Basic Physics Concepts

Measurement and units

Physics is a science of measurement. To measure a quantity means to compare it with an agreed standard of the same kind, called a unit. A result is always written as a number and a unit: length = 5 m.

A quantity that can be measured is a physical quantity — length, mass, time, force, temperature. Quantities like beauty or honesty cannot be measured and are not physical quantities.

• Fundamental (base) quantities are independent of one another: length, mass, time, electric current, temperature, amount of substance, luminous intensity.
• Derived quantities are combinations of base quantities: area, volume, speed, force, energy.

The SI system

The International System of Units (SI) is the modern metric system used all over the world. It has seven base units:

Two supplementary (dimensionless) angle units are the radian (plane angle) and the steradian (solid angle).

Since 2019 the SI units are defined by fixing the values of natural constants (for example the speed of light, Planck's constant and the Boltzmann constant). The kilogram is defined through Planck's constant, the second through the caesium-133 atomic transition frequency, and the metre through the distance light travels in a fixed fraction of a second (light travels 299,792,458 m in one second).

Other systems

• CGS: centimetre, gram, second.
• FPS: foot, pound, second (British).
• MKS: metre, kilogram, second — the basis of SI.

Derived units

Special and practical units

SI prefixes

Dimensions

The dimensions of a physical quantity show how it is made up of the base quantities mass [M], length [L] and time [T] (and others). For example:

Uses of dimensions: to check whether a formula is dimensionally correct (both sides must have the same dimensions), to convert units, and to derive the form of a relation. Dimensional analysis cannot give numerical constants (such as ½ or 2π) and cannot handle sums of functions such as sine.

Quantities with the same dimensions as each other: work, energy and torque all have $[M{L}^2{T}^{-2}]$; pressure, stress and energy density have $[M{L}^{-1}{T}^{-2}]$; strain, angle and refractive index are dimensionless.

Describing motion

• Rest and motion are relative: an object is in motion if its position changes with time compared with a reference point.
• Distance is the total length of the path travelled (scalar). Displacement is the shortest straight-line distance from the initial to the final position, with direction (vector). Distance ≥ |displacement|; they are equal for straight-line motion in one direction.
• Speed = distance / time (scalar). Velocity = displacement / time (vector).
• Average speed = total distance / total time. If an object covers equal distances at speeds $v_1$ and $v_2$, the average speed is $\frac{2v_1{v}_2}{v_1+v_2}$ (the harmonic mean), not the simple average.
• Acceleration = change in velocity / time; unit m/s². Retardation (deceleration) is negative acceleration.
• Uniform motion: equal distances in equal times (constant velocity). Non-uniform motion: velocity changes.

Equations of motion (constant acceleration)

where $u$ = initial velocity, $v$ = final velocity, $a$ = acceleration, $t$ = time, $s$ = displacement. The distance covered in the $n$-th second is $s_n=u+\frac{a}{2}(2n-1)$.

Graphs

• Distance–time graph: the slope gives speed. A straight line inclined upward = uniform speed; a horizontal line = at rest.
• Velocity–time graph: the slope gives acceleration; the area under the graph gives displacement. A horizontal line = uniform velocity; an inclined straight line = uniform acceleration.

Free fall

Near the Earth, all bodies fall with the same acceleration $g\approx 9.8{\text{m/s}}^2$ (in the absence of air resistance) — Galileo's result, shown by the feather-and-coin experiment in a vacuum (Newton's tube). For a body dropped from rest: $v=gt$, $h=\frac{1}{2}g{t}^2$. For a body thrown upward with speed $u$: maximum height $H=\frac{u^2}{2g}$; time of ascent $=u/g$; total time of flight $=2u/g$.

Circular motion

In uniform circular motion the speed is constant but the velocity changes direction, so the body is accelerating. The centripetal acceleration is $a=\frac{v^2}{r}$ directed to the centre, and the centripetal force is $F=\frac{m{v}^2}{r}$. Centrifugal force is the apparent outward force felt in a rotating frame. Examples of centripetal force: the tension in a string, gravitation for planets, friction for a car on a bend.

