Abstract:
Frequency is the rate at which energy vibrates, and it is a fundamental property of waves, whether mechanical waves (e.g., sound waves) or electromagnetic waves (e.g., light waves).
For mechanical waves:
The kinetic energy (Eₖ) of a mechanical wave is given by Eₖ = 1/4(μA²ω²λ), where μ is the linear mass density of the wave medium, A is the amplitude of the wave, ω is the angular frequency of the wave oscillator, and λ is the wavelength of the wave.
The frequency (f) of the kinetic energy (Eₖ) of a mechanical wave is given by f = √ {Eₖ / (π²μA²λ)}.
The period in degrees {T(deg)} of the kinetic energy (Eₖ) of a mechanical wave is given by T(deg) = 1/[360√{Eₖ/(π²μA²λ)}].
For electromagnetic waves:
The energy (E) of an electromagnetic wave is related to its frequency (f) through Planck's equation, E = hf, where h is Planck's constant.
The energy of electromagnetic waves is directly proportional to their frequency. As the frequency increases, the energy of the wave also increases.
Kinetic
energy | Radians and
degrees | Radian-degree
relation | Angular
frequency relation | Kinetic energy frequency | Period kinetic energy
| Wave potential energy |
Energy mechanical wave
| Energy EM wave | Maximum wave speed | Phase shift-frequency
| Relationships |
Time reading error
| Photon momentum
| An electron | Electron rest mass
| Electron energy | A photon | Photon energy | Electron energy | Energy in mass
Frequency is the rate at which energy vibrates.
Frequency, in terms of energy, refers to the rate at which
energy oscillates or vibrates. It is a fundamental property of waves, whether they
are mechanical waves, such as sound waves, or electromagnetic waves, such as
light waves.
1.0. For mechanical waves, e.g., sound waves, the
frequency is related to the kinetic energy of the wave. The kinetic energy (Eₖ) of a mechanical wave is given by the formula:
Eₖ = 1/4(μA²ω²λ);
Where:
Eₖ = Kinetic energy of the
mechanical wave
μ = Linear mass density of the wave medium (in kg/m)
A = Amplitude of the wave (in meters)
ω = Angular frequency of the wave oscillator (in radians
per second)
λ = Wavelength of the wave (in meters)
1.1. The radians and degrees:
Degree (°):
A degree is a unit of angular measurement. A full circle
is divided into 360 degrees, with each degree representing 1/360th of the
complete rotation. The symbol for degrees is "°." For example, a
right angle measures 90 degrees (90°), and a straight line measures 180 degrees
(180°).
Radian (rad):
A radian is another unit of angular measurement and is
defined as the angle subtended at the center of a circle by an arc equal in
length to the radius of the circle. In other words, if the length of the arc is
equal to the radius of the circle, then the angle formed is one radian. The
symbol for radians is "rad."
1.2. Relationship between Radian and Degree:
The relationship between radians and degrees is based on
the fact that a full circle (360 degrees) is equal to 2π radians. More
precisely:
1 full circle = 360 degrees = 2π radians
Degrees = Radians * (180/π)
Where: π (pi) is approximately 3.14159.
To convert from radians to degrees: Multiply by (180/π).
To convert from degrees to radians: Multiply by (π/180).
1.3. The angular frequency (ω) is related to the
frequency (f) by the equation:
ω = 2πf;
1.4. Angular Frequency (ω) of Eₖ:
We have: ω = 2πf;
ω = 2π√ {Eₖ / (π²μA²λ)};
1.5. The equation for the frequency (f) of the kinetic energy (Eₖ)
We have: Eₖ = 1/4(μA²ω²λ); ω = 2πf;
By substitution:
Eₖ = 1/4(μA² (2πf) ²λ);
f² = Eₖ / (π²μA²λ);
So, the equation for the frequency (f) of the kinetic
energy (Eₖ) of a mechanical wave is:
f = √ {Eₖ/ (π²μA²λ)};
Relationship of the variables:
- Wavelength (λ): λ = 2π / k;
- Speed of the Wave (v): v = fλ = ω / k;
- Frequency Times Wavelength (fλ): fλ = ω / k;
- Angular Frequency (ω): ω = 2πf;
- Wavenumber (k): k = 1 / λ;
- Kinetic Energy (Eₖ): Eₖ = 1/4 * μ * A² * ω² * λ
Example: In the equation for kinetic energy of a mechanical wave (Eₖ = 1/4 * μ * A² * ω² * λ), we can determine the values of the variables wavelength (λ), speed of the wave (v), frequency times wavelength (fλ), angular frequency (ω), wavenumber (k), and kinetic energy (Eₖ) by providing the values of amplitude (A), frequency (f), mass (m), and length (x) of the vibrating string.
