Electromagnetic Waves - Engineering Physics - CSE 2026 - Shoolini University

Section B: Electricity and Magnetism

Displacement Current

When a dielectric material is polarized, the charges are displaced from their original positions. This causes a current to flow through the material, known as the displacement current. This current is given by the equation: $$J = ∂D/∂t$$, where J is the displacement current, D is the electric displacement field and t is time.

  1. Displacement current is an electric current that arises when the electric displacement field changes with time. It is defined as the time rate of change of the electric displacement field: $$ I_d = ε_0 ∂D/∂t $$
  2. Displacement current is mathematically equivalent to conduction current, and it is treated as such in Maxwell's equations.
  3. Displacement current is important for understanding the behavior of capacitors and other devices that involve electric fields. It is also important for understanding electromagnetic radiation, as it is responsible for the production of fields that propagate away from the source.
  4. Displacement current can be thought of as a current of "virtual particles", which are not actually particles, but are instead a mathematical construct used to describe the behavior of electric fields.
  5. Displacement current is not associated with any flow of material, so it does not involve the transfer of energy or momentum. However, it does involve the transfer of electric charge, as the electric field is changing.

Dielectric Potential or Electric Field Potential

Dielectric potential, also referred to as the electric field potential, is the potential energy per unit charge that a dielectric material has when the material is placed in an electric field. It is an important property of dielectric materials that is used in many applications, such as electrical insulation and capacitors.

Dielectric potential is caused by the interaction between a dielectric material and an electric field. When a dielectric material is placed in an electric field, the dielectric material is polarized. This polarization of the material allows for the formation of electric dipoles, which creates a potential energy that is proportional to the electric field. This potential energy is the dielectric potential.

The magnitude of the dielectric potential is determined by the dielectric constant of the material. The dielectric constant is a measure of the material's ability to store electrical energy when placed in an electric field. Materials with a higher dielectric constant are able to store more electrical energy than materials with a lower dielectric constant.

The application of dielectric potential is found in many devices, such as capacitors and electrical insulation. Capacitors are devices that store energy in the form of an electric field, and the dielectric potential of the material used in the capacitor determines the amount of energy that can be stored. Electrical insulation is used to protect electrical components from electrical shock and the dielectric potential of the insulation material determines how effective the insulation is.

Dielectric potential is an important property of dielectric materials that is used in many applications. Knowing the dielectric constant of a material can help determine how much energy can be stored in a capacitor or how effective an electrical insulation material will be.

Dielectrics

What are Dielectrics

Dielectrics are materials that have the ability to store energy in an electric field. They are non-conducting materials and can be thought of as having a very high permittivity. A dielectric material has a large polarizability, which means that it is able to store energy in an electric field. This is done by separating positive and negative charges and creating an electric dipole.

Relationship between Electric Field & Potential

Electric fields and potentials are related to each other by the equation: $$E = -∇V$$, where E is the electric field, V is the electric potential and ∇ is the gradient operator. This equation states that the electric field is equal to the negative gradient of the electric potential. This means that when a potential is applied to a material, the electric field will be generated in the opposite direction, creating an electric dipole.

  1. The electric field is a vector quantity which is defined as the force exerted on a unit positive charge by the electric field. The electric field at any point is given by the equation: $$ E = -∇V $$ where E is the electric field, and V is the electric potential at the point.
  2. The electric potential at a point is the work done in bringing a unit positive charge from infinity to that point. The electric potential at a point is given by the equation: $$ V = -∫E.dr $$ where E is the electric field, and dr is a path element in the direction of the electric field.
  3. The electric potential at any point is equal to the potential energy of a unit positive charge at that point. The potential energy at a point is given by the equation: $$ U = qV $$ where U is the potential energy, q is the charge of the particle, and V is the electric potential of the point.
  4. The electric potential is related to the electric field by the equation: $$ E = -∇V $$ where E is the electric field, and V is the electric potential at the point.
  5. The electric field and potential are related to each other by the equation: $$ V = -∫E.dr $$ where E is the electric field, and dr is a path element in the direction of the electric field.

Dielectric Polarization

When an electric field is applied to a dielectric material, the material becomes polarized. This means that the positive and negative charges within the material are separated and an electric dipole is formed. This polarization of the material is known as dielectric polarization and the dipole created is known as the dipole moment. The dipole moment is proportional to the electric field applied and is given by the equation: $$ P = εE $$, where $P$ is the dipole moment, $ε$ is the permittivity of the material and $E$ is the electric field.

  1. Dielectric Polarization is a process in which charges in an insulator material are re-arranged by an applied electric field.
  2. The dielectric polarization can be expressed as the sum of electronic, ionic, and orientational contributions $P_e$, $P_i$, and $P_o$, respectively.
  3. The total polarization of a dielectric material is given by $$P = P_e + P_i + P_o$$.
  4. The electronic contribution to the polarization of a dielectric material is given by $$ P_e = -N_Ae^2/4πε_oK_BT $$ where $ N_A $ is Avogadro's number, e is the electron charge, $ε_o$ is the permittivity of free space, $ K_B$ is the Boltzmann's constant, and T is the temperature.
  5. The ionic contribution to the polarization of a dielectric material is given by $$ P_i = -N_iZ_i^2/4πε_oK_BT $$, where $N_i$ is the number of ions per unit volume, $Z_i$ is the ionic charge, ε_o is the permittivity of free space, $K_B$ is the Boltzmann's constant, and $T$ is the temperature.
  6. The orientational contribution to the polarization of a dielectric material is given by $$ P_o = N_mQ_o/4πε_oK_BT $$, where $ N_m $ is the number of molecules per unit volume, $ Q_o $ is the orientational dipole moment, $ ε_o $ is the permittivity of free space, $ K_B $ is the Boltzmann's constant, and $ T $ is the temperature.

Types of Polarization

There are four types of polarization that can occur within a dielectric material; most commonly electronic polarization and ionic polarization. Electronic polarization is caused by the movement of electrons within the material and ionic polarization is caused by the movement of ions.

