

f = f₀ × (V − V_observer) / (V − V_source)

Apparent wavelength: λ' = (V − V_S) / f₀

Sign convention: All velocities measured along the source-to-observer line. Velocities towards each other are positive in the numerator (observer) and the denominator (source), respectively.

HEAT & THERMODYNAMICS

Kinetic Theory of Gases:

Total translational KE of gas = (3/2)nRT = (3/2)PV

Pressure: P = (1/3)ρv_rms²

v_rms = √(3RT/M) = √(3KT/m) v_avg = √(8KT/πm) ≈ 1.59√(KT/m) v_most probable = √(2KT/m) ≈ 1.41√(KT/m)

Relation: v_mp < v_avg < v_rms

Degrees of Freedom:

Monoatomic: f = 3 Diatomic: f = 5 Polyatomic: f = 6

Maxwell's Law of Equipartition:

Total KE per molecule = (f/2)KT

Internal Energy of an Ideal Gas:

U = (f/2)nRT

Isothermal Process (T = const, ΔU = 0):

W = nRT ln(V_f/V_i) = 2.303 nRT log₁₀(V_f/V_i) ΔU = 0 Q = W

Isochoric Process (V = const, dW = 0):

dW = 0 ΔU = nC_v ΔT = (f/2)nRΔT Q = ΔU

Isobaric Process (P = const):

W = nRΔT = nR(T_f − T_i) ΔU = nC_v ΔT Q = ΔU + W = nC_p ΔT

Adiabatic Process (Q = 0):

PV^γ = constant TV^(γ−1) = constant W = nR(T_i − T_f)/(γ − 1) = (P_i V_i − P_f V_f)/(γ − 1)

Cyclic Process:

ΔU = 0 over complete cycle ΔQ = ΔW

Specific Heats:

C_v = (f/2)R C_p = (f/2 + 1)R γ = C_p/C_v = 1 + 2/f

Mayer's relation: C_p − C_v = R (for ideal gas only)

Monoatomic: γ = 5/3 ≈ 1.67 Diatomic: γ = 7/5 = 1.4 Triatomic: γ = 4/3 ≈ 1.33

Mixture of Non-Reacting Gases:

Mol. wt. = (n₁M₁ + n₂M₂) / (n₁ + n₂) C_v,mix = (n₁C_v1 + n₂C_v2) / (n₁ + n₂) C_p,mix = (n₁C_p1 + n₂C_p2) / (n₁ + n₂) γ_mix = C_p,mix / C_v,mix

Thermometers:

Liquid: T = [(ℓ − ℓ₀)/(ℓ₁₀₀ − ℓ₀)] × 100 Gas (constant volume): T = [(P − P₀)/(P₁₀₀ − P₀)] × 100; P = P₀ + ρgh Gas (constant pressure): T/T₀ = V/V₀ Electrical resistance: T = [(R − R₀)/(R₁₀₀ − R₀)] × 100

Thermal Expansion:

Linear: L = L₀(1 + αΔT) Area: A = A₀(1 + βΔT), β = 2α Volume: V = V₀(1 + γΔT), γ = 3α

Thermal Stress & Stored Energy:

Thermal stress: F/A = YαΔT Energy stored per unit volume = ½Y(αΔT)² Energy stored = ½AY(ΔL)²/L

Pendulum Clock Variation:

ΔT/T = ½αΔT T > T₀ → clock runs slow (time loss) T < T₀ → clock runs fast (time gain)

Calorimetry:

Specific heat: S = Q/(mΔT) Molar specific heat: C = Q/(nΔT) Water equivalent = m_w × S_w

Heat Transfer — Conduction:

dQ/dt = KA(dT/dx) Thermal resistance: R_th = ℓ/(KA)

Series (same A): ℓ_eq/K_eq = ℓ₁/K₁ + ℓ₂/K₂ + ... Parallel (same ℓ): K_eq A_eq = K₁A₁ + K₂A₂ + …

Absorption, Reflection, Transmission:

r + t + a = 1

Radiation:

Emissive power: E = ΔU/(AΔt) Spectral emissive power: E_λ = dE/dλ Emissivity: e = E(body at T) / E(blackbody at T)