Given,
Eₖ = 1/4(μA²ω²λ);
where, the frequency of the oscillation is f = ω /2π;
linear mass density µ = mass/length = m/x kg/m;
At a given time the distance between successive points, y = A, called the wavelength (λ), λ = 2π /k .
The speed of the wave, v = fλ = ω /k;
Given values of,
Amplitude, A = 1m;
Frequency, f = 60Hz = ω/2π = 1/λ;
Mass of the string, m = 0.06 kg;
Length of the string, x = 2 m;
Degrees = radians * (180/π);
To find, λ, fλ, ω/k, y, v, ω (in Degree); of the kinetic energy (Eₖ) of the mechanical wave?
Solution:
We have,
Amplitude (A): A = 1 m;
Frequency (f): f = 60 Hz;
Mass of the string (m): m = 0.06 kg;
Length of the string (x): x = 2 m;
Degrees = radians * (180/π);
Derived:
Linear mass density (µ): µ = m / x = 0.06 kg / 2 m = 0.03 kg/m;
Solution:
Wavelength (λ): The relationship f = 1/λ, so λ = 1 / f = 1 / 60 Hz ≈ 0.01667 meters;
Angular Frequency (ω): We have f = ω / 2π, so ω = 2π * f = 2π * 60 Hz ≈ 376.99 rad/s;
Wavenumber (k): We have λ = 2π / k, so k = 2π / λ ≈ 376.99 rad/m;
Speed of the Wave (v): v = fλ = ω / k = f / k = (2π * 60 Hz) / (2π / λ) ≈ 60 * 0.01667 m/s = 1 m/s;
Frequency Times Wavelength (fλ): fλ = ω / k = f / k = 1 m/s;
Angular Frequency (ω) in Degrees: Given that Degrees = radians * (180/π),
we convert ω from radians to degrees: ω_deg = ω * (180/π) ≈ 21599.16 degrees
Kinetic Energy (Eₖ): Using the given formula for kinetic energy, Eₖ = 1/4 * μ * A² * ω² * λ, substituting the known values: Eₖ = 1/4 * (0.03 kg/m) * (1 m)² * (376.99 rad/s)² * (0.01667 m) ≈ 1.424 J
1.6. The T(deg) (period in degrees) of the
kinetic energy (Eₖ) equation is:
T(deg) = T/360; where T is the period in seconds, the
reciprocal of the frequency (T = 1/f).
T(deg) = (1/f)/360 = [1/√{Eₖ/(π²μA²λ)}]/360;
So, the equation for T(deg) (period in degrees) of the
kinetic energy (Eₖ) of a mechanical wave is:
T(deg) = 1/[360√{Eₖ/(π²μA²λ)}];
1.7. The potential energy (Eₚ) of the mechanical wave, the equation is:
Eₚ = 1/4(μA²ω²λ);
1.8. The total energy (Eₜ) of the mechanical wave is:
Eₜ = Eₖ + Eₚ;
Eₜ = 1/4(μA²ω²λ) + 1/4(μA²ω²λ) =
1/2(μA²ω²λ);
2.0 The Energy of electromagnetic waves:
For electromagnetic waves (e.g., light waves), the energy
is related to frequency through Planck's equation:
E = hf;
Where:
E represents energy (in joules),
h is Planck's constant (approximately 6.626 x 10^-34 J ·
s), and
f is the frequency of the electromagnetic wave (in hertz).
The energy of an electromagnetic wave is directly
proportional to its frequency. As the frequency increases, the energy of the
wave also increases.
2.1 Maximum speed of any wave:
It's worth noting that the maximum speed of any wave,
including electromagnetic waves, is determined by the ratio of the Planck
length to the Planck time (ℓP/tP), which yields the speed of light in a vacuum
(c), approximately 299,792,458 meters per second (approximately 3.00 x 10^8
m/s).
