Diffraction of light also demonstrates the wave These spectra are evidence for quantised atomic energy levels. Standing waves in closed resonant tube of length L:
nature of light. Electrons move around a nucleus with discrete energies. When Overtones Harmonics λ f = v/λ
Gap Diffraction pattern has an electron jumps from high to low energy level, it loses energy Fundamental first 4L/1 1(v/4L)
width a bright central region in discrete amount equal to the difference between the two
levels and results in emission of a photon of the same energy. First third 4L/3 3(v/4L)
w between 2 dark fringes
Second fifth 4L/5 5(v/4L)
hf = EH – EL. de Broglie used the idea of standing matter
waves to explain the quantised energy levels of an atom. The For closed tubes only odd harmonics exist.
Obstacle Diffraction pattern has only matter waves that persist are those for which the Dynamic microphone: Sound moves the cone and the
width a dark shadow of the circumference of the orbit is an integral multiple of λ. attached coil of wire in a magnetic field to and fro.
obstacle with a bright Travelling sound wave through air is longitudinal because the Electromagnetic induction produces an emf (signal) at the
w fringe at the centre air molecules oscillate parallel to the direction of propagation of terminals of the coil. Ribbon (or velocity) microphone: Air
caused by constructive the sound wave. A sequence of high (compression) and low movement due to sound waves moves the metallic ribbon in
interference of light (rarefaction) air pressure is generated and it propagates a magnetic field. Electromagnetic induction generates emf
around the obstacle outwards from the source carrying the sound energy with it. between the ends of the ribbon. Condenser microphone:
The back plate and the front metallic membrane form a
Extent of diffraction ∝ λ . Significant effect when λ ≈ 1 . ∆p (at a particular time) ∆p (at a particular point)
capacitor (charged with a battery). Sound waves cause the
More diffraction when λ is longer and/or w is smaller. membrane to vibrate and change the spacing between the
x t plate and the membrane. This causes the output voltage
Photoelectric effect demonstrates particle-like nature of λ T (signal) to change. In electret-condenser microphone a
Light Accel./ f = 1 , speed of sound v = λ = fλ . permanently charged electret material is used for the
T T membrane, thus eliminating the need of a charging battery.
Setup V retard. Crystal microphone: uses a thin strip of piezoelectric
v( solid ) > v( water ) > v(hotair ) > v (coolair) . v is
voltage crystal attached to a diaphragm that is sent into vibration by
constant when sound travels in the same medium,
sound waves, causing the crystal to deform and produce a
I λ f voltage (signal).
∴ λ ∝ 1 and 2 = 1 . f is constant when sound travels A dynamic loudspeaker has the same basic construction as
f λ1 f 2
I (Constant freq.) I (Constant int.) a dynamic microphone. The input signal changes the current
H v2 λ2 in the coil and results in a varying magnetic force on the coil
from a medium into another, ∴v ∝ λ and = .
L intensity H L frequency v1 λ1 that is attached to the cone.
Vo V V Enclosure formed by baffles: to prevent the sound from the
Sound intensity I measures the amount of energy (J) arriving at back of the speaker cone cancelling the sound from the front
retard accel retard accel a square metre of surface in a second. It is defined as because of destructive interference due to phase difference.
Max EK E P Directional spread of sound waves (diffraction): Sound
Metal 1 Metal 2 Same gradient for I= = , E energy received, A area exposed, ∆t time diffracts when it passes by the edge of a barrier. Refer to
both metals = h
exposed. Unit: Js-1m-2 or Wm-2. For a small sound source in the diffraction of light. Extent of diffraction ∝ λ . w is the
f (Planck’s constant) open, the sound energy spreads outwards spherically,
φ2 width of obstacle or opening. High pitch (high f, short λ)
φ2 work function of metal 2 P 1 I b ra2 sound diffracts less than low pitch.
I= , P is the power of source, ∴ I ∝ , = .
Threshold frequencies fo1 and fo2 4πr 2
r 2 I a rb2 A loudspeaker is omni-directional, (i.e. it radiates sound
φ = hfo , max EK = qVo
When the distance r from the source is doubled, intensity I is a energy spherically in all directions) when λ > 4 , w is the
Failure of the wave model to explain the photoelectric w
quarter of the original value.
effect According to the wave model, light is a continuous diameter of speaker cone. The higher the frequency the less
wave and the intensity is related to its amplitude, which I omni-directional it becomes.