Kirchhoff's Law:

E(body) / E(blackbody) = a(body) [at same temperature]

Wien's Displacement Law:

λ_max × T = b (b = 0.282 cm·K)

Stefan-Boltzmann Law:

u = σT⁴ (σ = 5.67 × 10⁻⁸ W/m²K⁴) Net radiation: u = eσA(T⁴ − T₀⁴)

Newton's Law of Cooling:

dθ/dt = −k(θ − θ₀) θ = θ₀ + (θ_i − θ₀)e^(−kt)

ELECTROSTATICS

Coulomb's Law:

F⃗ = (1/4πε₀) × (q₁q₂/r²) × r̂ = Kq₁q₂/r² × r̂

Electric Field:

E⃗ = F⃗/q₀ Force on charge q in field E⃗: F⃗ = qE⃗

Electric Potential:

V_P = W_ext(∞→P) / q₀ V_A − V_B = −∫(A→B) E⃗ · dr⃗ E = −dV/dr = −∇V

I. Point Charge:

E = Kq/r² V = Kq/r

II. Infinite Line Charge (λ C/m):

E = 2Kλ/r = λ/(2πε₀r) (radially outward) V not defined absolutely V_A − V_B = 2Kλ ln(r_B/r_A)

III. Infinite Non-conducting Thin Sheet (σ):

E = σ/(2ε₀) (both sides, normal to surface) V_B − V_A = −σ(r_B − r_A)/(2ε₀)

IV. Uniformly Charged Ring (Q, radius R, at distance x on axis):

E_axis = KQx / (R² + x²)^(3/2) E_centre = 0 V_axis = KQ / √(R² + x²) V_centre = KQ/R

V. Infinite Conducting Sheet (σ):

E = σ/ε₀ (one side only)

VI. Uniformly Charged Hollow/Solid Conducting Sphere:

r ≥ R: E = KQ/r², V = KQ/r r < R: E = 0, V = KQ/R (constant inside)

VII. Uniformly Charged Solid Non-conducting Sphere:

r ≥ R: E = KQ/r², V = KQ/r r ≤ R: E = KQr/R³ = ρr/(3ε₀) V = (Kρ/6ε₀)(3R² − r²) [can also be written as K(Q/2R³)(3R² − r²)]

VIII. Thin Uniformly Charged Disc (σ, radius R):

E_axis = (σ/2ε₀)[1 − x/√(R² + x²)] V_axis = (σ/2ε₀)[√(R² + x²) − x]

Work Done by External Agent:

W_ext(A→B) = q(V_B − V_A)

Electrostatic Potential Energy:

U = qV (PE of point charge in external field) U_system = Σ(all pairs) Kq_i q_j / r_ij = (1/2)Σ q_i V_i

Energy Density:

u = ½ε₀E²

Self Energy:

Uniformly charged shell: U_self = KQ²/(2R) Uniformly charged solid non-conducting sphere: U_self = 3KQ²/(5R)

Electric Dipole (dipole moment P⃗ = qd⃗):

On axis: E = 2KP/r³ On equator: E = KP/r³ General point (r, θ): E = (KP/r³)√(1 + 3cos²θ)

Potential: V = KP cosθ / r² = K(P⃗ · r̂)/r²

In uniform electric field: Torque: τ⃗ = P⃗ × E⃗ Net force: F = 0 PE: U = −P⃗ · E⃗

In non-uniform electric field: τ⃗ = P⃗ × E⃗ Net force: F = P(∂E/∂r) U = −P⃗ · E⃗

Gauss's Law:

φ = ∮E⃗ · dA⃗ = q_enclosed / ε₀

Electric flux: φ = ∫E⃗ · dA⃗ = ∫E dA cosθ

Near Conductor Surface:

E = σ/ε₀

Electric Pressure on Conductor Surface:

P = σ²/(2ε₀)

Relation between E and V (Gradient):

E_x = −∂V/∂x, E_y = −∂V/∂y, E_z = −∂V/∂z E⃗ = −∇V = −(∂V/∂x)î − (∂V/∂y)ĵ − (∂V/∂z)k̂

CURRENT ELECTRICITY

Electric Current:

I_avg = Δq/Δt I_inst = dq/dt

Current in a Conductor:

I = neAv_d v_d = eEτ/m (τ = relaxation time)

Current Density:

J = I/A = n⃗ × (dI/dS) = σE

Resistance & Ohm's Law:

ρ = m/(ne²τ) (resistivity) σ = 1/ρ = ne²τ/m (conductivity) R = ρℓ/A V = IR (Ohm's law)

Units: R in ohm (Ω), ρ in ohm-meter (Ω·m), σ in siemens (Ω⁻¹m⁻¹)

Temperature Dependence:

R = R₀(1 + αΔT)

Current through Resistance:

I = (V₂ − V₁)/R

Electrical Power:

P = VI = I²R = V²/R

Energy & Heat:

Energy = ∫P dt H = VIt = I²Rt = V²t/R Joule to calorie: H(cal) = I²Rt / 4.2

Kirchhoff's Laws:

KCL (Junction law): ΣI_in = ΣI_out KVL (Loop law): ΣIR + ΣEMF = 0

Combination of Resistances:

Series: R_eq = R₁ + R₂ + R₃ + ... + R_n (R_eq is greater than any individual resistor) V = V₁ + V₂ + V₃ + ... + V_n Voltage divider: V_i = V × R_i / R_eq

Parallel: 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ + ... Current divider: I_i = I × R_eq/R_i

Wheatstone Network:

Balance condition (zero galvanometer current): P/Q = R/S → PS = QR

Grouping of Cells — Series:

E_eq = E₁ + E₂ + ... + E_N (with polarity signs) r_eq = r₁ + r₂ + ... + r_n

Grouping of Cells — Parallel:

E_eq = (E₁/r₁ + E₂/r₂ + ... + E_n/r_n) / (1/r₁ + 1/r₂ + ... + 1/r_n) 1/r_eq = 1/r₁ + 1/r₂ + ... + 1/r_n

Ammeter (Galvanometer + Shunt S in parallel):

I_G × R_G = (I − I_G) × S S = I_G R_G / (I − I_G) Ideal ammeter has zero resistance.

Voltmeter (Galvanometer + High R_S in series):

V = I_G(R_G + R_S) R_S = V/I_G − R_G. An ideal voltmeter has infinite resistance.

Potentiometer:

I = E/(r + R) V_AB = ER/(R + r) Potential gradient: x = V_AB/L = ER/[(R + r)L]

Applications: (a) Comparing EMFs: E₁/E₂ = ℓ₁/ℓ₂ (b) Finding current (if R known): I = xℓ/R (c) Finding internal resistance of cell: E = xℓ₁ (open circuit balance length) IR = xℓ₂ (closed circuit balance length) r = R(ℓ₁ − ℓ₂)/ℓ₂

A potentiometer is an ideal voltmeter — it draws zero current at the balance point. It can be used to calibrate the ammeter and voltmeter.

Metre Bridge:

X = R(100 − ℓ)/ℓ (Based on balanced Wheatstone bridge: P/Q = R/X, where P ∝ ℓ, Q ∝ (100−ℓ))

CAPACITANCE

Basic Relations:

q = CV q = charge on positive plate C = capacitance V = potential difference between plates

Energy Stored:

U = ½CV² = ½QV = Q²/(2C)

Energy Density:

u = ½ε₀E² = ½Kε₀E² (K = dielectric constant = ε_r)

Types of Capacitors:

(a) Parallel plate: C = Kε₀A/d A = area of plates, d = distance between plates

(b) Isolated spherical conductor (hollow or solid): C = 4πε₀R (R = radius)

(c) Spherical capacitor: C = 4πε₀(ab)/(b − a) With dielectric: C = 4πKε₀(ab)/(b − a)

(d) Cylindrical capacitor (per unit length, ℓ >> a,b): C/ℓ = 2πε₀/ln(b/a)

Capacitance depends on: (a) Area of plates, (b) Distance between plates, (c) Dielectric medium between plates

Electric Field Between Plates:

E = V/d = σ/ε₀

Force on Any Plate:

F = q²/(2ε₀A) = σ²A/(2ε₀)

Connecting Two Charged Capacitors (C₁, V₁ and C₂, V₂):