This is a fundamental constant in physics and serves as a
speed limit for the propagation of any energy or information in the universe.
2.2. Relationships - phase shift, frequency, wavelength,
period, T(deg)
i. Phase shift is a small difference between two waves; in math and
electronics, it is a delay between two waves that have the same period or
frequency. The phase shift is expressed in terms of angle, which can be
measured in degrees or radians, and the angle can be positive or negative. For
a shift to the right, the phase shift is positive, or negative for a shift to
the left.
ii. Frequency (f) is a measure of how many cycles or oscillations of a
wave occur in one second.
iii. Wavelength (λ) refers to the distance between two consecutive points
on a wave that are in phase.
iv. The period of the wave is represented by T, and the period in degrees
is represented by T(deg).
v. The period of the wave (T) is the time taken for one complete cycle
of the wave, measured in seconds.
vi. The period in degrees T(deg) represents the time taken for one
complete cycle of the wave to cover 360 degrees.
vii. The relationship between frequency (f) and wavelength (λ) is: f =
1/λ.
viii. The period is inversely proportional to the frequency and the
relationship between them is:
T = 1/f;
ix. The relationship between the period in degrees T(deg) and period (T)
is: T(deg) = T/360;
x. The period in degrees T(deg) is inversely proportional to the
frequency (f), and the relationship between them is:
T(deg) = (1/f) * 360.
xi. The angular frequency (ω) is directly proportional to the frequency
(f), and the relationship between them is:
ω = 2πf.
xii. The relationship between wavelength (λ) and period (T) for a wave is
directly proportional, it can be expressed as:
λ ∝ T.
xiii. To be more precise, the relationship between wavelength and period is
given by:
λ = v * T.
xiv. The wave equation describes the relationship between the speed of a
wave (v), its frequency (f), and its wavelength (λ). The equation is given as:
v = f * λ.
xv. The phase shift in degrees T(deg) is directly proportional to the
product of the time shift (Δt), the frequency (f), and 360. The relationship
between them is given by the equation:
T(deg) = (360 * Δt * f);
xvi. The phase shift in radians (ϕ) is directly proportional to the
time shift (Δt) in seconds. The relationship between them is given by the
equation:
ϕ = ω * Δt;
Where ω is the angular frequency of the oscillation (in
radians per second)
xvii. The conversion formulas between phase shifts in radians (ϕ) and degrees T(deg) are:
T(deg) = (ϕ * 180)/π;
ϕ = (T(deg) * π)/180;
2.3. The time interval T(deg) for 1° of phase is inversely proportional to
the frequency. If the frequency of a signal is given by f, then the time T(deg)
(in seconds) corresponding to 1° of phase is T(deg) = 1/(360f) = T/360.
Therefore, a one-degree (1°) phase shift on a 5 MHz signal shows a time shift
of 555 picoseconds.
2.4. A change in the frequency (f) of clock
oscillation due to a phase shift T(deg) in degrees will cause an error in time
(Δt) reading in a clock designed for a 360-degree time scale.
2.5. The symbol 'ρ' (rho) is commonly used to
represent the momentum of a photon, and it is related to the photon's
wavelength (λ) through the equation:
ρ = h/λ
Where:
ρ is the momentum of the photon, measured in kilogram
meters per second (kg·m/s).
h is Planck's constant, approximately equal to 6.62606868
× 10^-34 joules per second (J·s).
λ is the wavelength of the associated electromagnetic
wave, measured in meters (m).
This equation shows that the momentum of a photon is
inversely proportional to its wavelength. Photons with shorter wavelengths have
higher momentum, while photons with longer wavelengths have lower momentum. The
concept of photon momentum is important in various areas of physics, especially
in the context of particle interactions and the study of electromagnetic
radiation.
2.6. An electron is a subatomic particle and one of
the fundamental constituents of matter. It carries a negative electrical charge
and is an elementary particle, meaning it is not composed of smaller sub
particles. Electrons are classified as leptons, which are one of the six types
of elementary particles in the Standard Model of particle physics.
Properties of electrons include:
Charge: An electron carries a negative elementary charge
of approximately -1.602176634 × 10^-19 coulombs (C).