Sound intensity level L = 10 × log10 in dB,
measures the energy of the wave. Therefore an electron can 10−12 Frequency response of human ear: is most sensitive to
absorb any amount of light energy, depending on the time sound of frequency 4000Hz, e.g. of the three sounds, 100Hz,
∆L 4 kHz and 10 kHz, at the same dB level at the ear, the 4kHz
interval it is exposed to the light wave. The wave model If L
failed to explain why (1) maximum kinetic energy remained ∆L = 10 × log10 . I = 10 10 , = 10 10 . will sound the loudest to the listener. To make 100Hz and
the same when the intensity was changed; (2) maximum Ii Ii 10kHz the same loudness as 4 kHz, increase their dB level.
kinetic energy changed with the frequency of light used; If Loudness is measured in phons. The loudness of a sound is
(3)there was a threshold frequency for each metal used. When I is doubled, i.e. = 2 , ∆L = +3 dB. compared with the loudness of 1 kHz sound. The loudness
Einstein’s interpretation of photoelectric effect-the
Ii of a x dB 1 kHz sound is x phons. Sounds at different
photon model: A beam of light is a stream of particles If frequencies, which are as loud as the x dB 1 kHz sound have
1 a loudness of x phons. The following graph shows a curve of
called photons. Light of a single frequency f consists of When r is doubled, = , ∆L = − 6 dB.
photons of the same energy E = hf = hc/λ . Ii 4 equal loudness (30 phons) for different frequencies.
There are more photons in a more intense beam, hence Threshold of hearing 10-12 Wm-2 0 dB L(dB)
higher current. When photons strike a metal, some will be Normal conversation 10-6 60 80 Above curve,
absorbed by the electrons in the metal. To have louder than 30 phons 30-phon curve
photoelectrons emitted, the energy of each photon must be Car alarm 1 m away 10-2 100
high enough for the electrons to overcome the bonding Threshold of pain 1 120 30 Softer softer below curve
energy (i.e. the work function φ). As the photons penetrate Jet engine 30 m away 102 140
into the metal they collide with other electrons before they 0 20 1k 15 k f (Hz)
After reflection, f, λ and v remain the same. When the forward
are absorbed. Each collision lowers the photon frequency Frequency response curve of a microphone: is a graph of
and the reflected travelling waves superpose each other, a
(energy) slightly, the Compton effect. ∴ electrons at the output intensity level versus frequency for a constant input.
standing wave (a sequence of loud and soft sound at fixed
surface escape with higher (max) kinetic energy than those Zero dB is assigned to 1 kHz sound as the reference level.
positions quarter of a wavelength apart) is formed between the
inside metal, max EK = hf – φ for surface electrons. L(dB)
source and the wall. Pressure antinodes (max fluctuation in air
The Compton effect and photon momentum: The particle 10
pressure) give loud sound and pressure nodes (min fluctuation)
nature of light was further supported by the Compton effect. 0
give soft sound.
Photon momentum p = E = h . Wall – 10
c λ Loud- loud soft
The two models (wave and particle) of light appear to be 50 150 1 k 4k 6k f (Hz)
Speaker L S L S L S L
inconsistent with each other but both have been shown to be This graph indicates that the microphone responds equally
∆p well to frequencies between 150 and 4 kHz, more sensitive
valid depending on the circumstances. This dual nature of
light is known as wave-particle duality. to over 4 kHz, less sensitive to below 150 Hz.
Wave nature of matter: de Broglie proposed that a moving Frequency response of multi-speaker system:
material particle also has wave-particle duality. Wavelength λ L(dB) woofer tweeter
of particle is related to its momentum (like a photon). Every object has its own natural frequencies of vibration. If an
Fairly flat between
energy source at one of these frequencies interacts with the
h h h 20 and 20 kHz
de Broglie λ= = = . These equations are object, the latter will be set into vibration, i.e. a standing wave
p mv 2mEk is formed. The object is in resonance. The natural frequencies
of vibration are called resonant frequencies. 0 20 20 k f (Hz)
valid when λ in m, m in kg, v in ms-1, EK in J, h in Js. A single loudspeaker on its own (e.g. the woofer or tweeter)
The diffraction of electrons from the surface of a metal Standing waves in a stretched string of length L:
tends to ‘colour’ the sound it produces, i.e. some
crystal confirmed the wave nature of matter. Overtones Harmonics λ f = v/λ frequencies are louder than others due to resonance. An
An electron with the same λ as a photon has the same Fundamental first 2L/1 1(v/2L) ideal loudspeaker system would need to have the same
momentum as the photon, p = h/λ. First second 2L/2 2(v/2L) loudness at all frequencies, i.e. a fairly flat response curve.
When a gas or metal vapour is heated, the gas or vapour Second third 2L/3 3(v/2L) Some loudspeaker enclosures have tubes (called ports) put
glows and emits a characteristic diffraction pattern (obtained in them. Size and depth of ports can be changed to absorb
Note: v is the speed of travelling wave in the string.
with a diffraction grating) called an emission spectrum. sound of particular frequencies to produce a flat response.
When sunlight passes through a gas/vapour, some dark lines Standing waves in open resonant tube of length L: The
appear in its spectrum called absorption spectrum, caused vibration of the air column in the tube forms a standing wave.
by the absorption of certain wavelengths of sunlight by the Has the same pattern of harmonics as strings but v is the speed
atoms or molecules in the gas/vapour. of travelling sound wave in the tube.