The 261 formulas JEE Mains Physics actually tests

Significant figures in a result

sum or difference: fewest decimal places product or quotient: fewest significant figures

exact numbers (a count, the 2 in 2πr) never limit the answer

Note:Leading zeros never count; trailing zeros count only after a decimal point. A mean takes the decimal place of the least precise reading.

Dimensions to know by heart

force MLT⁻² energy, torque ML²T⁻² power ML²T⁻³ pressure, stress, modulus ML⁻¹T⁻² angular momentum ML²T⁻¹

M, L, T = mass, length, time

Note:Energy density and every elastic modulus have the dimensions of pressure; impulse has those of momentum.

Electromagnetic combinations

1/(μ₀ε₀) = c² √(μ₀/ε₀) ~ resistance ½ε₀E² = B²/(2μ₀) ~ energy per volume RC, L/R, √(LC) ~ time

μ₀, ε₀ = permeability and permittivity of free space

R, L, C = resistance, inductance, capacitance

~ = has the dimensions of

Homogeneity and the argument rule

Q = X + Y ⇒ [Q] = [X] = [Y] Q = A sin(Bx) ⇒ [B] = [x]⁻¹, [A] = [Q]

[ ] = dimensions of

Note:In (P + a/V²)(V − b) = RT: [a/V²] = [P] and [b] = [V]. The argument of sin, exp or log is dimensionless.

Changing the base units

n₂ = n₁ (M₁/M₂)ᵃ (L₁/L₂)ᵇ (T₁/T₂)ᶜ

n₁, n₂ = numerical values in the two systems

a, b, c = dimensions of the quantity in M, L, T

Note:To find unknown powers, write Q = k AᵃBᵇCᶜ and equate the powers of M, L and T.

Error in a product of powers

Q = a^p b^q / c^r ⇒ ΔQ/Q = p Δa/a + q Δb/b + r Δc/c

Δa/a = fractional (relative) error in a

p, q, r = sizes of the powers

Note:Every term is added, even for a quantity in the denominator; a square root counts ½. Constants such as 4π² carry no error. Multiply by 100 for a percentage.

Errors in sums and parallel combinations

Δ(A ± B) = ΔA + ΔB parallel resistors: ΔR/R² = ΔR₁/R₁² + ΔR₂/R₂²

ΔA, ΔB = absolute errors

Note:For values given as value ± error, find each relative error first, then ΔQ = Q × (sum of relative errors).

Errors in experiments

pendulum: Δg/g = Δl/l + 2 Δt/t one reading: Δx = LC a difference: Δ(x₂ − x₁) = 2 LC lens: Δf/f² = Δu/u² + Δv/v²

t = total time of n oscillations

LC = least count

Note:Timing many oscillations keeps Δt fixed while t grows, so the relative error falls.

Vernier and screw gauge

vernier: LC = 1 MSD − 1 VSD = (1 − m/N) MSD screw gauge: LC = pitch / circular divisions true reading = MSR + n × LC − (zero error)

N vernier divisions = m main-scale divisions

MSR = main-scale reading, n = coinciding division

Note:A negative zero error is subtracted as a negative, so it is added to the reading.

Equations of motion

v = u + at s = ut + ½at² v² = u² + 2as s = ((u + v)/2) t

u, v = initial and final velocity

a = constant acceleration

s = displacement in time t

Distance in the nth second

sₙ = u + (a/2)(2n − 1)

n = the second counted

Stopping distance

s = u²/(2a) s₂/s₁ = (u₂/u₁)²

a = size of the braking deceleration

Note:Same brakes, double the speed: four times the distance.

Average speed

v̄ = total distance / total time equal distances: v̄ = 2v₁v₂/(v₁ + v₂)

v₁, v₂ = speeds on the two legs

Note:Average velocity uses total displacement, not distance.

Relative velocity in a line

v_BA = v_B − v_A time to cross = (L₁ + L₂)/v_rel

L₁, L₂ = lengths of the two trains

v_rel = relative speed

Free fall from rest

h = ½gt² v = gt v = √(2gh)

h = height fallen

Thrown up or down

s = ut − ½gt² (up positive) H = u²/(2g) dropped from a tower: t = √(t₁t₂)

t₁ = time to land thrown up, t₂ = thrown down at the same speed

H = greatest height

Note:The two times at one height are roots of h = ut − ½gt², so their product is 2h/g. At the top v = 0 but the acceleration is still g.

Variable acceleration

v = dx/dt a = dv/dt = v dv/dx v = u + ∫a dt x = x₀ + ∫v dt

integrals run from 0 to t

Note:v = k√x gives a constant a = k²/2. Area under v–t is displacement; under a–t, change of velocity.

Resultant of two vectors

R = √(A² + B² + 2AB cos θ) tan α = B sin θ / (A + B cos θ)

θ = angle between A and B

α = angle of R from A

Note:Equal vectors: |A + B| = 2A cos(θ/2).

Dot, cross and projection

A · B = AB cos θ |A × B| = AB sin θ projection of A on B = (A · B)/|B|

θ = angle between A and B

Crossing a river

shortest time: t = d/v, drift = ud/v shortest path: sin α = u/v, t = d/√(v² − u²)

d = width, u = river speed

v = swimmer's speed in still water

α = heading upstream from straight across

Relative velocity and rain

v_AB = v_A − v_B vertical rain on a moving man: tan θ = v_man / v_rain

θ = angle of the umbrella from the vertical, tilted forward

Projectile on level ground

T = 2u sin θ / g H = u² sin²θ / (2g) R = u² sin 2θ / g

u = launch speed, θ = angle above the horizontal

Range relations

R_max = u²/g at 45° R/H = 4/tan θ θ and 90° − θ give the same R: R = 4√(H₁H₂)

H₁, H₂ = heights for the two angles

Note:For that pair, T₁T₂ = 2R/g and H₁ + H₂ = u²/(2g).

Path of a projectile

y = x tan θ − gx²/(2u² cos²θ) y = px − qx²: R = p/q, H = p²/(4q)

p, q = coefficients of the given path

Note:At the top only the horizontal velocity is left, so KE there = K cos²θ.

Thrown horizontally from a height

t = √(2h/g) x = u√(2h/g) v = √(u² + 2gh)

h = height of the launch

u = horizontal speed

Circular motion

v = ωr a_c = v²/r = ω²r a = √(a_t² + a_c²), a_t = dv/dt

a_c = centripetal, a_t = tangential acceleration

Note:Displacement over an angle θ on the circle: 2r sin(θ/2).

Curves and conical pendulum

flat road: v_max = √(μrg) banked, no friction: tan θ = v²/(rg) conical pendulum: tan θ = v²/(rg)

μ = coefficient of friction

θ = bank angle, or string angle from the vertical

Vertical circle on a string

T_bottom = mv_b²/r + mg T_top = mv_t²/r − mg v_b² = v_t² + 4gr full loop: v_b ≥ √(5gr)

v_b, v_t = speeds at the bottom and the top

Second law and apparent weight

F_net = ma lift or rope: N = m(g + a) accelerating up, m(g − a) accelerating down

N = scale reading or rope tension

Note:Free fall: N = 0. Slowing while going up is a downward acceleration.