  1. Dipole Polarization: Dipole polarization occurs when a dielectric material is subjected to an electric field. This occurs when the charges in the material, whether electrons or ions, are displaced from their equilibrium positions. The electric dipole moment produced as a result of this displacement is given by the equation $$ P=Qd $$, where $ Q $ is the charge and d is the distance between the charges.
  2. Chain Polarization: Chain polarization occurs when the dielectric material contains molecules that are linked by dipole bonds. It is this chain of dipoles that allows for the induced electric field to flow through the material. The electric dipole moment produced by this type of polarization is given by the equation $$ P=Ql $$, where $ Q $ is the charge, and $ l $ is the length of the chain.
  3. Space Charge Polarization: Space charge polarization occurs when an electric field is applied to a dielectric material and the applied field creates an imbalance of positive and negative charges. This results in a net charge density throughout the material, which is given by the equation $$ ρ=Q/V,$$ where $ Q $ is the total charge and $V$ is the volume of the material.
  4. Orientation Polarization: Orientation polarization occurs when the dipoles in a dielectric material are aligned by an external electric field. This type of polarization is responsible for the dielectric constant of a material. The electric dipole moment produced by orientation polarization is given by the equation $$ P=αE $$, where $ α $ is the polarizability of the material, and E is the applied electric field.

Maxwell's Equations

Maxwell's equations are a set of four equations that are used to describe the behavior of electric and magnetic fields. These equations can be used to calculate the electric field, magnetic field, electric potential and electric displacement field of a material.

  1. Maxwell's equations are a set of four partial differential equations that describe the behaviour of both electric and magnetic fields. The equations were formulated by James Clerk Maxwell in the mid-19th century and form the foundation of classical electromagnetism:
    1. Gauss's Law for Electric Fields: $$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$
    2. Gauss's Law for Magnetic Fields: $$\nabla \cdot \mathbf{B} = 0$$
    3. Faraday's Law: $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} $$
    4. Ampere's Law: $$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$
  2. Maxwell's equations are a cornerstone of modern physics, used in many different fields such as optics, electrodynamics, and quantum mechanics.
  3. The equations can be written in a variety of forms, but the most common are the integral form and the differential form.
  4. In the integral form, Maxwell's equations are written in terms of the electric and magnetic fields, which can be measured using an electric field meter and a magnetometer.
  5. In the differential form, the equations are written in terms of the electric and magnetic potentials, which can be calculated by solving the underlying partial differential equations.
  6. Maxwell's equations can also be used to calculate the force on a charge due to an electric field and a magnetic field. This force is known as the Lorentz force.
  7. The Lorentz force law is written as: $$ \mathbf{F} = q \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right) $$
  8. Maxwell's equations can also be used to calculate the energy stored in an electric field and a magnetic field. This energy is known as the electromagnetic energy.
  9. The electromagnetic energy density is written as: $$ u = \frac{1}{2} \epsilon_0 \mathbf{E}^2 + \frac{1}{2 \mu_0} \mathbf{B}^2 $$

Conclusion

In conclusion, dielectrics are materials that have the ability to store energy in an electric field. This occurs through dielectric polarization, where the material becomes polarized and an electric dipole is created. The dipole moment is proportional to the electric field applied and a displacement current is generated due to the movement of charges. There are two types of polarization that can occur within dielectrics, electronic and ionic, and Maxwell's equations can be used to calculate the electric field, magnetic field, electric potential and electric displacement field.

Some of the notable equations are:

Electromagnetic Waves

What do you mean by Electromagnetic Waves? What are the advantages of Electromagnetic Waves?

Electromagnetic Waves (EM Waves)

We studied Energy, a measure of the ability to do work, comes in many forms and can transform from one type to another. An example of stored or potential energy include batteries and water behind a dam and Objects in motion are examples of kinetic energy. Charged particles—such as electrons and protons—create electromagnetic fields when they move, and these fields transport the type of energy which we call an electromagnetic radiation, or light. The equation for electromagnetic energy density $u$ is $$ u = \frac{1}{2} \epsilon_0 \mathbf{E}^2 + \frac{1}{2 \mu_0} \mathbf{B}^2 $$ Electromagnetic radiations are composed of electromagnetic waves that are produced when an electric field comes in contact with the magnetic field. A changing magnetic field will induce a changing electric field and vice-versa—the two are linked. These changing fields form electromagnetic waves. The equation for electromagnetic wave is: $$\mathbf{E} \perp \mathbf{B}$$ Electromagnetic waves differ from mechanical waves in that they do not require a medium to propagate. Electromagnetic waves were first postulated by James Clerk Maxwell and subsequently confirmed by Heinrich Hertz. Heinrich Hertz, a German physicist, applied Maxwell's theories to the production and reception of radio waves. The unit of frequency of a radio wave -- one cycle per second -- is named the hertz, in honor of Heinrich Hertz. Electromagnetic waves are created as a result of vibrations between an electric and a magnetic field. It can also be said that electromagnetic waves are the composition of oscillating electric and magnetic fields. Electromagnetic waves are solutions of Maxwell's equations, which are the fundamental equations of electrodynamics. Maxwell's equations, which describe the relationship between the electric and magnetic fields is given as: $$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}, \quad \nabla \cdot \mathbf{B} = 0 $$ $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} $$ where $\rho$ is the charge density, $\mathbf{J}$ is the current density and $\epsilon_0$ and $\mu_0$ are the vacuum permittivity and vacuum permeability, respectively.

How EM waves are formed.

Electromagnetic waves are formed when an electric field comes in contact with a magnetic field. They are hence known as 'electromagnetic' waves. The electric field and magnetic field of an electromagnetic wave are perpendicular (at right angles) to each other. The equation for the electromagnetic wave can be expressed as: $$\mathbf{E} \perp \mathbf{B} $$ where $\mathbf{E}$ represents the electric field and $\mathbf{B}$ represents the magnetic field. The symbol $\perp$ represents "perpendicular to." This equation expresses that the electric field and magnetic field of an electromagnetic wave are perpendicular to each other.