Mass: The rest mass of an electron (when it is not moving)
is approximately 9.1093837 × 10^-31 kilograms (kg).
Location: Electrons are found outside the nucleus of
atoms. They orbit around the nucleus in specific energy levels, forming the
electron cloud of the atom.
Behavior: Electrons exhibit both particle-like and
wave-like behavior, known as wave-particle duality. This behavior is
fundamental in quantum mechanics.
Interaction: Electrons interact with other particles
through electromagnetic forces, which are mediated by photons, the particles of
light.
Energy Levels: Electrons occupy discrete energy levels in
an atom, and they can move between these levels by absorbing or emitting energy
in the form of photons.
Conductivity: The mobility of electrons allows them to
carry electric current in conductive materials like metals.
Electrons play a vital role in various physical phenomena,
including electricity, magnetism, chemical bonding, and the behavior of matter
at the atomic and subatomic scales. Understanding electrons and their
interactions with other particles is fundamental to many areas of physics,
chemistry, and engineering.
2.7 The electron rest mass (mₑ) is a fundamental constant in physics and represents the mass of an
electron when it is at rest or not moving with respect to an observer. Its
value is approximately:
Electron rest mass (mₑ) ≈ 9.1093837 × 10^-31 kilograms
(kg).
The electron is one of the elementary particles, and its
rest mass is an important parameter in various physical calculations and
theories. It is used to describe the mass of an electron in its rest frame and
is often used as a reference point to compare the masses of other particles.
2.8. The electron rest energy is the energy
equivalent of the electron's rest mass (mₑ) given by Einstein's famous
mass-energy equivalence equation:
For an electron, its rest mass (mₑ) is approximately 9.1093837 × 10^-31 kg, we can calculate its rest
energy (Eₑ):
Eₑ = mₑc²;
Eₑ ≈ (9.1093837 × 10^-31 kg) ×
(2.99792458 × 10^8 m/s)²
Eₑ ≈ 8.187105776 × 10^-14 J
≈ 8.19 × 10^-14 J (approximately)
2.9 The electron rest energy in terms of electron volts (eV):
1 electron volt (eV) ≈ 1.602176634 × 10^-19 J
Eₑ ≈ 8.19 × 10^-14 J ÷ (1.602176634
× 10^-19 J/eV) ≈ 0.511 MeV
So, the electron rest energy is approximately 0.511 MeV or
8.19 × 10^-14 J, depending on the preferred unit of measurement.
3.0. A photon is a fundamental particle in physics
that is associated with electromagnetic radiation, including visible light,
radio waves, microwaves, X-rays, and gamma rays. Photons are elementary
particles and are considered the force carriers of the electromagnetic force.
They play a crucial role in the interaction between charged particles and are
responsible for transmitting electromagnetic waves.
Key properties of photons include:
No Rest Mass: Photons are massless particles, meaning they
have zero rest mass. Unlike other particles such as electrons and protons,
which have rest masses, photons travel at the speed of light and do not
experience time or aging.
Energy and Frequency: Photons have energy (E) and are
characterized by their frequency (f) or wavelength (λ) of the associated
electromagnetic wave. The energy of a photon is related to its frequency by the
equation E = hf, where 'h' is Planck's constant.
Wave-Particle Duality: Photons exhibit both wave-like and
particle-like behavior, known as wave-particle duality. In some experiments,
photons behave like waves, and in others, they behave like discrete particles.
Quantum of Electromagnetic Energy: Photons carry quantized
packets or quanta of electromagnetic energy. The energy of a photon is directly
proportional to its frequency. Higher frequency (shorter wavelength) photons
have higher energies, and vice versa.
Propagation at the Speed of Light: Photons always travel
at the speed of light in a vacuum, which is approximately 2.99792458 × 10^8
meters per second (m/s). This constant speed of light is a fundamental constant
in physics and plays a key role in the theory of special relativity.
Interaction: Photons can interact with charged particles,
such as electrons, by transferring their energy and momentum. This interaction
is the basis for various phenomena, including the photoelectric effect, which
led to significant developments in quantum theory.
Photons are central to the study of quantum mechanics and
are essential in understanding the behavior of light and other electromagnetic
waves. Their wave-particle nature makes them a fascinating and essential aspect
of modern physics and is foundational to many areas, including quantum
mechanics, atomic and molecular physics, and quantum electrodynamics.