Impulse

F_avg Δt = Δp rebound: |Δp| = m(v + v′)

v, v′ = speeds before and after the bounce

Variable mass

thrust: F = v_rel dm/dt conveyor belt: P = v² dm/dt

v_rel = speed of the ejected mass relative to the body

dm/dt = mass flow rate

Equilibrium of forces

ΣF_x = 0, ΣF_y = 0 N = mg ± F sin θ smooth incline held by a horizontal force: F = mg tan θ

F sin θ = the part of a slanted force across the surface

Note:A chain hung at angle θ to the horizontal at both ends has tension (mg/2) cot θ at its lowest point.

Atwood machine

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

m₂ > m₁

Note:Block on a rough table pulled by a hanging block: a = (m_h g − μ_k m_t g)/(m_h + m_t).

String constraints

Σ Tᵢ aᵢ = 0 movable pulley: a_load = a_end / 2

Tᵢ, aᵢ = tension on and acceleration of each attached point

Friction

f_s ≤ μ_s N f_k = μ_k N least pull at angle θ: F = μmg/(cos θ + μ sin θ)

μ_s, μ_k = static and kinetic coefficients

Rough incline

push up: F = mg(sin θ + μ cos θ) hold: F = mg(sin θ − μ cos θ) slide down: a = g(sin θ − μ cos θ)

θ = incline angle

Note:Constant velocity down the slope: μ = tan θ. Rough time n times the smooth time: μ = tan θ (1 − 1/n²).

Pseudo force

F_pseudo = −m a₀ bob in an accelerating car: tan θ = a₀/g

a₀ = acceleration of the frame

θ = string angle from the vertical

Banked road with friction

v_max² = rg (tan θ + μ)/(1 − μ tan θ)

θ = bank angle, r = radius

Work

W = F · s = Fs cos θ = F_x s_x + F_y s_y + F_z s_z variable force: W = ∫F(x) dx

θ = angle between force and displacement

Work-energy theorem

W_net = ΣWᵢ = K_f − K_i = ½m(v₂² − v₁²)

K = kinetic energy

Note:Count the work of every force, gravity and friction included.

Kinetic energy and momentum

K = p²/(2m) p = √(2mK)

p = momentum

Potential energy and springs

F = −dU/dx W_cons = −ΔU spring: U = ½kx², W(x₁ → x₂) = ½k(x₂² − x₁²)

k = spring constant

x = extension from the natural length

Mechanical energy

½mv₁² + mgh₁ = ½mv₂² + mgh₂ full vertical circle on a string: v_bottom² ≥ 5gL

L = string length

Power

P = F · v P_avg = W/t constant power from rest: v = √(2P/m) t^(1/2), x = √(8P/(9m)) t^(3/2)

v = velocity at the instant

Momentum and sticking collisions

J = ∫F dt = Δp m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ KE lost on sticking: ½ (m₁m₂/(m₁ + m₂)) (u₁ − u₂)²

J = impulse

Ballistic pendulum

mu = (M + m)V V = √(2gh)

m, u = bullet's mass and speed

M = block, h = height it rises

Note:Momentum in the impact, energy in the swing; never energy across the impact.

Elastic collision with a target at rest

v₁ = (m₁ − m₂)u/(m₁ + m₂) v₂ = 2m₁u/(m₁ + m₂) K₂/K₁ = 4m₁m₂/(m₁ + m₂)²

m₁ moving at u hits m₂ at rest

Note:Equal masses exchange velocities.

Coefficient of restitution

e = (v₂ − v₁)/(u₁ − u₂) bounce height h′ = e²h K′/K = e²

e = 1 elastic, e = 0 perfectly inelastic

Centre of mass

x_cm = Σmᵢxᵢ / Σmᵢ continuous: ∫x dm / ∫dm piece removed: x_cm = (Mx₁ − mx₂)/(M − m)

M, x₁ = whole body; m, x₂ = removed piece

Motion of the centre of mass

v_cm = Σmᵢvᵢ / Σmᵢ M a_cm = F_ext

F_ext = net external force

Note:An explosion leaves the centre of mass on its old path.

Constant angular acceleration

ω = ω₀ + αt θ = ω₀t + ½αt² ω² = ω₀² + 2αθ

α = angular acceleration

Torque and equilibrium

τ = r × F equilibrium: ΣF = 0 and Στ = 0 about any point

r = position of the point of application

Standard moments of inertia

ring MR² hollow sphere ⅔MR² disc, solid cylinder ½MR² solid sphere ⅖MR² rod about centre ML²/12, about an end ML²/3

axis through the centre (normal to a ring or disc)

I = Mk², k = radius of gyration

Note:Ring about a diameter ½MR²; disc about a diameter ¼MR².

Axis theorems and removed pieces

I = I_cm + Md² I_z = I_x + I_y (flat body) I_rest = I_full − (I_hole + m_hole d²)

d = distance between parallel axes

Note:k² = k_cm² + d².

Rotational dynamics

τ = Iα P = τω K_rot = ½Iω² heavy pulley: a = (m₁ − m₂)g/(m₁ + m₂ + I/R²)

I, R = pulley's moment of inertia and radius

Angular momentum

L = r × mv, |L| = mvd I₁ω₁ = I₂ω₂ KE lost on locking: I₁I₂(ω₁ − ω₂)²/(2(I₁ + I₂))

d = perpendicular distance from the point to the line of motion

Note:Projectile about the launch point at the top: L = mu³ sin²θ cos θ/(2g).

Rolling without slipping

v_cm = ωR K = ½mv²(1 + k²/R²) speed of a point P: v_P = ω r_P

r_P = distance of P from the contact point

k = radius of gyration

Note:Translational : rotational KE = 1 : k²/R².

Rolling down a slope

a = g sin θ/(1 + k²/R²) v = √(2gh/(1 + k²/R²)) f = mg sin θ (k²/R²)/(1 + k²/R²)

f = friction needed for pure rolling

h = height descended

Note:Racing from rest down one slope: solid sphere first, then disc and solid cylinder, then hollow sphere, then ring.

Newton's law and potential energy

F = Gm₁m₂/r² U = −Σ Gmᵢmⱼ/rᵢⱼ over pairs ΔU (R to R + h) = mgh/(1 + h/R)

r = distance between centres

R = planet's radius

Note:Several masses: add the forces as vectors.

Field and potential of a uniform sphere

r ≥ R: E = GM/r², V = −GM/r r < R: E = GMr/R³, V = −GM(3R² − r²)/(2R³)

M, R = mass and radius

r = distance from the centre

Note:Inside a thin shell E = 0 and V = −GM/R.