Technological Applications of Electromagnetic Waves

Ever wonder how your mobile phone can carry your voice to someone else across the world? It does so by using electromagnetic waves. These are fluctuations in magnetic and electric fields that can travel through space. Since they have no tether to physical material, they can pass through walls, buildings, and even our bodies. Electromagnetic waves play a massive role in our lives. We can enjoy mobile phones, radio, TV, and more by controlling and manipulating these waves. Light testing equipment also checks for the presence of EM waves. So, EM waves are much more relevant than we might think. Delve into some of the technological applications of electromagnetic waves and discover how they’ve been making our lives easier for decades.

Radio Waves

First, we'll examine the most common application of EM waves—radio waves. A radio wave can broadcast FM and AM radio signals, but it can also do more. Your television uses radio waves to broadcast the signal from various TV stations (assuming you're watching analog TV instead of streaming). Finally, radio waves have applications in the military through radar. That's why radar and radio begin with the same three letters. The equation for a radio wave can be given as: $$\vec{E}(z,t) = \vec{E_0} \cos(\omega t - kz)$$ where $\vec{E_0}$ is the amplitude of the electric field, $\omega$ is the angular frequency, $t$ is time, $k$ is the wave number and $z$ is the propagation direction of the wave. The relationship between the electric field and magnetic field of a radio wave is described by: $$\vec{B}(z,t) = \frac{\mu_0}{\epsilon_0} \vec{E}(z,t)$$ where $\mu_0$ is the magnetic constant and $\epsilon_0$ is the electric constant. Finally, the velocity of a radio wave is given by: $$v = \frac{\omega}{k}$$ where $v$ is the velocity of the wave.

Microwaves

Another form of an EM wave is a microwave. As you probably guessed, these are the waves that a household microwave uses to heat food. That's not their only use, though. Microwaves are pretty similar to radio waves as they're also used for communication. TV stations use microwaves to send their signals out to longer distances. The frequency of microwaves lies between radio waves and infrared radiation. The equation for the wavelength of a microwave is given by: $$\lambda = \frac{c}{f}$$ where, $\lambda$ is the wavelength, $c$ is the speed of light (approx. 3 x 10^8 m/s), and $f$ is the frequency of the microwave. Microwaves can also be generated and transmitted through a waveguide. The equation for the cutoff frequency of a rectangular waveguide is given by: $$f_c = \frac{c}{2a} \sqrt{\frac{m^2}{n^2}-1}$$ where, $f_c$ is the cutoff frequency, $a$ is the width of the waveguide, $m$ and $n$ are the mode numbers. In microwave communication systems, the power received at the receiver is proportional to the product of the transmitted power and the inverse square of the distance between the transmitter and receiver. This can be represented mathematically as: $$P_{r} = \frac{P_{t} G_{t} G_{r} \lambda^2}{(4\pi)^2 d^2}$$ where, $P_r$ is the power received, $P_t$ is the power transmitted, $G_t$ and $G_r$ are the gains of the transmitter and receiver antennas, respectively, $\lambda$ is the wavelength of the microwave, $d$ is the distance between the transmitter and receiver.

Infrared Waves

An infrared wave generates heat, and it's also used in TV remote controls. In today's age, Bluetooth technology has primarily replaced infrared. Yet, infrared waves are necessary for creating heat-vision and night-vision cameras. That's because every living creature emits heat, which is infrared waves. That means anything that gets hot is producing a lot of infrared waves. An equation for the wavelength of an infrared wave can be represented by the following: $$ \lambda = \frac{c}{f} $$ where, $ \lambda $: Wavelength of the infrared wave $ c $: Speed of light in a vacuum, approximately equal to 3 x 10^8 m/s $ f $: Frequency of the infrared wave
Another equation that describes the relationship between the frequency and energy of an electromagnetic wave, including infrared, is given by Planck's law: $$ E = hf $$ where, $ E $: Energy of the electromagnetic wave $ h $: Planck's constant, approximately equal to 6.63 x 10^-34 Js $ f $: Frequency of the electromagnetic wave
By using these equations, we can determine various properties of infrared waves, such as their wavelength, frequency, and energy.

X- Rays

An x-ray is simply a form of EM wave used for internal photography. It uses a penetrating form of EM radiation to take pictures inside your body. The medical field uses the same EM radiation type as a cancer treatment. The radiation is concentrated in a high-energy form to eliminate cancer cells. The equation for the propagation of x-rays in a vacuum can be represented as follows: $$\frac{1}{c^2} \frac{\partial^2 E}{\partial t^2} = \nabla^2 E$$ where $E$ is the electric field, $c$ is the speed of light, and $\nabla^2$ is the Laplacian operator. The intensity of an x-ray wave can be represented as: $$I = \frac{c}{2} \epsilon_0 E^2$$ where $I$ is the intensity, $\epsilon_0$ is the electric constant, and $E$ is the electric field. The frequency of x-rays can be represented as: $$f = \frac{E}{h}$$ where $f$ is the frequency and $h$ is the Planck constant. Note that these equations are only valid for the propagation of x-rays in a vacuum and do not account for interactions with matter.

Characterstics of EM Waves

  1. Field of Electromagnetic wave

    There are two fields of the electromagnetic waves

    1. Electric Field
    2. Magnetic Field
  2. Field Angle of Electromagnetic Wave

    In the electromagnetic waves, the angle between the electric field and magnetic field is 90 degree or magnetic and electric fields are perpendicular to each other.

  3. Direction Of Electromagnetic Wave During Propagation

    The electric field and magnetic field is perpendicular to the direction of propagation

  4. Reflection of Electromagnetic Waves

    The Electromagnetic wave is having the characteristics of reflection just like the light waves. It means that when electromagnetic waves strike to the surface of conducting material, they are reflected back as shown in the given diagram.