Common potential: V = (C₁V₁ + C₂V₂)/(C₁ + C₂) = (Q₁ + Q₂)/(C₁ + C₂) Q₁' = C₁V = C₁(Q₁ + Q₂)/(C₁ + C₂) Q₂' = C₂V = C₂(Q₁ + Q₂)/(C₁ + C₂) Heat loss = ½ × C₁C₂(V₁ − V₂)²/(C₁ + C₂)

Series Combination:

1/C_eq = 1/C₁ + 1/C₂ + 1/C₃ + ... Voltage ratio: V₁:V₂:V₃ = 1/C₁ : 1/C₂ : 1/C₃

Parallel Combination:

C_eq = C₁ + C₂ + C₃ + ... Charge ratio: Q₁:Q₂:Q₃ = C₁:C₂:C₃

Charging of Capacitor (Initially Uncharged):

q = q₀(1 − e^(−t/τ)) q₀ = CV (steady state charge) τ = CR_eq (time constant) I = (q₀/τ)e^(−t/τ) = (V/R)e^(−t/τ)

Discharging of Capacitor:

q = q₀ e^(−t/τ) q₀ = initial charge I = −(q₀/τ)e^(−t/τ)

Capacitor with Dielectric:

C = KC₀ = Kε₀A/d (C₀ = capacitance without dielectric) E_inside = E₀/K = σ/(Kε₀) = V/d E₀ = σ/ε₀ (field without dielectric) Bound (induced) charge density: σ_b = σ(1 − 1/K)

Force on Dielectric:

Battery connected: F = ε₀b(K − 1)V²/(2d) Battery disconnected: F = Q²(dC/dx)/(2C²) Force = 0 when dielectric is fully inside.

MAGNETIC EFFECTS OF CURRENT & MAGNETIC FORCE

Magnetic Field due to Moving Point Charge:

B⃗ = (μ₀/4π) × q(v⃗ × r̂)/r²

Biot-Savart Law:

dB⃗ = (μ₀/4π) × I(dℓ⃗ × r̂)/r²

Magnetic Field — Straight Wire (Finite):

B = (μ₀I/4πr)(sinα₁ + sinα₂)

Magnetic Field — Infinite Straight Wire:

B = μ₀I/(2πr)

Circular Loop — At Centre:

B = μ₀NI/(2r)

Circular Loop — On Axis (distance x):

B = μ₀NIR²/[2(R² + x²)^(3/2)]

Solenoid — On Axis:

B = (μ₀nI/2)(cosθ₁ − cosθ₂) Infinite solenoid: B = μ₀nI (inside), B = 0 (outside)

Long Cylindrical Shell (radius R):

r < R: B = 0 r ≥ R: B = μ₀I/(2πr)

When v ⊥ B (circular motion):

r = mv/(qB) T = 2πm/(qB)

General Case (v at angle α to B — helical motion):

r = mv sinα/(qB) T = 2πm/(qB) Pitch = 2πmv cosα/(qB)

Force on Current-Carrying Wire:

F⃗ = I(ℓ⃗ × B⃗)

Magnetic Moment of Current Loop:

M⃗ = NIA⃗

Torque on Loop:

τ⃗ = M⃗ × B⃗

Magnetic Field due to a Single Pole:

B = (μ₀/4π)(m/r²)

Magnetic Field on Axis of Magnet:

B = (μ₀/4π)(2m/r³)

Magnetic Field on Equatorial Axis:

B = (μ₀/4π)(m/r³)

General Point (r, θ):

B = (μ₀m/4πr³)√(1 + 3cos²θ)

ELECTROMAGNETIC INDUCTION

Magnetic Flux:

φ = ∫B⃗ · dA⃗

Faraday's Law:

EMF = −dφ/dt

Lenz's Law:

Induced EMF opposes the cause producing it (conservation of energy).

Motional EMF:

Conducting rod rotating about one end: ε = ½Bωℓ²

EMF in Rotating Disc:

ε = ½Bωr² (between centre and edge)

Fixed Loop in Varying Magnetic Field:

If dB/dt ≠ = 0, the induced electric field along a circle of radius r: E = −(r/2)(dB/dt). This electric field is non-conservative (closed field lines).