Surface gravity

g = GM/R² = (4/3)πGρR

ρ = mean density

g at a height

g_h = g/(1 + h/R)² ≈ g(1 − 2h/R) (h ≪ R)

h = height above the surface

g at a depth

g_d = g(1 − d/R) equal g below and above: d ≈ 2h

d = depth below the surface

Note:g is zero at the centre and greatest at the surface.

Effect of the earth's spin

g′ = g − ω²R cos²λ

λ = latitude

ω = the earth's angular speed

Note:No change at the poles; the largest drop at the equator.

Escape velocity

v_e = √(2GM/R) = √(2gR) = R√(8πGρ/3)

independent of the launched mass and direction

Note:Launched at k·v_e (k < 1), the body rises to R/(1 − k²) from the centre.

Circular orbit

v₀ = √(GM/r) T = 2π√(r³/GM) L = m√(GMr)

r = orbit radius from the centre

Note:Near the surface, v_e = √2 × v₀.

Satellite energy

KE = GMm/(2r) U = −GMm/r E = −GMm/(2r) to move r₁ → r₂: ΔE = (GMm/2)(1/r₁ − 1/r₂)

E = total energy, negative for a bound orbit

Two bodies orbiting each other

m₁r₁ = m₂r₂ T = 2π√(d³/(G(m₁ + m₂)))

d = separation, r₁ + r₂ = d

Kepler's laws

T² = (4π²/GM) r³ dA/dt = L/(2m) = constant v_max r_min = v_min r_max

r = orbit radius or semi-major axis

Young's modulus

Y = (F/A)/(ΔL/L) ΔL = FL/(AY)

F/A = stress, ΔL/L = strain

Note:Y is a property of the material, not of the wire's length or thickness.

Comparing two wires

ΔL₁/ΔL₂ = (F₁/F₂)(L₁/L₂)(d₂/d₁)²(Y₂/Y₁)

d = diameter

Wires in series

ΔL = T(L₁/(A₁Y₁) + L₂/(A₂Y₂)) equal lengths and areas: Y_eq = 2Y₁Y₂/(Y₁ + Y₂)

T = tension, the same in both

Breaking stress and own weight

T_max = σ_b A largest upward acceleration: a_max = σ_b A/m − g own weight: ΔL = MgL/(2AY)

σ_b = breaking stress

Note:Longest wire that can hang without breaking: L_max = σ_b/(ρg), whatever its area.

Bulk modulus

B = −ΔP/(ΔV/V) ΔV = ΔP V/B Δρ = ρΔP/B at depth h: ΔV/V = ρgh/B

ρ = density of the liquid above

Shear modulus and Poisson's ratio

η = (F/A)/(x/h) Y = 2η(1 + σ) = 3B(1 − 2σ)

x/h = shear strain

σ = Poisson's ratio

Natural length from two loads

T = k(l − l₀) l₀ = (T₂l₁ − T₁l₂)/(T₂ − T₁)

l₁, l₂ = lengths under tensions T₁, T₂

Energy stored

energy per volume = ½ × stress × strain = ½Yε² U = ½F ΔL

ε = strain

Pressure at a depth

P = P₀ + ρgh

P₀ = pressure at the surface

ρgh = gauge pressure

Pascal's law

F₁/A₁ = F₂/A₂ F₁d₁ = F₂d₂

d = distance each piston moves

Floating bodies

V_sub/V = ρ_body/ρ_liquid

V_sub = volume under the liquid

Note:Buoyant force = weight of the liquid displaced.

Continuity and Bernoulli

A₁v₁ = A₂v₂ P + ρgh + ½ρv² = constant

along one streamline

Note:Lift on a wing: F = ½ρ(v_top² − v_bottom²)A, with A the total wing area.

Torricelli's law

v = √(2gh) range on the floor: x = 2√(hy)

h = depth of the hole below the surface

y = height of the hole above the floor

Note:The range is greatest for a hole at mid-depth, and then equals the water's height.

Viscosity and Reynolds number

F = ηA v/d Re = ρvd/η Stokes: F = 6πηrv

η = coefficient of viscosity

d = gap between the layers, or pipe diameter

Terminal velocity

v_T = 2r²(ρ − σ)g/(9η)

ρ = sphere's density, σ = fluid's density

Note:v_T ∝ r². n equal drops merging: R = n^(1/3) r, so v′ = n^(2/3) v.

Surface energy

split into n drops: W = 4πR²T(n^(1/3) − 1) blow a bubble r₁ → r₂: W = 8πT(r₂² − r₁²)

T = surface tension

a soap bubble has two surfaces

Excess pressure

drop or air bubble in a liquid: ΔP = 2T/r soap bubble: ΔP = 4T/r

r = radius

Capillary rise

h = 2T cos θ/(ρgr)

θ = contact angle

r = tube radius

Note:h ∝ T/(ρr). A tilted tube keeps the same vertical height; hot water rises less than cold.

Linear temperature scales

(X − X_ice)/(X_steam − X_ice) = C/100 = (F − 32)/180 = (K − 273)/100

X = reading on any linear scale

Thermal expansion

ΔL = LαΔT ΔA = A(2α)ΔT ΔV = V(3α)ΔT

α = coefficient of linear expansion

Note:Ideal gas at constant pressure: γ = 1/T.

Thermal stress

σ = YαΔT F = YAαΔT energy per volume = ½Y(αΔT)²

rod clamped so it cannot expand

Note:The force does not depend on the rod's length.

Heat and latent heat

Q = msΔT Q = mL heater: ηPt = msΔT

s = specific heat, L = latent heat

η = fraction of the power that heats

Note:Falling water: ΔT = gh/s, whatever the mass. On a heating curve, slope dT/dQ = 1/(ms).

Mixing ice and water

heat lost = heat gained: m_w s_w(T_w − T) = m_i s_i(0 − T_i) + m_i L + m_i s_w(T − 0)

T_i = ice's starting temperature (°C)

T = final temperature

Note:First check whether the water has enough heat to melt all the ice.

Conduction

H = KAΔT/L R = L/(KA) series: R = R₁ + R₂ parallel: 1/R = 1/R₁ + 1/R₂

K = thermal conductivity

H = heat current

Note:Junction of two in series: θ = (θ₁R₂ + θ₂R₁)/(R₁ + R₂). Spherical shell: R = (r₂ − r₁)/(4πKr₁r₂).

Stefan and Wien

P = eσAT⁴ net loss: eσA(T⁴ − T₀⁴) λ_m T = b

T in kelvin, e = emissivity

b = Wien's constant

Note:For a sphere P ∝ er²T⁴; hotter means a shorter peak wavelength.

Newton's law of cooling

(T₁ − T₂)/t = k((T₁ + T₂)/2 − T_s) exactly: T − T_s = (T₀ − T_s)e^(−kt)

T_s = surroundings

T₁ → T₂ = fall in time t

First law

ΔQ = ΔU + W W = ∫P dV

W = work done BY the gas, positive on expansion

Internal energy and heat capacities

ΔU = nC_VΔT (any process) C_V = (f/2)R C_P − C_V = R γ = 1 + 2/f

f = degrees of freedom

How heat splits at constant pressure

Q : ΔU : W = C_P : C_V : R

one isobaric process

Note:So ΔU = Q/γ and W = Q(1 − 1/γ).