  5. Refraction of Electromagnetic Waves

    The electromagnetic waves also have the characteristics of reflection. It means that when the electromagnetic waves inters from one medium to another, it bends towards the normal or away from the normal while traveling in the next medium. When the electromagnetic waves inter form rare medium to dens medium, it bends towards the normal and when the electromagnetic inter from dens to rare medium it bends away from the normal.

  6. Intensity of Electromagnetic Waves

    The intensity of electromagnetic waves depends on the electrical field strength of the wave. It means that if the field strength of electromagnetic waves is more, its intensity will be more and vice versa.
    The electrical field strength is measured in V/m volt per meter. For the microwave it's measured in micro volt per meter µv/m.

  7. Attenuation of Electromagnetic Waves

    When the electromagnetic wave is propagated its energy is wasted while travelling from source to load. This loss of energy is known as attenuation of the electromagnetic waves. The attenuation occurs due to buildings, tree's, hills, rivers etc. which observe the electromagnetic energy during propagation.

  8. Polarization of Electromagnetic Waves

    There are the following types of electromagnetic waves polarization.

    1. Vertical Polarization

      If the position of the electric field is vertical with respect to ground during propagation, it is called vertical polarization of electromagnetic waves.

    2. Horizontal Polarization

      If the position of the electric field is horizontal with respect to ground during propagation, it is called horizontal polarization of the electromagnetic waves.

    3. Linear polarization

      If the position of electric field is sometimes vertical and sometimes horizontal with equal intervals and the direction of propagation does not change, it is called linear polarization.

    4. Circular Polarization

      If the position of electric field changes continuously during propagation of electromagnetic waves and its intensity remains the same, its is called circular polarization.

    5. Elliptical polarization

      If the position and intensity of electrical field changes continuously during propagation, it is called elliptical polarization.

  9. Field Position of Electromagnetic Waves

    As we know that the electrical field and magnetic field are perpendicular to each other and at the same time these fields are perpendicular to the direction of propagation, if we change the direction of any field it well result in corresponding change of the other field. It means that if E-field is vertical, the H-field will be horizontal. Now if we change the position of the E-field from vertical to horizontal, the position of H0field (magnetic field) will also change from horizontal to vertical. If we change the direction of E-field or H-field the direction of propagation will also change if we change the direction of E0field and H-field at the same time the direction of propagation will remain the same.

  10. Velocity of Electromagnetic Waves

    Velocity of electromagnetic waves in the air is the same as the velocity of light. It is 3 X 108 m/s or 3 X 1010 cm/s.

  11. Separation of fields of EM Wave

    Electrical field and magnetic are always related to each other and whenever there is electric field, the magnetic field will also be present over there.

  12. Energy of EM Waves

    When the electromagnetic waves are propagated, the energy of these waves is equally distributed in electric field and magnetic field throughout the propagation.

Polarization

Polarization is a measurement of the electromagnetic field's alignment. In the figure above, the electric field (in red) is vertically polarized. Think of a throwing a Frisbee at a picket fence. In one orientation it will pass through, in another it will be rejected. This is similar to how sunglasses are able to eliminate glare by absorbing the polarized portion of the light. Polarization of electromagnetic waves can be described mathematically using the Jones vector formalism. The Jones vector represents the electric field of an EM wave and is given by: $$ \mathbf{E} = E_0 \begin{bmatrix} \cos(\omega t - kz) \ \sin(\omega t - kz) \end{bmatrix} $$ where $E_0$ is the magnitude of the electric field, $\omega$ is the angular frequency, $t$ is time, and $k$ is the wave number. The wave is said to be vertically polarized when the Jones vector is of the form $$ \mathbf{E} = E_0 \begin{bmatrix} 1 \ 0 \end{bmatrix} $$ and horizontally polarized when it is of the form $$ \mathbf{E} = E_0 \begin{bmatrix} 0 \ 1 \end{bmatrix} $$ In general, the polarization state of an EM wave can be described by a superposition of these two basic polarization states. The degree to which an EM wave is vertically or horizontally polarized is described by the relative phase between these two components. The degree of polarization can be quantified using the degree of polarization or the polarization purity.

Frequency

The number of crests that pass a given point within one second is described as the frequency of the wave. One wave—or cycle—per second is called a Hertz (Hz), after Heinrich Hertz who established the existence of radio waves. A wave with two cycles that pass a point in one second has a frequency of 2 Hz. Frequency can be defined as: $$ f = \frac{1}{T} $$ where $f$ is the frequency in Hertz (Hz), and $T$ is the period in seconds. The frequency of a wave determines its wavelength and velocity. Higher frequency waves have shorter wavelengths and travel faster, whereas lower frequency waves have longer wavelengths and travel slower. In the context of electromagnetic waves, frequency is a measure of the number of oscillations of the electric and magnetic fields per second.

Wavelength

Electromagnetic waves have crests and troughs similar to those of ocean waves. The distance between crests is the wavelength. The shortest wavelengths are just fractions of the size of an atom, while the longest wavelengths scientists currently study can be larger than the diameter of our planet! More of that in Electromagnetic Spectrum. The wavelength ($\lambda$) of an electromagnetic wave is defined as the distance between two consecutive crests (or troughs) of the wave. It can be mathematically represented as $$\lambda = \frac{c}{f}$$, where $c$ is the speed of light in vacuum and $f$ is the frequency of the wave. The unit of wavelength is meter (m). The wavelength of an electromagnetic wave determines its properties, such as its ability to penetrate different materials, its interaction with charged particles, and its absorption or reflection by various objects.

Magnetic materials

Magnetic materials have several unique properties that make them invaluable for a variety of applications. There are several different types of magnetic materials, each with its own unique characteristics and behavior. These materials can be broadly classified into two main categories: ferromagnetic and non-ferromagnetic materials which can be further classified into four main categories: Diamagnetic, Paramagnetic, Ferromagnetic, and Ferrites.

Each type of material has its own distinct characteristics and potential uses, from medical and industrial applications to producing ultrasonic waves.