Self Inductance:

Nφ = LI EMF = −L(dI/dt)

Self-Inductance of Solenoid:

L = μ₀n²πr²ℓ = μ₀n²(Volume)

Inductor — Electrical Equivalence:

V_A − V_B = L(dI/dt)

Energy Stored in Inductor:

U = ½LI²

Growth of Current in Series RL Circuit:

I = (E/R)(1 − e^(−Rt/L)) Time constant: τ = L/R Final current = E/R (independent of L) After one τ: I = 63% of E/R Higher τ → slower rate of change of current

Decay of Current in RL Circuit:

I = I₀ e^(−Rt/L) After one τ: I = 37% of I₀ = I₀/e

Mutual Inductance:

N₂φ₂ = MI₁ EMF₂ = −M(dI₁/dt)

Equivalent Self Inductance:

Series (no mutual): L = L₁ + L₂ Series (same winding, mutually coupled): L = L₁ + L₂ + 2M Series (opposite winding, mutually coupled): L = L₁ + L₂ − 2M

Parallel (no mutual): 1/L = 1/L₁ + 1/L₂

Coupling: M = k√(L₁L₂), k ≤ 1 (coupling constant)

Transformer:

E_S/E_P = N_S/N_P = I_P/I_S Step-up: N_S > N_P → E_S > E_P

LC Oscillations:

ω = 1/√(LC)

ALTERNATING CURRENT

AC vs DC: AC changes direction periodically. DC maintains constant direction.

RMS Value:

f_rms = √[(1/(t₂−t₁)) ∫(t₁ to t₂) f² dt]

For sinusoidal: V_rms = V_m/√2, I_rms = I_m/√2

Average Power:

P = V_rms × I_rms × cosφ = ½V_m I_m cosφ cosφ = power factor

Impedance & Reactance:

Z = V_m/I_m = V_rms/I_rms (impedance) Inductive reactance: X_L = ωL Capacitive reactance: X_C = 1/(ωC)

Purely Resistive Circuit:

V and I in phase I_m = V_m/R I_rms = V_rms/R P = V_rms²/R = V_rms × I_rms

Purely Capacitive Circuit:

I leads V by π/2 I_m = V_m/(1/ωC) = V_m ωC X_C = 1/(ωC) P = 0 (since cosφ = cos90° = 0)

Purely Inductive Circuit:

V leads I by π/2 X_L = ωL P = 0

GEOMETRICAL OPTICS

Reflection:

i = r (angle of incidence = angle of reflection)

Plane Mirror Image Properties: (a) Object distance = Image distance from mirror (b) Line joining object and image is normal to mirror (c) Image is same size as object (d) For real object → virtual image; for virtual object → real image

Velocity of Image (Plane Mirror):

(v_I)_x,mirror = −(v_O)_x,mirror (perpendicular to mirror) (v_I)_y,mirror = (v_O)_y,mirror (parallel to mirror) (v_I)_z,mirror = (v_O)_z,mirror (parallel to mirror)

Spherical Mirror Formula:

1/v + 1/u = 1/f = 2/R

Concave mirror: f, R are negative (x-coordinate). Convex mirror: f, R are positive. Real image: v is negative. Virtual image: v is positive

Lateral Magnification:

m = −v/u

Longitudinal Magnification (small object along axis):

m_long = −v²/u²

Image Velocity (along principal axis):

v_image = −(v/u)² × v_object (Image always moves opposite to object direction w.r.t. mirror along axis)

Newton's Formula:

xy = f² (x, y = distances of object and image from principal focus)

Optical Power of Mirror:

P = 1/f (f in metres, P in dioptres)

Refraction — Snell's Law:

n₁ sin i = n₂ sin r

Deviation due to Refraction:

δ = |i − r|

Principle of Reversibility: Incident and refracted rays are mutually reversible.

