Formulas & shortcuts

These four units underlie all the rest: the electrical units start from the joule and the watt.

The volt comes out two ways — joule per coulomb or watt per ampere — with the same result.

All magnetic units follow from the Ampère force and the law of electromagnetic induction: once you know the weber, the tesla and the henry follow easily.

From this, s = v · t and t = s / v.

If vₓ and aₓ have the same sign, the speed grows; if the signs are opposite, the body slows down. So a “negative acceleration” does not always mean braking!

The coordinate equation of uniformly accelerated motion. The displacement is sₓ = x − x₀ = v₀ₓt + aₓt²/2; if the body does not turn back, this is also the distance.

The average velocity is the mean of the initial and final velocities — only for uniformly accelerated motion!

Another form: v² − v₀² = 2as. Braking distance: s = v₀² / (2|a|).

It comes from the difference sₙ = s(n) − s(n − 1).

The slope of the x–t graph is the velocity projection: the steeper the line, the greater the speed; a falling line means vₓ < 0.

The slope of the vₓ–t graph is the acceleration projection.

The area of the trapezium under vₓ–t. For uniform motion it is a rectangle: sₓ = vₓ · t. In the same way the rectangle under aₓ–t gives Δvₓ = aₓ · t.

The coordinate equation of uniformly accelerated motion; its graph is a parabola that opens downwards when aₓ < 0. The formula is explained in detail in the lesson «Acceleration and uniformly accelerated motion».

Connecting springs. If you cut a spring into n equal parts, each part has stiffness n · k.

They follow from a = μg and the formulas of uniformly decelerated motion.

The force of gravity and how it depends on height.

When v = √(g · R), W = 0 at the top of a convex bridge — weightlessness.

The formulas of free fall (v₀ = 0).

Equations of motion for a body thrown straight up.

Symmetry: the body comes back with speed v₀.

Horizontal throw: the time of flight does not depend on v₀. The landing speed is v = √(v₀² + (gt)²).

The range is greatest at 45°; angles that add up to 90° (30° and 60°) give the same range.

The second formula follows from the first: F_A = P − P_l = ρ_l g V and P = ρ_b g V; dividing them cancels g and V.

The buoyant force on a floating body always equals its weight and does not depend on the liquid’s density: in a denser liquid the body sinks less, in a less dense one more.

P — power, in watts (W); W — work done, in J

Rule of moments: a lever is in equilibrium when the clockwise moments equal the anticlockwise moments. The forces are inversely proportional to their arms. If the lever’s own weight is given, it acts at the centre of gravity (the middle of a uniform lever).

This is the rule of moments itself: about the centre of gravity the moments of the gravity forces add up to zero, so the system balances when supported at this point.

Friction and the rope’s weight are neglected. A block and tackle (a system of pulleys) reduces the force n times, while the end of the rope travels n times farther.

The golden rule for an ideal (frictionless) inclined plane: the work F · l equals the work P · h needed to lift the load straight up. The longer (gentler) the plane, the smaller the force.

Always η < 100%: the golden rule holds only for an ideal machine. For the same useful work, the lower the efficiency, the larger the force needed.

Current is measured with an ammeter, which is connected in series.

From this, U = I · R and R = U / I.

The permeability shows how many times a substance strengthens (or weakens) the field.

1 T = 1 N / (A · m). The Earth’s field is about 5 · 10⁻⁵ T; a medical MRI scanner uses 1.5–3 T. B does not depend on the wire or the current: a larger current gives a larger F_max, but B stays the same.

The period depends on neither the speed nor the radius: a faster particle draws a bigger circle but goes round it in the same time. Frequency ν = 1 / T = |q| · B / (2π · m).

For ions of equal charge R ~ √m: an ion 4 times heavier moves on a circle of twice the radius.

f — frequency, in Hz

Water n ≈ 1.33, glass ≈ 1.5, diamond ≈ 2.42

P — lens power, in dioptres (D); f — focal length, in metres. For a diverging lens f and P are negative.

Perfectly inelastic collision

Perfectly elastic head-on collision, second body initially at rest

The force that causes it is F = m · v² / R. It is not a new kind of force but the net force supplied by friction, tension, gravity and so on.

For a circular orbit this follows from T = 2πr / v and v² = G · M / r.

Amount of substance. It gives the number of molecules N = (m / M) · N_A.

Mass of one molecule. The volume per molecule is V₀ = V / N, and the molecule’s size is estimated as d ≈ ∛V₀.

The basic equation of the kinetic theory of an ideal gas. Since (N/V) · m₀ = ρ, it can also be written p = (1/3) · ρ · v².