Ferromagnetic materials are those that are strongly attracted to magnetic fields and can be magnetized to produce their own magnetic field. Examples of ferromagnetic materials include iron, cobalt, and nickel. These materials have a high magnetic permeability, which is a measure of the ease with which they can be magnetized. The magnetic permeability of a ferromagnetic material is typically much higher than that of a non-ferromagnetic material.

The magnetic permeability, $\mu$, of a material is a measure of the ease with which it can be magnetized: $$ \mu = \frac{B}{H} $$ where $B$ is the magnetic flux density and $H$ is the magnetic field strength.

The magnetic memory or hysteresis of a ferromagnetic material is defined as its ability to retain its magnetization, even in the absence of an external magnetic field: $$ B = f(H) $$ where $B$ is the magnetic flux density and $H$ is the magnetic field strength.

One of the most important properties of ferromagnetic materials is their ability to retain their magnetization, even in the absence of an external magnetic field. This is known as magnetic memory or hysteresis, and it is the result of the alignment of the magnetic moments of the atoms within the material. The magnetic moment of an atom is a measure of its magnetic properties, and is related to its angular momentum and spin. In ferromagnetic materials, the magnetic moments of the atoms are aligned in the same direction, resulting in a strong overall magnetic field.

Non-ferromagnetic materials, on the other hand, are not strongly attracted to magnetic fields and cannot be magnetized to produce their own magnetic field. Examples of non-ferromagnetic materials include copper, aluminum, and gold. These materials have a low magnetic permeability and do not exhibit magnetic memory.

One of the key characteristics of non-ferromagnetic materials is their susceptibility to magnetic fields. The susceptibility of a material is a measure of its response to an external magnetic field, and is defined as the ratio of the magnetization of the material to the applied magnetic field. In non-ferromagnetic materials, the susceptibility is typically very small, meaning that they are not greatly affected by external magnetic fields.

The magnetic susceptibility, $\chi$, of a material is a measure of its response to an external magnetic field and is defined as the ratio of the magnetization of the material to the applied magnetic field: $$ \chi = \frac{M}{H} $$ where $M$ is the magnetization and $H$ is the magnetic field strength.

Another important property of magnetic materials is their magnetic anisotropy, which is a measure of the degree to which the magnetic properties of a material depend on the direction of the applied magnetic field. In materials with high magnetic anisotropy, the magnetic properties are highly dependent on the direction of the applied field, while in materials with low magnetic anisotropy, the magnetic properties are less sensitive to the direction of the applied field.

The magnetic anisotropy, $K$, of a material is a measure of the degree to which its magnetic properties depend on the direction of the applied magnetic field: $$ K = \frac{E_{anisotropy}}{V} $$ where $E_{anisotropy}$ is the energy of magnetic anisotropy and $V$ is the volume of the material.

Overall, the different types of magnetic materials can be distinguished based on their magnetic permeability, magnetic memory, susceptibility, and magnetic anisotropy. These properties are crucial for understanding the behavior of magnetic materials and for designing applications that make use of their unique characteristics. Let us learn more about them in detail.

Diamagnetic Materials

Diamagnetic materials are materials that are weakly repelled by a magnetic field. They have no permanent magnetic dipole moment and are, therefore, not attracted to a magnet. Examples of diamagnetic materials are water, bismuth, and some metals. These materials have low susceptibility to magnetization, so they are not commonly used in practical applications. The susceptibility, $\chi$, of a diamagnetic material is low, meaning it is weakly repelled by a magnetic field: $$ \chi \approx 0 $$ A material with no permanent magnetic dipole moment is not attracted to a magnet. This property can be expressed mathematically as: $$ \vec{m} = 0 $$ where $\vec{m}$ is the magnetic moment of the material.

Paramagnetic Materials

Paramagnetic materials are materials that are weakly attracted to a magnetic field. They have a permanent magnetic dipole moment and are, therefore, attracted to a magnet. Examples of paramagnetic materials are aluminum, titanium, and some rare-earth elements. These materials have greater susceptibility to magnetization than diamagnetic materials and are commonly used in industrial applications. The susceptibility, $\chi$, of a paramagnetic material is greater than that of a diamagnetic material, meaning it is weakly attracted to a magnetic field: $$ 0 < \chi \ll 1 $$ A material with a permanent magnetic dipole moment, $\vec{m}$, is attracted to a magnet. This property can be expressed mathematically as: $$ \vec{m} \neq 0 $$ where $\vec{m}$ is the magnetic moment of the material.

Ferromagnetic Materials

Ferromagnetic materials are materials that are strongly attracted to a magnetic field. They have a permanent magnetic dipole moment and are, therefore, strongly attracted to a magnet. Examples of ferromagnetic materials are iron, cobalt, and nickel. These materials have very high susceptibility to magnetization and are commonly used in motors, generators, and transformers. The susceptibility $\chi$ of a paramagnetic material is given by the equation: $$ \chi = \frac{M}{H} $$ where $M$ is the magnetization and $H$ is the magnetic field. For ferromagnetic materials, the susceptibility is much greater than for paramagnetic materials, meaning that they are much more strongly affected by magnetic fields. The susceptibility of ferromagnetic materials is related to the magnetic permeability $\mu$ through the relationship: $$ \chi = \frac{\mu - 1}{\mu_0} $$ where $\mu_0$ is the vacuum permeability. The magnetic permeability of ferromagnetic materials is much higher than that of paramagnetic materials, making them much more effective at producing magnetic fields. The magnetic anisotropy of ferromagnetic materials is also much greater, meaning that the direction of the magnetic field is much more strongly influenced by the direction of the applied field.

Ferrites

Ferrites are a type of magnetic material that combines the properties of both ferromagnetic and paramagnetic materials. They have a permanent magnetic dipole moment and are, therefore, attracted to a magnet. Examples of ferrites are barium ferrite and strontium ferrite. These materials have higher magnetic anisotropy than ferromagnetic materials, which makes them useful for applications such as motors, generators, and transformers. Some of the notable formulas are: $$ M = K * H $$ where $M$ is the magnetic moment of the ferrite, $K$ is the magnetic anisotropy constant, and $H$ is the magnetic field strength.