Projectile motion

A projectile launched at angle $\theta$ with speed $u$ has: time of flight $T=\frac{2u\sin \theta }{g}$, maximum height $H=\frac{u^2{\sin }^2\theta }{2g}$, range $R=\frac{u^2\sin 2\theta }{g}$. The range is maximum at 45° and is the same for angles $\theta$ and $90°-\theta$.

Nature of light

Light is a form of energy that produces the sensation of sight. It is an electromagnetic wave and shows both wave and particle behaviour (wave–particle duality). It travels in straight lines in a uniform medium (rectilinear propagation — the reason for shadows and eclipses) and needs no medium.

• Speed of light in vacuum $c\approx 3\times 10^8$ m/s (299,792,458 m/s). It is the maximum speed possible.
• Light takes about 8 minutes 20 seconds to reach the Earth from the Sun and about 1.3 seconds from the Moon.
• Light is a transverse wave (it can be polarised). Sound is a longitudinal wave.
• Luminous objects emit light of their own (Sun, stars, lamp); non-luminous objects are seen by reflected light (Moon, planets).
• Transparent (glass, clear water) transmits light; translucent (frosted glass, oiled paper) partly; opaque (wood, metal) blocks it.

Reflection

When light falls on a surface and bounces back, it is reflected.

Laws of reflection:

1. The angle of incidence equals the angle of reflection ($i=r$).
2. The incident ray, the reflected ray and the normal at the point of incidence all lie in the same plane.

Plane mirror

The image is virtual, erect, the same size as the object, at the same distance behind the mirror as the object is in front, and laterally inverted (left–right reversed — the reason ambulance is written reversed). A plane mirror rotated by an angle $\theta$ turns the reflected ray through $2\theta$. To see one's full image, a mirror of half the person's height is enough.

Spherical mirrors

• Concave mirror (converging): reflecting surface curved inward. Convex mirror (diverging): curved outward.
• Terms: pole (P), centre of curvature (C), radius of curvature (R), principal axis, principal focus (F) and focal length (f). For small apertures: $f=\frac{R}{2}$.

Mirror formula:

Magnification: $m=\frac{h^{\prime }}{h}=-\frac{v}{u}$. (Sign convention: distances measured from the pole; taken positive in the direction of the incident light. A concave mirror has negative $f$; a convex mirror has positive $f$.)

A convex mirror always gives a virtual, erect, diminished image between P and F, whatever the object position.

Uses: concave mirrors — shaving mirrors, dentists' mirrors, headlights, reflectors, solar cookers, telescopes (reflecting). Convex mirrors — rear-view mirrors of vehicles (wide field of view, erect image), street lights, security mirrors.

Refraction

Refraction is the bending of light when it passes from one transparent medium to another, because its speed changes.

• Light bends towards the normal when it enters a denser medium (slower) and away from the normal when entering a rarer medium.
• Snell's law: $\frac{\sin i}{\sin r}=n$ (constant for a pair of media), the refractive index.
• Refractive index of a medium $n=\frac{c}{v}$ (speed of light in vacuum / speed in the medium). It is a pure number, at least 1. Typical values: air ≈ 1.0003, water ≈ 1.33, glass ≈ 1.5, diamond ≈ 2.42 (highest among common substances; the reason for its brilliance).
• Effects of refraction: a pencil in water appears bent; a pond appears shallower than it is (apparent depth = real depth / $n$); stars twinkle (varying refractive index of the atmosphere); the Sun is seen a little before sunrise and after sunset; a swimming pool looks shallower.
• Lateral shift through a glass slab: the emergent ray is parallel to the incident ray but displaced.

Total internal reflection (TIR)

When light travels from a denser to a rarer medium and the angle of incidence exceeds the critical angle $C$, all light is reflected back into the denser medium. The critical angle is given by $\sin C=\frac{1}{n}$ (about 41.8° for glass–air, 48.6° for water–air, 24.4° for diamond–air).

Conditions: light must pass from a denser to a rarer medium, and the angle of incidence must be greater than the critical angle.

Applications and examples:

• Optical fibres (communication, endoscopy) — light is trapped by repeated TIR in a glass or plastic core.
• Mirage (desert, hot roads), shining of air bubbles in water, brilliance of a diamond, totally reflecting prisms (periscope, binoculars), shining of a test tube in water.