The mean kinetic energy of a molecule of any ideal gas depends only on temperature. At the same p and T all gases have the same number density.

The speed grows as √T and falls as √M: at the same temperature lighter molecules are faster.

The Mendeleev–Clapeyron equation (ideal-gas law). It gives the density of a gas: ρ = m / V = p · M / (R · T).

Clapeyron’s equation (the combined gas law), m = const.

The three isoprocess laws (m = const).

η_max is the efficiency of an ideal (Carnot) engine; no real engine working between the same temperatures can do better.

The Mendeleev–Clapeyron equation for a vapour; for saturated vapour p = p_s, ρ = ρ_s. Boyle’s law cannot be applied to saturated vapour — its mass changes when it is compressed.

Relative humidity. A comfortable humidity for people is about 40–60 %.

Strain and stress — the elastic force per unit cross-sectional area.

Stiffness depends on the material, the cross-section and the length, not on the applied force.

The area of the triangle under the F(Δl) graph. When the force is given, the form F² / (2k) is handiest.

For complete wetting: the surface tension force σ · 2πr holds up the weight of the column ρgπr²h. h is inversely proportional to the radius.

Only for conducting balls of the same size and shape. Mind the signs: +8 and −2 → (8 − 2) / 2 = +3.

For point charges (bodies much smaller than the distance between them). By Newton’s third law F₁₂ = F₂₁ — even when the charges differ, they act on each other with the same force.

The field strength is a vector: it points away from a positive charge and towards a negative one. A charge q in the field feels F = q · E; the force on a negative charge (an electron) points opposite to E.

When the field does positive work, the potential energy decreases. For like charges Wp > 0 (an external force must do work to bring them closer), for unlike charges Wp < 0. The zero level is at infinity.

φ = 0 at infinity. In the field of a positive charge the potential is positive and grows towards the charge; in the field of a negative charge it is negative. φ ~ 1/r while E ~ 1/r²: when the distance doubles, the potential halves and the field strength drops 4 times. At the same point |φ| = E · r.

The potential of a negative charge is taken as negative. Where the potential is zero the field strength need not be zero, and vice versa — see Example 4.

1 V is the voltage between two points when the field does 1 J of work moving 1 C of charge. A positive charge released in a field moves from high to low potential, a negative charge (an electron) from low to high potential — in both cases the field does positive work.

For a uniform field only. The field points in the direction in which the potential decreases. Using this formula in a capacitor is covered in the lesson «Capacitance and capacitors».

For a charged conducting sphere in air. Outside the sphere the field is as if all the charge sat at the centre. Inside E = 0, and the potential is the same as on the surface. The field strength is greatest just outside the surface: E = k · q / R².

ε = 1 for a vacuum, ≈ 1 for air, ≈ 2 for kerosene and paraffin, ≈ 7 for glass, 81 for water. In a dielectric the field strength and potential of a point charge, and the force between charges, are ε times smaller than in a vacuum.

The speed is proportional to √U: four times the voltage gives twice the speed and four times the kinetic energy. If the field slows the particle down, the voltage that stops it is U = m · v₀² / (2 · |q|).

A particle entering midway between the plates misses them if y < d / 2. At the exit the velocity component perpendicular to the plates is a · t, and the deflection angle of the velocity is given by tan β = a · t / v₀.

1 F = 1 C/V. The farad is a very large unit, so µF (10⁻⁶ F), nF (10⁻⁹ F) and pF (10⁻¹² F) are used. Capacitance depends neither on the charge nor on the voltage: when q grows, U grows by the same factor and the ratio stays the same.

The capacitance of a parallel-plate capacitor is directly proportional to the plate area and inversely proportional to the gap; it depends only on the geometry and the dielectric.

For two capacitors in series C = C₁ · C₂ / (C₁ + C₂) — “product over sum”.

The three forms are the same formula (substitute q = C · U). Which one to use depends on what stays constant: if U is constant, use C · U² / 2; if q is constant, use q² / (2C).

The energy density grows as the square of the field strength: double the field and every cubic metre holds four times the energy.

Ohm's law for a complete circuit. With R = 0 you get the short-circuit current I = ℰ / r.

Electricity meters measure energy in kilowatt-hours: 1 kW · h = 3.6 · 10⁶ J.

In a series circuit the voltage is shared in proportion to the resistances: U₂ / U₁ = Rₜ / R. The element whose resistance drops gets a smaller share of the voltage.

The minus sign expresses Lenz’s rule: the induced current creates a magnetic field that opposes the change in flux that caused it.