And, $$ H = \frac {M}{\mu_0 * M_s} $$ where $H$ is the magnetic field strength, $\mu_0$ is the permeability of free space, and $M_s$ is the saturation magnetization.

And, $$ \Delta E = -K * sin^2(\theta) $$ where $\Delta E$ is the energy change, $K$ is the magnetic anisotropy constant, and $\theta$ is the angle between the magnetic moment and the magnetic anisotropy axis.

And, $$ V = B * S * H $$ where $V$ is the magnetic potential energy, $B$ is the magnetic flux density, $S$ is the surface area, and $H$ is the magnetic field strength.

And, $$ \frac {d\Phi}{dt} = -M * \frac {dH}{dt} $$ where $\Phi$ is the magnetic flux, $M$ is the magnetic moment of the ferrite, and $\frac {dH}{dt}$ is the rate of change of the magnetic field strength.

Magnetic Anisotropy

Magnetic anisotropy is a property of a material that describes its ability to respond differently to a magnetic field when it is applied in different directions.

The following equation describes the magnetic energy $E$ as a function of the magnetic anisotropy constant $K$ and the angle $\theta$ between the magnetic moment and the magnetic anisotropy axis. It represents the energy required to rotate the magnetic moment from the anisotropy axis to a different direction. $$ E = -K * cos^2(\theta) $$ where $E$ is the magnetic energy, $K$ is the magnetic anisotropy constant, and $\theta$ is the angle between the magnetic moment and the magnetic anisotropy axis.

Let us represent the rate of change of magnetic moment $\frac{dM}{dt}$ as a function of the gyromagnetic ratio $\gamma$, the magnetic field strength $H$, and the time derivative of magnetic moment $\frac{\partial M}{\partial t}$. It shows the relationship between the rate of change of magnetic moment and the external magnetic field as: $$ \frac {dM}{dt} = \gamma * H * \frac {\partial M}{\partial t} $$ where $\frac {dM}{dt}$ is the rate of change of magnetic moment, $\gamma$ is the gyromagnetic ratio, $H$ is the magnetic field strength, and $\frac {\partial M}{\partial t}$ is the time derivative of magnetic moment.

This derivative of the magnetic energy with respect to angle $\theta$ can be denoted as below. It shows how the magnetic energy changes as the angle between the magnetic moment and the magnetic anisotropy axis changes. $$ \frac {\partial E}{\partial \theta} = 2 * K * cos(\theta) * sin(\theta) $$ where $\frac {\partial E}{\partial \theta}$ is the derivative of magnetic energy with respect to angle $\theta$, $K$ is the magnetic anisotropy constant, and $\theta$ is the angle between the magnetic moment and the magnetic anisotropy axis.

Let us show the relationship between the magnetic moment $M$, saturation magnetization $M_s$, and the angle $\theta$. It represents the magnetic moment as a function of the saturation magnetization and the angle between the magnetic moment and the magnetic anisotropy axis. $$ M = M_s * sin(\theta) $$ where $M$ is the magnetic moment of the material, $M_s$ is the saturation magnetization, and $\theta$ is the angle between the magnetic moment and the magnetic anisotropy axis.

Finally, this following equation represents the anisotropy field $H_k$ as the derivative of magnetic energy with respect to magnetic moment. It shows how the magnetic energy changes as the magnetic moment changes. $$ H_k = -\frac {\partial E}{\partial M} $$ where $H_k$ is the anisotropy field, and $\frac {\partial E}{\partial M}$ is the derivative of magnetic energy with respect to magnetic moment.

Magnetostriction

Magnetostriction is the change in the physical dimensions of a material due to a change in its magnetic field. These two properties are important components of magnetic materials and are used in the production of ultrasound.

Magnetostriction is a phenomenon that occurs in ferromagnetic materials whereby the material changes shape in response to an applied magnetic field. The phenomenon is the result of a coupling between a material’s magnetic and elastic properties and can be used to generate ultrasonic waves. This article will discuss the concept of magnetostriction and its applications in production of ultrasonic waves.

Magnetostriction is the result of a coupling between a material’s magnetic and elastic properties. When a ferromagnetic material is subjected to an external magnetic field, the magnetic domains within the material elongate or contract, resulting in a change in shape of the material. This phenomenon is known as magnetostriction and is related to the magnetic susceptibility of the material, which is a measure of how easily a material can be magnetized. The magnitude of the magnetostriction depends on the magnitude of the magnetic field and the susceptibility of the material.

The phenomenon of magnetostriction is described by the following formula: $$ \Delta L = \lambda \cdot H $$ where $\Delta L$ is the change in length of the material, $\lambda$ is the magnetostriction coefficient, and $H$ is the applied magnetic field.

One of the most common applications of magnetostriction is the production of ultrasonic waves. Ultrasonic waves are sound waves with a frequency greater than 20 kHz, and they can be used for a variety of purposes such as medical imaging and non-destructive testing. Magnetostriction can be used to produce ultrasonic waves by passing an alternating current through a coil of wire that is wrapped around a ferromagnetic material, such as iron or nickel. The alternating current creates a changing magnetic field which induces magnetostriction in the material, causing it to change shape at the same frequency as the current. This change in shape produces ultrasonic waves, which can be used for various applications.

The equation for producing ultrasonic waves using magnetostriction can be represented as follows: $$ \frac{\partial^2 u}{\partial t^2} = \frac{1}{\rho} \left(\frac{\partial^2 \sigma}{\partial x^2} + \frac{\partial^2 \sigma}{\partial y^2} + \frac{\partial^2 \sigma}{\partial z^2}\right) $$ where $u$ represents the displacement of the material, $t$ is time, $\rho$ is the density of the material, $\sigma$ is the stress produced due to magnetostriction and the partial derivatives represent the spatial derivatives in the $x$, $y$ and $z$ directions.