Lenses

A lens is a transparent material bounded by two curved surfaces.

• Convex (converging) lens: thicker at the centre; converges parallel rays; real focus.
• Concave (diverging) lens: thinner at the centre; spreads rays; virtual focus.

Lens formula: $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$; magnification $m=\frac{v}{u}$.

Power of a lens: $P=\frac{1}{f}$ (with $f$ in metres). SI unit: dioptre (D). A convex lens has positive power, a concave lens negative. Power of a combination of thin lenses in contact: $P=P_1+P_2$.

A concave lens always forms a virtual, erect, diminished image.

The human eye

• Cornea (transparent front; most of the refraction), aqueous humour, iris (coloured; controls the size of the pupil), pupil, eye lens (adjusts focus), ciliary muscles, vitreous humour, retina (light-sensitive screen with rods for dim light and cones for colour), optic nerve.
• Accommodation is the ability of the eye lens to change its focal length to focus objects at different distances. The near point of a normal eye is 25 cm; the far point is at infinity.
• The image on the retina is real and inverted; the brain interprets it upright. Persistence of vision is about 1/16 of a second (basis of cinema).

Defects of vision

Dispersion, scattering and colours

• Dispersion: splitting of white light into its colours by a prism — VIBGYOR (violet, indigo, blue, green, yellow, orange, red). Violet bends the most and red the least; red has the longest wavelength and violet the shortest. Newton first showed that white light is a mixture of colours.
• Rainbow: formed by refraction, internal reflection and dispersion in raindrops, seen with the Sun behind the observer.
• Scattering: the scattering of light is proportional to $1/{\lambda }^4$ (Rayleigh) — blue is scattered much more than red. Therefore the sky is blue, the Sun looks red at sunrise and sunset (blue scattered away over the long path), danger signals are red (least scattered), and the sky appears dark to an astronaut (no atmosphere).
• Twinkling of stars is due to atmospheric refraction; planets do not twinkle because they are close and are extended sources.

Colours

• Primary colours of light: red, green, blue (RGB). Mixing them gives white. Secondary colours: yellow (R + G), cyan (G + B), magenta (R + B).
• Primary pigments (subtractive): cyan, magenta, yellow (CMY) — used in printing.
• The colour of an opaque object is the colour it reflects; a red rose in green light looks black. A transparent object has the colour it transmits.
• Complementary colours add up to white (e.g. red and cyan).

Optical instruments

Other phenomena

• Interference, diffraction and polarisation confirm the wave nature of light. Young's double-slit experiment shows interference. Polaroid sunglasses reduce glare.
• Photoelectric effect (Einstein, Nobel Prize 1921) demonstrates the particle nature: light consists of photons of energy $E=h\nu$.
• Laser (Light Amplification by Stimulated Emission of Radiation) gives a highly monochromatic, coherent, directional beam; used in surgery, cutting, barcode scanners, communication, holography.
• Raman effect (C. V. Raman, Nobel Prize 1930) is the change in wavelength of scattered light; National Science Day is 28 February.
• Eclipses: a solar eclipse occurs at new moon when the Moon lies between the Sun and the Earth; a lunar eclipse at full moon when the Earth lies between the Sun and the Moon.

Frequently tested facts

• Speed of light ≈ 3 × 10⁸ m/s; sunlight takes about 8 min 20 s to reach the Earth.
• Plane mirror: virtual, erect, same size, laterally inverted; convex mirror in vehicles; concave mirror in headlights and shaving.
• Refractive index $n=c/v$; diamond has the highest among common substances (about 2.42).
• TIR: optical fibres, mirage, diamond's sparkle.
• Power of a lens $=1/f$ (dioptre); convex lens positive, concave negative.
• Myopia → concave lens; hypermetropia → convex lens.
• Red is scattered least and violet most; the sky is blue due to scattering.
• Primary colours of light: red, green, blue.

What is sound?

Sound is a form of energy produced by vibrating bodies (a vibrating string, a tuning fork, vocal cords). It needs a material medium (solid, liquid or gas) to travel — sound cannot travel through a vacuum. This is why an astronaut on the Moon cannot hear another person directly, and why a bell in an evacuated jar cannot be heard.