Isothermal work

W = nRT ln(V₂/V₁) ΔU = 0, Q = W

V₁ → V₂ at temperature T

Process PVˣ = constant

W = (P₁V₁ − P₂V₂)/(x − 1) = nR(T₁ − T₂)/(x − 1) C = C_V + R/(1 − x)

C = molar heat capacity of the process

Note:x = 0 gives C_P; x = γ gives 0; x = 1 (isothermal) gives an infinite C.

Adiabatic relations

PV^γ = const TV^(γ − 1) = const P^(1 − γ)T^γ = const

γ = C_P/C_V

Note:An adiabatic curve is steeper than an isothermal one through the same point.

Adiabatic work

W = nR(T₁ − T₂)/(γ − 1) = (P₁V₁ − P₂V₂)/(γ − 1) = −ΔU

Q = 0

Cyclic process

W_net = Q_net = enclosed area ΔU = 0 over a cycle

area on the P–V graph

Note:Clockwise on P–V: the gas does net positive work. An elliptical loop encloses πab.

Engines and refrigerators

η = W/Q₁ = 1 − T₂/T₁ Q₁ = W + Q₂ COP = Q₂/W = T₂/(T₁ − T₂)

T₁, T₂ = hot and cold reservoirs (kelvin)

Q₁ in, Q₂ out (engine)

Note:For a Carnot cycle Q₂/Q₁ = T₂/T₁.

Engines in series and entropy

η = 1 − T₃/T₁ = η₁ + η₂ − η₁η₂ ΔS = Q/T heating: ΔS = ms ln(T₂/T₁)

T₁ → T₂ → T₃ = the three reservoirs

Note:Equal work from the two engines: T₂ = (T₁ + T₃)/2.

Ideal gas equation

PV = nRT = NkT P₁V₁/T₁ = P₂V₂/T₂

N = number of molecules, k = Boltzmann constant

Mixtures and joined vessels

n = PV/(RT) joined vessels: Σ PᵢVᵢ/Tᵢ = constant

moles are conserved

Note:On a given path P(V), the hottest state is where d(PV)/dV = 0.

Pressure from molecular impacts

P = ⅓ρv_rms² PV = ⅔E_trans

ρ = gas density

E_trans = total translational KE

Kinetic energy and temperature

mean KE per molecule = (3/2)kT E_trans = (3/2)nRT

depends on T alone, not on the gas

Note:Equal temperatures, equal mean KE, so v_rms ∝ 1/√m.

Molecular speeds

v_rms = √(3RT/M) v̄ = √(8RT/(πM)) v_p = √(2RT/M)

M = molar mass in kg/mol

Note:v_p < v̄ < v_rms; each scales as √(T/M). Also v_rms = √(3P/ρ).

Mean free path

λ = 1/(√2 πd²n) = kT/(√2 πd²P) collision frequency = v̄/λ

d = molecular diameter

n = molecules per volume

Heat capacities from f

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

f = 3 monatomic, 5 rigid diatomic or linear, 6 rigid non-linear

Note:Equipartition: ½kT per degree of freedom per molecule.

Internal energy

U = n(f/2)RT = (f/2)PV ΔU = nC_vΔT

f = degrees of freedom

A mixture as one gas

f_mix = (n₁f₁ + n₂f₂)/(n₁ + n₂) γ_mix = 1 + 2/f_mix

n₁, n₂ = moles of each gas

SHM equation

x = A sin(ωt + φ) v = Aω cos(ωt + φ) a = −ω²x

A = amplitude, φ = initial phase

Speed at a displacement

v = ω√(A² − x²) v_max = Aω a_max = ω²A

x = displacement from the mean

Time between two positions

t = Δθ/ω = (Δθ/2π) T

Δθ = phase covered

Note:Mean to A/2 takes T/12; A/2 to A takes T/6; mean to A/√2 takes T/8.

Adding SHMs of one frequency

A = √(A₁² + A₂² + 2A₁A₂ cos Δφ) a sin ωt + b cos ωt has amplitude √(a² + b²)

Δφ = phase difference

Cut and combined springs

k ∝ 1/l series: 1/k = 1/k₁ + 1/k₂ parallel: k = k₁ + k₂

l = length of the spring piece

Spring–mass period

T = 2π√(m/k) two masses on one spring: ω = √(k/μ), μ = m₁m₂/(m₁ + m₂)

μ = reduced mass

Any restoring force or torque

F = −Cx ⇒ ω = √(C/m) τ = −κθ ⇒ ω = √(κ/I)

C, κ = restoring constants

Simple pendulum

T = 2π√(L/g_eff) ΔT/T = ½ ΔL/L

g_eff = g ± a in a lift; g(R/(R + h))² at a height

Note:In free fall g_eff = 0 and the pendulum does not swing.

Energy in SHM

E = ½kA² = ½mω²A² U = ½kx² K = ½k(A² − x²)

k = mω²

Note:K = U at x = A/√2. KE and PE each oscillate at twice the motion's frequency.

Amplitude changes and damping

after a sudden change at x: A′² = x² + v′²/ω′² damping: A = A₀e^(−bt/2m), E = E₀e^(−bt/m)

v′, ω′ = new speed and angular frequency

b = damping constant

Note:Mass m placed gently on M at the mean: A′ = A√(M/(M + m)). Added at an extreme: A unchanged.

Reading a wave equation

v = ω/k = fλ ω = 2πf k = 2π/λ Δφ = (2π/λ) Δx

k = wave number

Δx = separation of two points

A travelling wave

y = A sin(ωt − kx + φ₀) y = f(x ∓ vt)

x − vt: moving towards +x

x + vt: moving towards −x

Particle speed and intensity

v_p,max = Aω v_p,max / v = Ak = 2πA/λ point source: I = P/(4πr²)

v = wave speed

P = power of the source

Wave speed

string: v = √(T/μ) solid: v = √(Y/ρ) gas: v = √(γP/ρ) = √(γRT/M)

μ = mass per length

M = molar mass

Note:In a gas v ∝ √T and does not change with pressure at fixed temperature.

Two waves of one frequency

A² = A₁² + A₂² + 2A₁A₂ cos φ φ = (2π/λ) × path difference

φ = phase difference

String fixed at both ends

fₙ = (n/2L)√(T/μ) fₙ₊₁ − fₙ = v/(2L) standing wave: y = 2A cos kx sin ωt

n = harmonic number

L = vibrating length

Organ pipes

open: fₙ = nv/(2L), all harmonics closed: f = (2n − 1)v/(4L), odd harmonics only

L = pipe length

Note:Open pipe: kth overtone = (k + 1)th harmonic. Closed pipe: kth overtone = (2k + 1)th harmonic.