Magnetostriction is a phenomenon that occurs in ferromagnetic materials whereby the material changes shape in response to an applied magnetic field. It is related to the magnetic susceptibility of the material and can be used to generate ultrasonic waves. The process of producing ultrasonic waves using magnetostriction involves passing an alternating current through a coil of wire that is wrapped around a ferromagnetic material which induces magnetostriction in the material, causing it to change shape at the same frequency as the current. This change in shape produces ultrasonic waves, which can be used for various applications. Magnetostriction can be represented as follows: $$\Delta L = \lambda_m \cdot \left(\frac{\partial B}{\partial t}\right)$$ where: $\Delta L$ is the change in length of the ferromagnetic material $\lambda_m$ is the magnetostriction constant, which is a measure of the sensitivity of the material to an applied magnetic field $B$ is the magnetic field strength $t$ is time The change in shape of the ferromagnetic material can then be used to produce ultrasonic waves, with frequency $f$ given by: $$f = \frac{1}{2 \pi} \cdot \frac{\partial \Delta L}{\partial t}$$

Ultrasound

Ultrasound is a form of sound that has a frequency higher than the human ear can hear. It is used for a variety of medical and industrial applications, such as imaging and non-destructive testing. Ultrasound is produced using magnetostrictive materials such as ferrites and ferromagnetic materials. These materials are organized in a specific pattern and are subjected to a magnetic field in order to produce the ultrasonic waves.

Let us relate the velocity of sound $v$ in a magnetostrictive material to its elastic modulus $K$ and density $\rho$. It shows that the velocity of sound is proportional to the square root of the ratio of elastic modulus to density. $$ v = \sqrt {\frac {K}{\rho}} $$ where $v$ is the velocity of sound, $K$ is the elastic modulus, and $\rho$ is the density of the magnetostrictive material.

When the wavelength of ultrasound $\lambda$ is related to the velocity of sound $v$ and frequency $f$, It represents the wavelength as the ratio of the velocity of sound to the frequency. $$ \lambda = \frac {v}{f} $$ where $\lambda$ is the wavelength of the ultrasound, $v$ is the velocity of sound, and $f$ is the frequency of the ultrasound.

If we relate the frequency of ultrasound $f$ to the velocity of sound $v$ and wavelength $\lambda$; It represents the frequency as the ratio of the velocity of sound to the wavelength. $$ f = \frac {v}{\lambda} $$ where $f$ is the frequency of the ultrasound, $v$ is the velocity of sound, and $\lambda$ is the wavelength of the ultrasound.

When we relate the power of ultrasound $P$ to the density of the magnetostrictive material $\rho$, velocity of sound $v$, and cross-sectional area $A$; It represents the power as half the product of the density, velocity of sound squared, and cross-sectional area. $$ P = \frac {1}{2} * \rho * v^2 * A $$ where $P$ is the power of the ultrasound, $\rho$ is the density of the magnetostrictive material, $v$ is the velocity of sound, and $A$ is the cross-sectional area of the material.

When the force applied on a magnetostrictive material $F$ is related to the rate of change of pressure $\frac{dp}{dt}$. It represents the force as the derivative of pressure with respect to time. $$ F = \frac {dp}{dt} $$ where $F$ is the force applied on the magnetostrictive material, and $\frac {dp}{dt}$ is the rate of change of pressure.

If we find the relation between then change in length $\Delta L$ of a magnetostrictive material to the coefficient of linear expansion $\lambda$, initial length $L$, change in temperature $\Delta T$, and initial temperature $T$. It shows that the change in length is proportional to the initial length, coefficient of linear expansion, and the ratio of change in temperature to initial temperature. $$ \Delta L = \lambda * L * \frac {\Delta T}{T} $$ where $\Delta L$ is the change in length of the magnetostrictive material, $\lambda$ is the coefficient of linear expansion, $L$ is the initial length of the material, $\Delta T$ is the change in temperature, and $T$ is the initial temperature.

Conclusion

In conclusion, magnetic materials possess a unique set of properties that render them indispensable for a wide range of applications. These materials can be broadly categorized into four main classes: Diamagnetic, Paramagnetic, Ferromagnetic, and Ferrites. Each class of magnetic materials is characterized by its distinct magnetic properties, including magnetic susceptibility, magnetic permeability, and magnetic remanence, among others. These properties are critical in determining the suitability of magnetic materials for various applications.

For instance, diamagnetic materials are characterized by their low magnetic susceptibility and are typically used in applications where low magnetic fields are required. On the other hand, ferromagnetic materials possess a high magnetic susceptibility and are often used in applications that require high magnetic fields, such as in electromagnets and transformers. Paramagnetic materials exhibit intermediate magnetic susceptibility, and are used in a wide range of applications, including in magnetic refrigeration and magnetic storage.

Magnetic anisotropy and magnetostriction are two important properties of magnetic materials that are critical in the production of ultrasound. Magnetic anisotropy refers to the directional dependence of magnetic properties, and is used in determining the direction of magnetic domains in magnetic materials. Magnetostriction refers to the change in the dimensions of magnetic materials in response to an applied magnetic field, and is used in the production of ultrasonic waves through the generation of elastic stress.

To quantitatively describe magnetic anisotropy and magnetostriction, the following formulas can be used:

Magnetic Anisotropy Energy (MAE) $$ MAE = K * \left( \frac{sin^2\theta}{2} \right) $$ where $ K $ is the magnetic anisotropy constant and $ \theta $ is the angle between the magnetic field and the easy axis of magnetization.

Magnetostriction Energy $$ \Delta L = \lambda * L * H $$ where $ \Delta L $ is the change in length, $ \lambda $ is the magnetostriction coefficient, $ L $ is the length of the magnetic material, and $ H $ is the applied magnetic field.

In summary, magnetic materials are vital for various applications and possess a rich set of magnetic properties, including magnetic anisotropy and magnetostriction, which are critical in the production of ultrasound and other applications.

Super Conductivity

Superconductivity is a phenomenon that occurs in certain materials when they are cooled to very low temperatures. It is characterized by the total lack of electrical resistance, the ability to expel magnetic fields and become diamagnetic. These properties are evident in the form of sudden drop in electrical resistance to zero and strong diamagnetic response when the material is cooled below its critical temperature.