The charge does not depend on how fast the flux changes: push the magnet in quickly and the current is larger but lasts a shorter time — the same charge passes through the galvanometer. If Φ is given as the total flux through all the turns, N is not written separately: q = ΔΦ / R.

For a long coil (solenoid). Inductance grows as the square of the number of turns, and an iron core increases it hundreds of times.

The minus sign is Lenz’s rule: when the current grows (ΔI > 0) the EMF acts against the source, when it falls the EMF acts along the current. This gives a second definition: 1 H is the inductance of a circuit in which an EMF of 1 V appears when the current changes steadily by 1 A every second.

It looks like kinetic energy mv²/2: inductance plays the role of “electrical mass” and current the role of “velocity”. Its partner for a capacitor is W_E = CU²/2 (see “Capacitance and capacitors”).

The energy density is proportional to B²: a field twice as strong holds 4 times more energy in every cubic metre.

√2 ≈ 1.41, 1/√2 ≈ 0.71. The mains has U = 220 V, so U_m = 220 · √2 ≈ 311 V.

As f grows, X_L grows in proportion and X_C falls in inverse proportion; R does not depend on frequency.

With only a resistor Z = R, only a coil Z = X_L, only a capacitor Z = X_C. When X_L = X_C, Z = R is smallest and the current is largest — this is resonance (see “Electromagnetic oscillations and waves”).

Power is released only in the resistance. With only a resistor cos φ = 1 and P = U · I; for an ideal coil or capacitor cos φ = 0 and P = 0.

A transformer works only with alternating current: direct current gives a constant flux, and no EMF appears in the secondary.

Large transformers reach an efficiency of 99 %. In an ideal transformer (η = 1), U₁I₁ = U₂I₂, so I₁ / I₂ = N₂ / N₁: it lowers the current by the same factor as it raises the voltage.

Raising the voltage n times cuts the loss n² times. That is why a step-up transformer raises the voltage at the power station and step-down transformers lower it again near the consumers.

In an ideal circuit (no resistance) the total energy is conserved. So the W_M(W_E) graph is the straight line W_M = W − W_E: it cuts the same length W off both axes. Since q_m = C · U_m, we can also write W = C · U_m² / 2.

Thomson’s formula: the period depends only on L and C, not on the amplitude of the charge. The charge changes as q = q_m · cos ωt and the current as i = −I_m · sin ωt; I_m = ω · q_m, U_m = q_m / C. The current reaches its maximum a quarter of a period after the charge.

When a wave passes into another medium its frequency stays the same, while its speed and wavelength change. The wavelength radiated (or received) by an LC circuit is λ = c · T = 2πc · √(L · C).

Radar is based on the reflection of radio waves. The radar sends short pulses; the next pulse must be sent only after the reflected signal has come back.

Moving away from a point source, the energy spreads over a sphere of area 4πr²: twice the distance means a quarter of the intensity. The radiated intensity is proportional to the fourth power of the frequency — that is why radio needs high-frequency carrier waves.

In components: vₓ = dx/dt, a_x = dvₓ/dt = d²x/dt², etc.; the speed is |v| = √(vₓ² + v_y² + v_z²).

Trajectory equation: solve x = v₀ cos θ · t for t and substitute into y(t) — the result is a parabola.

T comes from y(t) = 0, H from v_y = 0, and L = vₓ · T.

If the external torque is zero, L = const: I₁ω₁ = I₂ω₂.

The answer depends on neither mass nor radius — only on the shape (k).

Velocity v = −Aω₀ sin(ω₀t + φ₀), acceleration a = −ω₀²x; the total energy E = kA²/2 is constant.

The maximum is at ω_res = √(ω₀² − 2β²) ≈ ω₀; for weak damping A_max ≈ F₀ / (2mβω₀), which is Q times the static displacement.

I. Gauss's law: charges are the sources of the electric field.

III. Faraday's law: a changing magnetic field creates a circulating electric field.

IV. Ampère–Maxwell law: currents and changing electric fields create a circulating magnetic field.

Wave intensity: I = c · ε₀ · E₀² / 2 (W/m²).

Lengths contract only along the direction of motion; perpendicular dimensions are unchanged.

For v ≪ c they reduce to the Galilean ones: x′ = x − vt, t′ = t, u = u′ + v.

In atomic physics energy is measured in electronvolts: 1 eV = 1.6 · 10⁻¹⁹ J.

Each element emits only photons matching differences between its own levels — its spectral “fingerprint”.

Mean lifetime τ = 1/λ = T / ln 2 ≈ 1.44 T.