Resonance tube

lₙ + e = (2n − 1)λ/4 l₂ − l₁ = λ/2 e = 0.3d

e = end correction

d = tube diameter

Note:The difference of two resonance lengths cancels the end correction.

Beats

f_beat = |f₁ − f₂| = v|1/λ₁ − 1/λ₂|

f₁, f₂ = two close frequencies

Note:Loading a fork with wax lowers its frequency.

Doppler effect

f′ = f (v ± v_o)/(v ∓ v_s) echo from a wall approached at u: f′ = f(v + u)/(v − u)

v = speed of sound

v_o, v_s = speeds of observer and source

Note:Choose each sign so that motion towards the other raises f′. Light from a receding source: Δλ/λ = v/c.

Coulomb's law

F = kq₁q₂/r², k = 1/(4πε₀) = 9 × 10⁹ N m² C⁻² in a medium: F_m = F/K

q₁, q₂ = charges, signs included

r = separation

K = dielectric constant

Note:A separation r in the medium gives the force of r√K in vacuum. Identical spheres touched share (q₁ + q₂)/2 each, signs included.

Fields of standard shapes

point: E = kq/r² line: E = λ/(2πε₀r) = 2kλ/r sheet: E = σ/(2ε₀) just outside a conductor: E = σ/ε₀

λ = charge per length

σ = charge per area

Note:Sheets +σ and −σ: σ/ε₀ between them, zero outside. A sheet's field does not fall with distance.

Ring on its axis and arc at its centre

ring: E = kQz/(z² + R²)^(3/2), largest at z = R/√2 arc: E = (2kλ/R) sin(φ/2)

z = distance along the axis

R = radius

φ = angle the arc spans at the centre

Note:Half ring: 2kλ/R. Full ring at its centre: zero.

Flux and Gauss's law

φ = E · A = EA cos θ ∮ E · dA = q_enc/ε₀

A = area vector, along the normal

q_enc = charge inside the closed surface

Note:Charge at a cube's centre: q/6ε₀ through each face. Charges outside add nothing to the net flux.

Spheres, shells and cylinders

solid sphere: E = kQr/R³ = ρr/(3ε₀) inside, kQ/r² outside shell: E = 0 inside long cylinder: E = ρr/(2ε₀) inside

R = radius of the body

ρ = charge per volume

Note:The solid sphere's field is largest at its surface.

Potential and its link to E

V = Σ kqᵢ/rᵢ shell: V = kQ/R inside, kQ/r outside E = −dV/dr

rᵢ = distance from each charge

R = shell radius

Note:n identical drops merging: potential × n^(2/3). Spheres joined by a wire: q ∝ R, σ ∝ 1/R.

Work and potential energy

W_ext = q(V_B − V_A) U = Σ kqᵢqⱼ/rᵢⱼ (each pair once)

W_ext = work by an agent moving q slowly from A to B

Note:Work by the field is the negative of this.

Short dipole

p = qd E_axial = 2kp/r³ E_equatorial = kp/r³ V = kp cos θ/r²

p points from −q to +q

θ = angle from the axis

Note:The axial field points along p, the equatorial field opposite to it; V is zero on the equatorial line.

Dipole in a uniform field

τ = p × E (pE sin θ) U = −pE cos θ W = pE(cos θ₁ − cos θ₂)

θ = angle between p and E

Note:Net force zero in a uniform field. Turning from aligned to reversed costs 2pE.

Charge crossing a field

y = qEL²/(2mv²) tan θ = qEL/(mv²)

L = length of the plates

v = entry speed along the plates

Note:The velocity part across the field never changes.

Capacitance

plates: C = Kε₀A/d sphere: C = 4πε₀R spherical: C = 4πε₀R₁R₂/(R₂ − R₁)

A = plate area, d = gap

R₁, R₂ = inner and outer radii

Series and parallel

series: 1/C = Σ 1/Cᵢ parallel: C = Σ Cᵢ

series: same Q

parallel: same V

Note:Steady DC: no current flows through a capacitor's branch.

Slabs in the gap

across the gap: C = ε₀A/(d − t + t/K) side by side: C = (ε₀/d) Σ KᵢAᵢ

t = slab thickness

Aᵢ = area each dielectric covers

Note:A metal sheet gives ε₀A/(d − t). A boundary parallel to the plates is series; perpendicular to them, parallel.

Stored energy

U = ½CV² = Q²/(2C) = ½QV u = ½Kε₀E²

u = energy per unit volume

Note:Slab inserted with the battery on (V fixed): U → KU. Battery removed (Q fixed): U → U/K.

Joining two capacitors

V = (C₁V₁ + C₂V₂)/(C₁ + C₂) ΔU = C₁C₂(V₁ − V₂)²/(2(C₁ + C₂))

like plates joined

ΔU = energy lost as heat

Note:Unlike plates joined: use C₁V₁ − C₂V₂ and (V₁ + V₂)².

Current and drift

I = dq/dt = neAv_d v_d = eEτ/m = μE J = nev_d = σE

n = free electrons per m³

τ = mean time between collisions

μ = mobility

Note:At a fixed voltage, v_d does not depend on the area; at a fixed current, v_d ∝ 1/A.

Resistance from shape

R = ρl/A σ = 1/ρ

l = length along the current

A = area across it

Stretching a wire

length × n at constant volume: R → n²R same mass: R ∝ 1/r⁴ ΔR/R ≈ 2Δl/l

r = radius of the wire

Note:Increased BY twice its length means the new length is three times the old.

Resistance and temperature

R_T = R₀(1 + αΔT)

R₀ = resistance at the reference temperature

α = temperature coefficient

Note:Metals: α > 0. Semiconductors: α < 0.

Series, parallel and loops

R_s = Σ Rᵢ 1/R_p = Σ 1/Rᵢ loop tapped at fractions x and 1 − x: R = R_loop · x(1 − x)

R_loop = resistance of the whole loop

Note:A wire cut into n equal pieces, all in parallel: R/n².

Voltage and current dividers

V₁ = V R₁/(R₁ + R₂) I₁ = I R₂/(R₁ + R₂)

series pair across V

parallel pair carrying I (the OTHER resistor on top)

Kirchhoff's laws

junction: Σ I_in = Σ I_out loop: Σ ε = Σ IR

IR drops along the current, rises against it

Note:A current that comes out negative flows the other way.

Wheatstone and meter bridge

balance: P/Q = R/S meter bridge: P/Q = l/(100 − l)

P, Q, R, S = the four arms

l = null point in cm from P's end

Note:At balance the middle arm carries no current whatever its resistance.

A real cell

I = ε/(R + r) V = ε − Ir (discharging), ε + Ir (charging) P_max = ε²/(4r) at R = r

ε = emf, r = internal resistance

R = external resistance

Cells together

series: ε_eq = Σ ±εᵢ, r_eq = Σ rᵢ parallel: ε_eq = (Σ εᵢ/rᵢ)/(Σ 1/rᵢ), 1/r_eq = Σ 1/rᵢ

minus sign for a reversed cell

Note:m rows of n identical cells: I = nε/(R + nr/m).