3.1. Photons are massless particles, which mean
they do not have rest mass (m). As massless particles, they travel at the speed
of light in a vacuum (approximately 2.99792458 × 10^8 meters per second) and
always move at this constant speed.
Given the energy-frequency equivalence for photons, which
is E = hf, where 'E' is the photon's energy and 'f' is the wave frequency; we
can calculate the kinetic energy of a photon using this relationship.
Since photons have no rest mass, their energy (E) is
entirely kinetic energy, which is the energy associated with their motion.
Thus, the kinetic energy (Eₖ) of a photon can be represented
by E = hf:
Eₖ = hf
Where:
Eₖ is the kinetic energy of the
photon, measured in joules (J).
h is Planck's constant, approximately 6.62606868 × 10^-34
J s.
f is the wave frequency of the photon, measured in hertz
(Hz).
The kinetic energy of a photon is directly proportional to
its frequency. Photons with higher frequencies (shorter wavelengths) have
higher kinetic energies, while photons with lower frequencies (longer
wavelengths) have lower kinetic energies.
3.2. The change in potential energy (Δmₑ) of an electron after absorbing a photon:
When an electron absorbs a photon, the energy of the
photon increases as the energy of the photon is transferred to the electron.
This change in energy is known as the change in potential energy (Δmₑ) of the electron. However, it's important to note that electrons do
not have "rest" potential energy in the same sense as in classical
mechanics, where potential energy is associated with the position of an object
in a gravitational field. In the context of quantum mechanics, the potential
energy of an electron typically refers to the interaction potential energy in
an atomic or molecular system.
In the case of an electron absorbing a photon, we can
calculate the change in energy (Δmₑ) by considering the change in the
electron's total energy before and after the absorption process.
Before absorption, the electron's total energy (Eᵢ) is its rest energy (mₑc²) since electrons at rest have
no kinetic energy:
Eᵢ = mₑc²
After absorption, the electron's total energy (E_f) will
be its rest energy plus the energy of the absorbed photon (E_photon):
E_f = mₑc² + E_photon
The change in energy (Δmₑ) is then
the difference between the final and initial total energies:
Δmₑ = E_f - Eᵢ
Δmₑ = (mₑc² + E_photon) - mₑc²
Δmₑ = E_photon
So, the change in potential energy (Δmₑ) of an electron after absorbing a photon is simply equal to the
energy of the absorbed photon. The absorbed energy increases the electron's
total energy, and depending on the context, it can lead to various phenomena
such as excitation of the electron to higher energy levels in an atom or
electron ejection in the case of the photoelectric effect.
3.3. Invariant mass (mₑ) of an electron after absorbing photon:
When an electron absorbs photon energy (hf), its total
energy increases by that amount. The absorbed energy can cause the electron to
transition to a higher energy level in an atom or molecule, leading to an
excited state. This phenomenon is fundamental in various physical processes,
including the interaction of light with matter.
In terms of the electron's mass, the absorption of a
photon (hf) does not directly impact its rest mass (mₑ). The rest mass of an electron remains the same before and after
absorbing the photon, as the rest mass is an intrinsic property of the electron
and is not affected by the energy it possesses.
The energy absorbed by the electron increases its total
energy, but it does not change its rest mass. The kinetic energy of the
electron can increase if it absorbs a photon, but the rest mass energy remains
constant.
After absorbing the photon, the electron will have an
increased total energy given by:
Total Energy after absorption (E_f) = E₀ + Eᵏ + hf
Where:
E₀ is the rest mass energy of the electron, which is
approximately 8.19 × 10^−14 J (corresponding to the rest energy of 0.511 MeV).
Eᵏ is the kinetic energy of the
electron before absorbing the photon.
hf is the energy of the absorbed photon.
The electron's mass (mₑ) is still represented by its rest
mass, and any increase in energy due to the absorbed photon will manifest as
additional kinetic energy or potential energy, depending on the specific
context and the electron's environment.
The consequences of electron-photon interactions can be
varied, absorption of a photon can lead to electronic excitation, while in the
photoelectric effect, and an absorbed photon can eject an electron from a
material. The behavior of electrons when interacting with photons is crucial.
#frequency #energy #wave