Notable Equations:
$j = \sigma * E$ relates the current density $j$ to the conductivity $\sigma$ and electric field $E$. $$ j = \sigma * E $$ where $j$ is the current density, $\sigma$ is the conductivity, and $E$ is the electric field.
$R = \frac {V}{I}$ gives the electrical resistance $R$ of a material in terms of voltage $V$ and current $I$. $$ R = \frac {V}{I} $$ where $R$ is the electrical resistance, $V$ is the voltage, and $I$ is the current.
$T_c = \frac {\Theta_D}{1 - (\frac {T}{T_c})^2}$ gives the critical temperature $T_c$ of a superconducting material in terms of the Debye temperature $\Theta_D$ and the temperature $T$ of the material. $$ T_c = \frac {\Theta_D}{1 - (\frac {T}{T_c})^2} $$ where $T_c$ is the critical temperature, $\Theta_D$ is the Debye temperature, and $T$ is the temperature of the superconducting material.
$H_{c1} = \frac {\Phi_0}{2\pi * \xi^2}$ gives the lower critical field $H_{c1}$ in terms of the magnetic flux quantum $\Phi_0$ and the coherence length $\xi$. $$ H_{c1} = \frac {\Phi_0}{2\pi * \xi^2} $$ where $H_{c1}$ is the lower critical field, $\Phi_0$ is the magnetic flux quantum, and $\xi$ is the coherence length.
$H_{c2} = \frac {\Phi_0}{2\pi * \xi * \lambda}$ gives the upper critical field $H_{c2}$ in terms of the magnetic flux quantum $\Phi_0$, the coherence length $\xi$, and the penetration depth $\lambda$. $$ H_{c2} = \frac {\Phi_0}{2\pi * \xi * \lambda} $$ where $H_{c2}$ is the upper critical field, $\Phi_0$ is the magnetic flux quantum, $\xi$ is the coherence length, and $\lambda$ is the penetration depth.
$M = -\frac {\partial^2 F}{\partial H^2}$ gives the magnetic moment $M$ in terms of the free energy $F$ and magnetic field strength $H$. $$ M = -\frac {\partial^2 F}{\partial H^2} $$ where $M$ is the magnetic moment, $F$ is the free energy, and $H$ is the magnetic field strength.

Types of Superconductors

Superconductors can be divided into two broad categories, Type 1 and Type 2.

Type 1 Superconductors

Type 1 superconductors are ideal diamagnetic materials and display a complete exclusion of magnetic fields. This type of superconductivity is often referred to as perfect diamagnetism. Type 1 superconductors are usually composed of elements such as lead, tin and indium.

  1. Type 1 Superconductors are materials that can conduct electricity with no resistance below a certain temperature, known as the critical temperature (Tc).
  2. Type 1 Superconductors are typically composed of pure metals such as lead, tin, and mercury.
  3. The critical temperature of Type 1 Superconductors can vary depending on the material, but is typically within the range of 4-15 Kelvin (-269 to -258 Celsius).
  4. Type 1 Superconductors exhibit the Meissner effect, where the material will completely expel any external magnetic fields when cooled below the critical temperature.
  5. The critical temperature of type 1 superconductors can be predicted using the BCS theory, where Tc is approximated using the equation Tc =1.13Δ₀/kₐ, where Δ₀ is the superconducting gap at 0K and kₐ is the Boltzmann constant.
  6. Type 1 superconductors can be used to create superconducting magnets, which can generate extremely powerful magnetic fields for use in magnetic resonance imaging and particle accelerators.

Type 2 Superconductors

Type 2 superconductors are also diamagnetic, but display a less perfect exclusion of magnetic fields. This type of superconductor is usually composed of metals such as niobium, vanadium and titanium. Type 2 superconductors are often referred to as "dirty" superconductors as they are less perfect than Type 1.

  1. Type 2 superconductors are materials that can carry an electrical current without any resistance below a certain critical temperature, typically between 30 and 100K.
  2. Type 2 superconductors are composed of two materials, a metal and a non-metal, and are usually layered in an alternating pattern.
  3. Type 2 superconductors are described by the Ginzburg-Landau theory, which states that the critical temperature of a superconductor is related to its material properties such as the London penetration depth, λ, the coherence length, ξ, and the Ginzburg-Landau parameter, κ, according to the following equation: Tc = φ/2πλξκ
  4. Type 2 superconductors are characterized by an intermediate temperature regime between the fully-superconducting and normal states, known as the mixed state, where the superconductivity is partially suppressed.
  5. In the mixed state, the Meissner effect is less pronounced, and the material exhibits flux pinning due to the presence of small magnetic vortices.

Application of Superconductors

Superconductors are of great interest to scientists and engineers due to their many potential applications, such as in electric motors, generators, and other electronic devices. Superconducting materials have also been used to create extremely sensitive magnetic field detectors, which are used in a variety of fields including medical imaging, astrophysics, and mineral exploration.

Conclusion

Despite the many potential applications of superconductors, their use is still limited by the fact that they require temperatures near absolute zero in order to operate. For this reason, scientists and engineers continue to search for new materials that are able to operate at higher temperatures. If such materials can be discovered, then the potential applications of superconductors will be greatly expanded.

Additional Reading:

  1. Maxwell Field Equations
  2. Time Variation
  3. Wave Equation
  4. Characterstics of EM Wave Propagation
  5. Energy integral - Power Flow anf Poynting Vector
  6. Snell's Law
  7. Signal Distortion
  8. Ground Wave Propagation
  9. Attenuation
  10. VLF / ELF Wave Propagation
  11. Tropospheric Super Refraction and Scattering
  12. EM Waves in Magnetoionic Medium - Appleton-Hartree Formula
  13. Ray Theory
  14. Equatorial Ionosphere Anomaly
  15. Electron Density Profiling
  16. Conductivities and currents in D-region and E-region
  17. Middle Atmospheric Dynamics
  18. Whistler Propagation
  19. Planetary Molecular Absorptions