Potentiometer

ε = kl, k = V_wire/L ε₁/ε₂ = l₁/l₂ r = R(l₁ − l₂)/l₂

k = potential gradient

l₁ = open-circuit balance, l₂ = balance with R across the cell

Power and ratings

P = VI = I²R = V²/R rated: R = V₀²/P₀, P = P₀(V/V₀)² H = I²Rt

V₀, P₀ = rated voltage and power

Note:In series the lower-rated bulb glows more; in parallel the higher-rated one does.

RC and LR circuits

RC: q = Q₀(1 − e^(−t/RC)), discharge q = q₀e^(−t/RC) LR: i = (E/R)(1 − e^(−tR/L))

τ = RC or L/R

Note:Time to fall to 1/n: τ ln n. Energy halves in ½τ ln 2. Just after switching an inductor is open; long after, a capacitor is open.

Biot–Savart law

dB = (μ₀/4π) I dl sin θ/r²

θ = angle between dl and the line to the point

Note:μ₀/4π = 10⁻⁷ T m/A. A point on the line of the wire itself gets zero.

Straight wires

infinite: B = μ₀I/(2πd) finite: B = (μ₀I/4πd)(sin α₁ + sin α₂)

d = perpendicular distance

α₁, α₂ = angles measured from the perpendicular

Note:Semi-infinite wire, point on the perpendicular through its end: μ₀I/(4πd), half the infinite value.

Arcs and circular coils

arc at centre: B = μ₀Iθ/(4πR) coil centre: B = μ₀NI/(2R) axis: B = μ₀NIR²/(2(R² + x²)^(3/2))

θ = arc angle in radians

x = distance along the axis

Ampère's law

∮ B · dl = μ₀I_enc solid wire: B = μ₀Ir/(2πa²) inside, μ₀I/(2πr) outside

a = radius of the wire

Note:Hollow tube: zero inside. Coaxial cable with equal and opposite currents: zero outside.

Solenoid and toroid

solenoid: B = μ₀nI (core: μ₀μ_r nI) toroid: B = μ₀NI/(2πr)

n = turns per metre

N = total turns

Note:H = nI has no μ₀ in it.

Lorentz force

F = q(E + v × B) magnetic part: qvB sin θ selector: v = E/B

θ = angle between v and B

Note:The magnetic force does no work, so speed never changes under it alone.

Circle, helix and cyclotron

r = mv/(qB) = √(2mK)/(qB) T = 2πm/(qB) pitch = 2πmv cos θ/(qB)

K = kinetic energy

θ = angle between v and B

Note:T does not depend on the speed. Cyclotron: K_max = q²B²R²/(2m).

Force on currents

F = I L × B (ILB sin θ) parallel wires: F/L = μ₀I₁I₂/(2πd)

L = length in the field

d = gap between the wires

Note:Like currents attract. A bent wire acts like the straight line joining its ends.

Moment and torque of a coil

m = NIA τ = m × B (NIAB sin θ)

θ = angle between the coil's normal and B

Note:Largest torque when the coil's plane is parallel to B.

Galvanometer, ammeter, voltmeter

NIAB = Cθ shunt: S = I_gG/(I − I_g) series: R = V/I_g − G

C = torsional constant

G = coil resistance, I_g = full-scale current

Note:Current sensitivity θ/I = NAB/C; voltage sensitivity θ/V = NAB/(CR).

Short bar magnet

M = m(2l) B_axial = (μ₀/4π)(2M/r³) B_equatorial = (μ₀/4π)(M/r³)

m = pole strength

2l = magnetic length

M points from S to N

Note:Bending keeps m: a semicircle gives 2M/π, an L at the middle gives M/√2.

Dipole in a uniform field

τ = MB sin θ U = −MB cos θ W = MB(cos θ₁ − cos θ₂)

θ = angle between M and B

Note:Stable to unstable costs 2MB.

Earth's field and dip

B_H = B cos δ B_V = B sin δ tan δ = B_V/B_H

δ = angle of dip

Note:In a plane at α to the magnetic meridian: tan δ′ = tan δ/cos α. Two perpendicular planes: cot²δ = cot²δ₁ + cot²δ₂.

Oscillating magnet

T = 2π√(I/(MB_H))

I = moment of inertia of the magnet

Note:One needle at two places: n² ∝ B cos δ.

Magnetising a material

M = χH B = μ₀(H + M) = μ₀μ_r H μ_r = 1 + χ

M here = magnetisation (moment per volume)

H = magnetic intensity, A/m

χ = susceptibility

Susceptibility and temperature

paramagnet: χ = C/T ferromagnet above T_C: χ = C/(T − T_C)

C = Curie constant

T_C = Curie temperature

Note:Diamagnet: −1 ≤ χ < 0, independent of temperature.

Flux and Faraday's law

Φ = NBA cos θ ε = −dΦ/dt

θ = angle between B and the normal

Note:The minus sign is Lenz's law: the induced current opposes the change.

Charge through the circuit

Q = NΔΦ/R

ΔΦ = change of flux through one turn

Note:Pulled out of the field: ΔΦ = BA. Field reversed or coil flipped: 2BA.

Motional emf and a rod on rails

ε = Blv F = B²l²v/R terminal speed v_t = mgR/(B²l²)

l = length between the rails

R = whole circuit's resistance

Note:A wing or a horizontal rod moving horizontally cuts the vertical component B sin δ.

Rotating rod or disc

ε = ½Bωl²

l = rod length (or disc radius)

ω = angular speed

Note:Fan blades are in parallel, so the emf is that of one blade.

Rotating coil

ε = NBAω sin ωt ε₀ = NBAω

ω = 2π × revolutions per second

Note:Plane perpendicular to B: flux largest, emf zero.

Self-inductance

ε = −L dI/dt solenoid: L = μ₀n²Al = μ₀N²A/l

n = turns per metre

l = solenoid length

Note:L depends on geometry and core, not on the current.

Mutual inductance

ε₂ = −M dI₁/dt coil on a solenoid: M = μ₀nN₂A series coils: L = L₁ + L₂ ± 2M

N₂ = turns of the outer coil

plus when the fluxes aid

Note:M ≤ √(L₁L₂).

Inductor energy and LR growth

U = ½LI² u = B²/(2μ) I = (E/R)(1 − e^(−t/τ)), τ = L/R

u = energy per unit volume

Note:A fraction f of the final energy needs I = √f × E/R.

RMS values

I_rms = I₀/√2 V_rms = V₀/√2 d.c. + a.c.: I_rms = √(I_dc² + I₀²/2)

I₀, V₀ = peak values

Note:Meters and supply ratings are rms. The rms of a sum is not the sum of the rms values.