Apparent Depth & Shift:

Apparent depth: d' = d/n_rel n_rel = n(medium of refraction)/n(medium of incidence) Apparent shift = d(1 − 1/n_rel)

Composite slab: Apparent depth = t₁/n₁ + t₂/n₂ + t₃/n₃ + ... Apparent shift = t₁(1 − 1/n₁) + t₂(1 − 1/n₂) + …

Total Internal Reflection:

Critical angle: sin C = n_rarer/n_denser = 1/n_rel Conditions: (a) Light from denser to rarer medium, (b) i > C

Prism:

δ = (i + e) − A r₁ + r₂ = A

At minimum deviation (δ_min): i = e, r₁ = r₂ = A/2 n = sin[(A + δ_min)/2] / sin(A/2)

Thin prism (A < 10°, small i): δ = (n_rel − 1)A

Dispersion:

Angular dispersion: θ = (n_v − n_r)A Mean deviation: δ_y = (n_y − 1)A Dispersive power: ω = (n_v − n_r)/(n_y − 1) = θ/δ_y

Cauchy's formula: n(λ) = a + b/λ² (a, b are positive constants)

If n_y is not given: n_y = (n_v + n_r)/2

Direct Vision Combination (dispersion without deviation):

(n_v + n_r − 2)A/2 = (n'_v + n'_r − 2)A'/2 Simplified: δ₁ = −δ₂ (net deviation = 0)

Achromatic Combination (deviation without dispersion):

(n_v − n_r)A = (n'_v − n'_r)A' (net dispersion = 0)

Refraction at Spherical Surface:

n₁/u + n₂/v = (n₂ − n₁)/R

Transverse magnification: m = (v − R)/(u − R) = n₁v/(n₂u)

Thin Lens:

Lens maker's formula: 1/f = (n_rel − 1)(1/R₁ − 1/R₂) Lens formula: 1/v − 1/u = 1/f Magnification: m = v/u

Combination of Lenses:

1/F = 1/f₁ + 1/f₂ + 1/f₃ + …

WAVE OPTICS

Interference:

I = I₁ + I₂ + 2√(I₁I₂) cos(Δφ) I_max = (√I₁ + √I₂)² I_min = (√I₁ − √I₂)²

Incoherent sources: I = I₁ + I₂

Young's Double Slit Experiment (YDSE):

Path difference: Δ = d sinθ ≈ dy/D (for d << D, y << D)

Bright fringes (maxima): Δ = nλ y = nλD/d (n = 0, ±1, ±2, ...)

Dark fringes (minima): Δ = (2n−1)λ/2 y = (2n−1)λD/(2d) (n = ±1, ±2, ...)

Fringe width: β = λD/d (λ = wavelength in medium)

If I₁ = I₂ = I₀: I = 4I₀ cos²(Δφ/2)

Highest Order Maxima:

n_max = ⌊d/λ⌋ (floor function) Total bright fringes: 2n_max + 1

Highest Order Minima:

n_max = ⌊d/λ + ½⌋ Total dark fringes: 2n_max

YDSE with Two Wavelengths (λ₁ and λ₂):

Nearest coincidence of bright fringes: y = LCM(β₁, β₂) where β₁ = λ₁D/d, β₂ = λ₂D/d

Nearest coincidence of dark fringes: y = n₁β₁/2 = n₂β₂/2 (find appropriate n₁, n₂)

Optical Path Difference:

Δ_opt = μ × Δ_geometric Shift due to slab (thickness t): Δφ = (μ − 1)t Central fringe shift: Δy = (μ − 1)tD/d

YDSE with Oblique Incidence (angle θ₀ to axis):

Central maxima at O' where Δp = 0 (not at geometric centre).

Above O': Δ = d(sinθ + sinθ₀) Between O and O': Δ = d(sinθ₀ − sinθ) Below O: Δ = −d(sinθ + sinθ₀)

Thin-Film Interference — Reflected Light:

Constructive: 2μd = (n + ½)λ Destructive: 2μd = nλ

Thin-Film Interference — Transmitted Light:

Constructive: 2μd = nλ Destructive: 2μd = (n + ½)λ

(Conditions are swapped between reflected and transmitted.)