• Sound travels as a longitudinal wave: particles of the medium vibrate parallel to the direction of the wave, producing alternate compressions (regions of high pressure) and rarefactions (low pressure).
• Sound in air is a mechanical, longitudinal wave (unlike light, which is electromagnetic and transverse). In solids sound can also travel as a transverse wave.

Wave terms

Frequency depends only on the source and remains the same when the wave passes from one medium to another; the speed and wavelength change.

Speed of sound

• The speed depends on the medium and its temperature; it does not depend on the loudness or frequency.
• Solids > liquids > gases: about 5000 m/s in steel/iron, about 1500 m/s in sea water, about 1480 m/s in fresh water, and about 343 m/s in air at 20 °C (331 m/s at 0 °C).
• In air, the speed increases by about 0.6 m/s for each °C rise. It is larger in humid air than in dry air (water vapour is lighter). It is affected by pressure only through density, and in a gas at constant temperature does not change with pressure.
• Sound is slower than light; in a thunderstorm we see lightning first and hear the thunder later. Distance to lightning ≈ speed of sound × time delay.

Characteristics of sound

• Intensity is the energy flowing per second through unit area (W/m²). Loudness is a subjective measure of intensity.
• Audible range for human beings: 20 Hz to 20,000 Hz (20 kHz).
• Infrasonic sound: below 20 Hz (earthquakes, whales, elephants); ultrasonic sound: above 20 kHz.
• Typical levels: whisper about 30 dB; normal conversation about 60 dB; heavy traffic about 80 dB; a loud horn about 110 dB; the threshold of pain is about 120–130 dB. Prolonged exposure above about 85 dB can damage hearing.

Electric charge

• Matter contains positive charges (protons), negative charges (electrons) and neutral particles (neutrons). Charge is quantised: the smallest free charge is that of the electron, $e=1.6\times 10^{-19}$ C. The SI unit of charge is the coulomb (C).
• Like charges repel; unlike charges attract. Charge is conserved.
• Coulomb's law: $F=k\frac{q_1{q}_2}{r^2}$, with $k=9\times 10^9{\text{N m}}^2{\text{C}}^{-2}$.
• Conductors (metals, graphite, acids, salt solutions, the human body) allow charge to flow; insulators (rubber, glass, plastic, dry wood, air) do not; semiconductors (silicon, germanium) lie in between.
• Charging: by friction (glass rod rubbed with silk becomes positive, ebonite/plastic rubbed with fur becomes negative), by conduction and by induction.
• The gold-leaf electroscope detects charge. Lightning is a huge electric discharge between clouds or between a cloud and the ground; a lightning conductor (arrester) on tall buildings safely conducts the charge to earth (Benjamin Franklin).

Electric current and potential difference

• Electric current is the rate of flow of charge: $I=\frac{Q}{t}$. The SI unit is the ampere (A): 1 A = 1 C/s. By convention the direction of current is from the positive to the negative terminal (opposite to the flow of electrons).
• Ammeter measures current; it is connected in series and has a very low resistance.
• Potential difference (voltage) between two points is the work done per unit charge: $V=\frac{W}{Q}$. SI unit: volt (V). Voltmeter measures it; it is connected in parallel and has a very high resistance.
• A cell is a source of EMF; a battery is a combination of cells. The EMF is the potential difference across the terminals when no current is drawn.

Ohm's law

At constant temperature, the current through a conductor is proportional to the potential difference across it:

$R$ is the resistance (unit: ohm, Ω). Conductors that obey Ohm's law are ohmic (metals at constant temperature); a diode and a filament lamp are non-ohmic.

Resistance and resistivity

Resistance is directly proportional to length and inversely proportional to the cross-section area, and depends on the material ($\rho$ = resistivity, unit Ω·m) and temperature. For metals, resistance increases with temperature; for semiconductors and insulators it decreases.

• Good conductors: silver (best), copper, gold, aluminium. Nichrome, constantan and manganin have high resistivity (used in heaters, resistance boxes). Tungsten has a high melting point (3422 °C) and is used for bulb filaments. Superconductors have zero resistance below a critical temperature.
• Conductance $G=1/R$ (unit siemens).

Combination of resistors