Reactances

X_L = ωL = 2πfL X_C = 1/(ωC) = 1/(2πfC)

f = frequency

Note:In L the voltage leads by π/2; in C the current leads by π/2.

Series LCR impedance

Z = √(R² + (X_L − X_C)²) V² = V_R² + (V_L − V_C)²

V_R, V_L, V_C = voltages across each part

Phase and power factor

tan φ = (X_L − X_C)/R cos φ = R/Z

φ = angle between voltage and current

Note:X_L > X_C: current lags. X_C > X_L: current leads.

Average power

P = V_rms I_rms cos φ = ½V₀I₀ cos φ = I_rms² R

cos φ = power factor

Note:Pure L or pure C: zero power, a wattless current.

Resonance

ω₀ = 1/√(LC) f₀ = 1/(2π√(LC)) at resonance: Z = R, I = V/R

R does not set the resonant frequency

Quality factor and bandwidth

Q = ω₀L/R = (1/R)√(L/C) Δω = R/L = ω₀/Q

Δω = gap between the half-power frequencies

LC oscillations

ω = 1/√(LC) I_max = Q₀/√(LC) = V₀√(C/L)

Q₀, V₀ = starting charge and voltage

Note:q²/(2C) + ½Li² = Q₀²/(2C) at every instant.

Transformer

V_s/V_p = N_s/N_p ideal: I_s/I_p = N_p/N_s V_sI_s = η V_pI_p

η = efficiency

Displacement current

i_d = ε₀ dΦ_E/dt = C dV/dt

Φ_E = electric flux

Note:Between capacitor plates it equals the conduction current in the leads.

Wave speed

c = 1/√(μ₀ε₀) v = c/√(μ_rε_r) n = √(μ_rε_r) v = ω/k, λ = 2π/k

k = wave number

Note:Entering a medium, the frequency stays; speed and wavelength fall by n.

E and B in a wave

E₀ = cB₀ B = (k × E)/ω travel along E × B

E, B and the direction of travel are mutually perpendicular

Note:B carries exactly the same phase as E.

Energy density

⟨u⟩ = ½ε₀E₀² = B₀²/(2μ₀)

⟨u⟩ = average over a cycle

Note:The electric and magnetic shares are equal.

Intensity

I = ½cε₀E₀² = cB₀²/(2μ₀) point source: I = P/(4πr²)

P = power radiated

Momentum and radiation pressure

p = U/c absorbed: P = I/c reflected: P = 2I/c

U = energy delivered

P here = pressure

Note:Force = pressure × area.

Order of the spectrum

γ-rays < X-rays < ultraviolet < visible < infrared < microwaves < radio (rising λ)

frequency rises the other way

Note:Photon energy E = hc/λ ≈ 1240/λ eV with λ in nm.

Plane mirrors

δ = 180° − 2i mirror turned by θ: ray turns by 2θ images between mirrors at θ: 360°/θ − 1

i = angle of incidence

Note:The image-count rule holds when 360°/θ is even.

Mirror formula

1/v + 1/u = 1/f f = R/2 m = −v/u = f/(f − u)

u, v = object and image distances from the pole

Note:Cartesian sign convention throughout: distances from the pole or optical centre, positive along the incident light, heights positive upward. So a concave mirror has f < 0 and a convex one f > 0.

Image speed

dv/dt = −m² du/dt

m = magnification at that instant

Note:Across the axis the image moves at m times the object's speed.

Snell's law and the slab

n₁ sin i = n₂ sin r n = c/v lateral shift d = t sin(i − r)/cos r

angles measured from the normal

t = slab thickness

Apparent depth

d_app = d/μ shift = d(1 − 1/μ) layers: d_app = Σ dᵢ/μᵢ

viewed from the rarer medium, near the normal

Note:Looking from the denser side at an object in the rarer one: d_app = μd.

Critical angle

sin C = n_rarer/n_denser (= 1/μ into air) circle of light: r = h tan C = h/√(μ² − 1)

h = depth of the source

Note:Total internal reflection needs the denser side and i > C.

One spherical surface

μ₂/v − μ₁/u = (μ₂ − μ₁)/R m = μ₁v/(μ₂u)

μ₁ = medium the light comes from, μ₂ = the one it enters

Lens-maker's formula

1/f = (μ − 1)(1/R₁ − 1/R₂) in a medium: 1/f = (μ_l/μ_m − 1)(1/R₁ − 1/R₂)

R₁ = face the light meets first

μ_l = lens, μ_m = surrounding medium

Note:Equiconvex: f = R/(2(μ − 1)). Plano-convex: f = R/(μ − 1).

Thin lens

1/v − 1/u = 1/f m = v/u = f/(f + u) P = 1/f

P in dioptres with f in metres

Lenses together

in contact: P = P₁ + P₂ separated by d: P = P₁ + P₂ − dP₁P₂

d = separation

Note:Step by step instead: total m = m₁m₂.

Silvered lens

P = 2P_L + P_M F = 1/P plane face silvered: F = f_L/2

P_L = lens power

P_M = 2/R, the silvered face as a mirror

Note:The system behaves as a concave mirror of focal length F.

Prism

r₁ + r₂ = A δ = i + e − A μ = sin((A + δ_m)/2)/sin(A/2)

A = prism angle

δ_m = minimum deviation, where i = e

Note:Grazing emergence: r₂ = C.

Thin prism and dispersion

δ = (μ − 1)A dispersive power ω = (μ_v − μ_r)/(μ_y − 1)

μ_y = mean (yellow) index

Note:No deviation: (μ₁ − 1)A₁ = (μ₂ − 1)A₂. No dispersion: (μ_v − μ_r) A equal for both prisms.

Microscopes and telescopes

simple: M = D/f (1 + D/f, image at D) compound: M = (L/f_o)(D/f_e) telescope: M = f_o/f_e, length f_o + f_e

D = 25 cm

L = tube length

Light in a medium

f unchanged v = c/μ λ = λ₀/μ

λ₀ = wavelength in vacuum

Two coherent beams

I = I₁ + I₂ + 2√(I₁I₂) cos φ equal beams: I = 4I₀ cos²(φ/2)

φ = phase difference

Note:Incoherent beams simply add: I₁ + I₂.

Brightest and darkest

I_max/I_min = ((√I₁ + √I₂)/(√I₁ − √I₂))² = ((r + 1)/(r − 1))²

r = amplitude ratio = √(I₁/I₂)

Note:Intensity through a slit is proportional to its width unless the stem says amplitude.

Path and phase

φ = (2π/λ)Δx Δx = yd/D

y = position on the screen

d = slit gap, D = screen distance

Double-slit fringes

β = λD/d angular width λ/d bright: y = nλD/d dark: y = (n − ½)λD/d

β = fringe width

Note:In a liquid: β/μ. Two wavelengths coincide where n₁λ₁ = n₂λ₂.

Sheet over one slit

shift = (μ − 1)tD/d fringes shifted N = (μ − 1)t/λ

t = sheet thickness

Note:The pattern moves towards the covered slit; the fringe width stays.