GRAVITATION

Newton's Law of Gravitation:

F = Gm₁m₂/r² G = 6.67 × 10⁻¹¹ Nm²/kg²

Vector form: F⃗₁₂ = −(Gm₁m₂/r²)r̂₁₂ and F⃗₂₁ = −(Gm₁m₂/r²)r̂₂₁

Gravitational Field & Potential:

E⃗ = F⃗/m = −(GM/r²)r̂ V = −GM/r E = −dV/dr

Ring (mass M, radius a, at distance x on axis):

V = −GM/√(a² + x²) E = −GMx/(a² + x²)^(3/2) E is maximum at x = a/√2

Thin Circular Disc (mass M, radius a):

V = −(2GM/a²)[√(a² + r²) − r] E = (2GM/a²)[1 − r/√(a² + r²)]

Solid Sphere (mass M, radius a):

Outside (r ≥ a): V = −GM/r, E = GM/r² Inside (r ≤ a): V = −(GM/2a)(3 − r²/a²), E = GMr/a³ At centre: V = −3GM/(2a), E = 0

Thin Spherical Shell (mass M, radius a):

Outside (r ≥ a): V = −GM/r, E = GM/r² Inside (r < a): V = −GM/a (constant), E = 0

Variation of g:

With altitude: g_h = g(1 − 2h/R_e) for h << R_e Exact: g_h = GM/(R_e + h)²

With depth: g_d = g(1 − d/R_e)

Surface: g_pole > g_equator (equatorial radius > polar radius)

Satellite (Orbital) Velocity:

v₀ = √[GM/(R_e + h)] = √[gR_e²/(R_e + h)] For h << R_e: v₀ = √(gR_e) ≈ 7.92 km/s

Time Period of Satellite:

T = 2π(R_e + h)^(3/2) / (R_e√g)

Energy of Satellite (at distance r from centre):

KE = GMm/(2r) PE = −GMm/r Total E = −GMm/(2r)

Kepler's Laws:

1st Law: Planets move in elliptical orbits with the Sun at one focus.

2nd Law (Law of Areas): Areal velocity = ½r²(dθ/dt) = L/(2m) = constant ½r²ω = constant

3rd Law (Law of Periods): T² ∝ R³ → T²/R³ = constant

FLUIDS, ELASTICITY, SURFACE TENSION, VISCOSITY

Hydraulic Press (Pascal's Law):

f/a = F/A → F = f(A/a)

Hydrostatic Pressure:

P = P₀ + ρgh P_A = P_B = P_C (same horizontal level, same liquid — hydrostatic paradox)

Fluid in Accelerating Systems:

Elevator (upward acceleration a₀): P = ρh(g + a₀) Buoyancy: B = m(g + a₀)

Horizontal acceleration: tanθ = a₀/g (free surface tilt) p₁ − p₂ = ρa₀ℓ (pressure difference along acceleration)

Rotating cylinder: h = ω²r²/(2g)

Equation of Continuity (Incompressible):

A₁v₁ = A₂v₂ In general: Av = constant

Bernoulli's Theorem:

P + ½ρv² + ρgh = constant (along a streamline)

Torricelli's Theorem (Speed of Efflux):

v = √(2gh) General (with vessel area A₁, hole area A): v = √[2gh / (1 − A²/A₁²)]

Elasticity:

Stress = F/A Strain: Longitudinal strain = ΔL/L Volume strain = ΔV/V Shear strain = tanφ ≈ φ (or x/ℓ)

Young's Modulus:

Y = (F/A)/(ΔL/L) = FL/(AΔL)

Potential Energy per Unit Volume:

u = ½ × stress × strain = ½Y(strain)²

Inter-atomic Force Constant:

k = Yr₀ (r₀ = inter-atomic distance)

Newton's Law of Viscosity:

F = ηA(dv/dx) η = coefficient of viscosity

Stoke's Law:

F = 6πηrv

Terminal Velocity:

v_t = 2r²(ρ − σ)g / (9η) (ρ = density of sphere, σ = density of fluid)

Surface Tension:

T = F/L = W/ΔA (surface energy per unit area)

Excess Pressure:

Inside a bubble (2 free surfaces): ΔP = 4T/r Inside a drop (1 free surface): ΔP = 2T/r Air bubble in liquid: ΔP = 2T/r

Capillary Rise:

h = 2T cosθ / (rρg)

MODERN PHYSICS

Work Function:

W₀ = hν₀ = hc/λ₀ Minimum for Caesium: 1.9 eV

Einstein's Photoelectric Equation:

hν = W₀ + KE_max hc/λ = hc/λ₀ + eV_s

KE_max = eV_s (V_S = stopping potential). Stopping potential is independent of intensity (ν constant). Photocurrent ∝ Intensity (ν constant)

Photon Properties:

Energy: E(eV) = 12400/λ(Å) Momentum: p = h/λ = E/c Intensity: I = ½ε₀E²c

Radiation Pressure (Normal Incidence):

Fully absorbed (a=1): F = IA/c, P = I/c Fully reflected (r=1): F = 2IA/c, P = 2I/c Partial (a + r = 1): F = IA(1+r)/c, P = I(1+r)/c

At angle θ to normal:

Absorbed: F = (IA/c)cosθ, P = (I/c)cos²θ Reflected: F = (2IA/c)cos²θ, P = (2I/c)cos²θ

de Broglie Wavelength:

λ = h/(mv) = h/p = h/√(2mKE)

Bohr Model — Hydrogen-like Atoms (atomic number Z):

Radius: r_n = a₀n²/Z (a₀ = 0.529 Å) Speed: v_n = v₀Z/n (v₀ = 2.19 × 10⁶ m/s) Energy: E_n = −13.6 Z²/n² eV

Spectral Lines:

1/λ = RZ²(1/n₁² − 1/n₂²)

Total transitions from state n: n(n−1)/2

Spectral Series:

Lyman: n₁ = 1, n₂ = 2,3,4,... → Ultraviolet Balmer: n₁ = 2, n₂ = 3,4,5,... → Visible Paschen: n₁ = 3, n₂ = 4,5,6,... → Infrared Brackett: n₁ = 4, n₂ = 5,6,7,... → Infrared Pfund: n₁ = 5, n₂ = 6,7,8,... → Infrared

Reduced Mass Effect (Nuclear Motion Correction):

μ = mM/(m + M) (M = nuclear mass, m = electron mass) r_n = (a₀n²/Z) × (m_e/μ)

X-Rays:

Minimum wavelength: λ_min = hc/(eV) = 12400/V(volt) Å Moseley's law: √ν = a(Z − b) (a, b are positive constants, independent of Z)

Nuclear Physics:

Nuclear radius: R = R₀A^(1/3) (R₀ = 1.1 × 10⁻¹⁵ m, A = mass number) Binding energy: B = [Zm_p + Nm_n − M]c²

Alpha Decay:

ᴬ_Z X → ᴬ⁻⁴_(Z−2) Y + ⁴₂He Q = [m(X) − m(Y) − m(He)]c²

Beta-Plus Decay:

ᴬ_Z X → ᴬ_(Z−1) Y + e⁺ + ν Q = [m(X) − m(Y) − 2m_e]c² (in atomic masses)

Electron Capture:

An atomic electron is captured → X-rays emitted

Radioactive Decay Law:

N = N₀ e^(−λt) (λ = decay constant) Activity: A = λN = A₀ e^(−λt) Specific activity = Activity per unit mass

Half-Life:

T₁/₂ = 0.693/λ

Mean (Average) Life:

T_avg = 1/λ = T₁/₂/0.693

Two Decay Modes (Effective Half-Life):

1/T = 1/T₁ + 1/T₂

What Makes Vedantu’s JEE Advanced Physics Formula Sheet Different?

This isn't a generic formula list. This is a JEE Advanced-specific formula designed by experienced physics educators who understand what the exam actually tests. Here's what sets it apart:

• Complete coverage, nothing is missing: Every formula from every chapter in the JEE Advanced Physics syllabus is included. We have verified this against the last 10 years of JEE Advanced papers to ensure there are no gaps.

• Organised for revision, not for reading: The formulas are arranged in a logical flow within each chapter; definitions first, then derived results, then special cases and advanced results. This mirrors how your brain recalls information, making revision sessions faster and more effective.

• JEE Advanced level depth: This formula sheet includes results tested in JEE Advanced that are often missing from standard textbooks and coaching material. 

• Ready to download: The JEE Advanced physics formulas PDF version of this page is available for free download, formatted for clean printing on A4 sheets so you can carry a physical copy to your study table or exam centre.

Subject Wise Links JEE Advanced 2026 Formulas