Thin film, normal incidence

one phase reversal: reflected bright at 2μt = (n − ½)λ, dark at 2μt = nλ

t = film thickness

Note:With no reversal, or two, swap bright and dark.

Single slit

minima: a sin θ = nλ central maximum: 2λ/a (angle), 2λD/a (on the screen)

a = slit width

Note:Secondary maxima are half as wide as the central one.

Resolving power

telescope: Δθ = 1.22λ/D microscope: RP = 2μ sin θ/(1.22λ)

D = aperture diameter

Malus' law

unpolarised in: I = I₀/2 then I = I′ cos²θ

θ = angle between successive axes

Note:A sheet at θ between crossed polaroids passes (I₀/8) sin²2θ.

Brewster's law

tan i_B = μ₂/μ₁ i_B + r = 90°

i_B = polarising angle

Note:The reflected light is completely polarised, perpendicular to the plane of incidence.

Photon energy and momentum

E = hν = hc/λ ≈ 1240/λ eV (λ in nm) p = h/λ = E/c

1 eV = 1.6 × 10⁻¹⁹ J

Note:Use the hc the stem prints if it differs.

Photons per second and force

n = P/E = Pλ/(hc) force: P/c absorbed, 2P/c reflected

P = power of the beam

Threshold

φ = hν₀ = hc/λ₀

φ = work function

Note:No emission below ν₀, however bright the light.

Einstein's equation

hν = φ + K_max K_max = eV₀ = ½mv_max²

V₀ = stopping potential

Note:V₀ depends on frequency and metal, not on intensity; the saturation current grows with intensity.

Stopping potential against frequency

V₀ = (h/e)ν − φ/e

slope h/e, the same for every metal

meets the ν-axis at ν₀

Two wavelengths, one metal

e(V₁ − V₂) = hc(1/λ₁ − 1/λ₂)

φ cancels

de Broglie wavelength

λ = h/p = h/√(2mK) = h/√(2mqV) electron: λ = 1.227/√V nm

V = accelerating voltage

Note:Equal λ means equal momentum. Same K: λ ∝ 1/√m.

Particle at temperature T

λ = h/√(3mkT)

k = Boltzmann constant

Rutherford scattering

r₀ = (1/4πε₀)(2Ze²/K) b = (r₀/2) cot(θ/2)

K = alpha's kinetic energy

b = impact parameter, θ = scattering angle

Note:e²/(4πε₀) = 1.44 MeV fm.

Bohr's postulates

mvr = nh/(2π) hν = E_upper − E_lower

n = 1, 2, 3, …

Radius and speed

r_n = 0.529 n²/Z Å v_n = 2.19 × 10⁶ Z/n m/s

Z = atomic number

Note:Period ∝ n³/Z²; current of the orbit ∝ Z²/n³; field at the nucleus ∝ Z³/n⁵; moment μ = neh/(4πm).

Energy levels

E_n = −13.6 Z²/n² eV K = −E, U = 2E

n = 1 is the ground state

Note:Ionisation energy from level n = 13.6 Z²/n² eV.

Rydberg formula

1/λ = RZ²(1/n_f² − 1/n_i²) R = 1.097 × 10⁷ m⁻¹

n_f = lower level, n_i = upper level

Note:Lyman ends on 1, Balmer on 2, Paschen on 3. Series limit: λ = n_f²/(RZ²).

Transition energy

ΔE = 13.6 Z²(1/n_f² − 1/n_i²) eV λ (nm) = 1240/ΔE (eV)

ΔE = photon energy

Number of lines

N = n(n − 1)/2

n = highest level reached

Note:A single atom gives at most n − 1 photons.

X-ray cut-off and recoil

λ_min = hc/(eV) ≈ 1240/V nm v_recoil = E/(Mc)

V = tube voltage

M = mass of the whole atom

Nuclear radius

R = R₀A^(1/3), R₀ ≈ 1.2 fm

A = mass number

Note:Every nucleus has the same density, since A cancels.

Mass defect and binding energy

Δm = Zm_p + (A − Z)m_n − M BE = Δm c², 1 u c² = 931.5 MeV

M = nuclear mass

Note:Compare stability by BE/A, not by total BE.

Q-value

Q = (Σ m_reactants − Σ m_products)c² = Σ BE_products − Σ BE_reactants

Q > 0: energy released

Note:A free proton or neutron has zero binding energy.

Alpha decay at rest

K_α = Q(A − 4)/A K_daughter = 4Q/A

A = parent's mass number

Counting decays

n_α = ΔA/4 n_β = 2n_α − ΔZ

ΔA, ΔZ = parent minus daughter

Decay law

N = N₀e^(−λt) = N₀(½)^(t/T½) A = λN

λ = decay constant

A = activity

Half-life and mean life

T½ = ln 2/λ ≈ 0.693/λ τ = 1/λ ≈ 1.44 T½

τ = mean life

Note:1 Ci = 3.7 × 10¹⁰ Bq.

Two routes

λ = λ₁ + λ₂ T = T₁T₂/(T₁ + T₂)

T₁, T₂ = half-lives of each route alone

Energy from a sample

E = (m/M)N_A Q reactions per second = P/Q

M = molar mass

P = power

Note:1 MeV = 1.6 × 10⁻¹³ J.

Carriers

n_e n_h = n_i²

n_i = intrinsic carrier density

Note:n-type: electrons are the majority. p-type: holes are. Both stay neutral.

LED and photodiode

λ (nm) ≈ 1240/E_g (eV)

E_g = band gap

Note:LED: forward biased. Photodiode: reverse biased. Solar cell: no bias.

Diodes in a loop

I = (V − Σ V_D)/Σ R

V_D = drop of each conducting diode

Note:About 0.7 V for silicon and 0.3 V for germanium unless stated. A reverse-biased ideal diode is an open branch.

Rectifiers

half-wave: output at f full-wave: 2f bridge peak: V_m − 2V_D

f = input frequency

Zener regulator

I_s = (V_in − V_Z)/R_s I_L = V_Z/R_L I_Z = I_s − I_L P_Z = V_Z I_Z

R_s = series resistor

Note:Check breakdown first: without the Zener, does the load voltage exceed V_Z?

Transistor currents

I_E = I_B + I_C α = β/(1 + β) β = α/(1 − α)

α = I_C/I_E

β = I_C/I_B

Common-emitter gains

β = ΔI_C/ΔI_B A_V = β R_L/r_i A_P = βA_V

r_i = ΔV_BE/ΔI_B, input resistance

R_L = load

Note:The output is 180° out of phase with the input.

De Morgan's laws

NOT(A·B) = Ā + B̄ NOT(A + B) = Ā·B̄

· = AND, + = OR

Note:NAND or NOR with tied inputs is a NOT gate.

Simplifying rules

A + AB = A A + ĀB = A + B A ⊕ B = AB̄ + ĀB

⊕ = XOR, 1 when the inputs differ