UNIT 5

ELECTROMAGNETISM AND MAGNETIC PROPERTIES OF MATERIALS

5.1 LAWS OF ELECTROSTATICS

ELECTROSTATICS

Electrostatics, the study of electromagnetic phenomena that occur when there are no moving charges—i.e., after a static equilibrium has been established. Charges reach their equilibrium positions rapidly, because the electric force is extremely strong.

Electrostatics, as the name implies, is the study of stationary electric charges. A rod of plastic rubbed with fur or a rod of glass rubbed with silk will attract small pieces of paper and is said to be electrically charged. The charge on plastic rubbed with fur is defined as negative, and the charge on glass rubbed with silk is defined as positive.

There are some important laws we used in electrostatics.

Coulomb’s Law:

The magnitude of the electric force that a particle exerts on another particle is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The direction of the force is along the line joining the particles.

Mathematically, the electric force F that a particle of charge q’ exerts on a particle of charge q at a distance r is given by the formula

F =k

Where k is a constant of proportionality. This formula not only gives the magnitude of the force, but also the direction, if we interpret a positive value of the force F as repulsive and a negative value as attractive.

When a small positive test charge is brought near a large positive charge, it experiences a force directed away from the large charge. If the test charge is far from the large charge, the electrostatic force given by Coulomb's law is smaller than when it is near. This data of direction and magnitude of an electrostatic force, due to a fixed charge or set of fixed charges, constitutes an electrostatic field. The electric field is defined as the force per unit charge exerted on a small positive test charge (q 0) placed at that point. Mathematically, 

E =

Note that both the force and electric field are vector quantities.

Figure 1:  is a pictorial representation of the electric fields surrounding a positive charge and a negative charge. These lines are called field lines or lines of force.

Figure 1: Electric field lines of (a) positive and (b) negative point charges.

Figure 2  shows the electric fields for opposite charges, similar charges, and oppositely charged plates.

Figure 2:Electric field lines of (a) two opposite charges, (b) two like charges, and (c) two oppositely charged plates.

Electric flux

Electric flux is defined as the number of field lines that pass through a given surface. In Figure , lines of electric flux emerging from a point charge pass through an imaginary spherical surface with the charge at its center.

Figure 3: Electric flux

Gauss’s law of electrostatics

The law was published posthumously in 1867 as part of a collection of work by the famous German mathematician Carl Friedrich Gauss.

Gauss’s law for the electric field describes the static electric field generated by a distribution of electric charges.

It states that the electric flux through any closed surface is proportional to the total electric charge enclosed by this surface.

Let’s calculate the electric flux through a spherical surface around a positive point charge q, since we already know the electric field in such a situation. Recall that when we place the point charge at the origin of a coordinate system, the electric field at a point P that is at a distance r

From the charge at the origin is given by

is the radial vector from the charge at the origin to the point P. We can use this electric field to find the flux through the spherical surface of radius r, as shown in Figure 4

.

Figure 4: A closed spherical surface surrounding a point charge q.

Then we apply Φ=∫S  ⋅ dA

To this system and substitute known values. On the sphere, and r=R

So for an infinitesimal area dA,

We now find the net flux by integrating this flux over the surface of the sphere

Where the total surface area of the spherical surface is 4πR2. This gives the flux through the closed spherical surface at radius r as

Φ=q/ϵ0

A remarkable fact about this equation is that the flux is independent of the size of the spherical surface. This can be directly attributed to the fact that the electric field of a point charge decreases as 1/r2 with distance, which just cancels the r2 rate of increase of the surface area.

Key Takeaways

• Coulomb’s Law states that the magnitude of the electric force that a particle exerts on another particle is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The direction of the force is along the line joining the particles.
• Electric flux is defined as the number of field lines that pass through a given surface.
• Gauss’s law of electrostatics states that the electric flux through any closed surface is proportional to the total electric charge enclosed by this surface.

5.2 ELECTRIC CURRENT AND THE CONTINUITY EQUATION

The amount of electric current traveling per unit cross-section area is called as current density and expressed in amperes per square meter.

Current density is a vector quantity having both a direction and a scalar magnitude. The electric current flowing through a solid having units of charge per unit time is calculated towards the direction perpendicular to the flow of direction. It is all about the amount of current flowing across the given region.

In case of a steady current is flowing through a conductor, same current flows through all the cross-sections  of the conductor, even though the cross-sections may differ in area. Electric current is a macroscopic entity.

We here discuss electric current passing through a conductor and not about electric current at a point.

Current density J, at any point inside a conductor, is a vector quantity whose magnitude  is equal to the current per unit area through an infinitesimal area at that point, the area being held perpendicular to the direction of flow of charge and its direction is along the direction of flow of positive charge.

Figure 5: Current density

More the current in a conductor, the higher will be the current density. However, the current density alters in different parts of an electrical conductor and the effect takes place with alternating currents at higher frequencies.

Electric current always creates a magnetic field. Stronger the current, more intense is the magnetic field. Varying AC or DC creates an electromagnetic field and this is the principle based on which signal propagation takes place.

Current Density Formula

The formula for Current Density is given as,

J = I / A

Where,

I = current flowing through the conductor in Amperes

A = cross-sectional area in m2.

Current density is expressed in A/m2.

In case of non-uniform flow, current I through any cross-section is

Here the integral denotes the surface integral over the entire cross-section.

Equation of continuity

Consider a closed surface S enclosing a Volume V through which currents are flowing. Let I be the current passing through the surface at any time t. Consider a small area of the closed surface. The outward current flowing through    at a time t is

Figure 6: Current density through surface S

DI =       …………..(1)

The total current flowing out through the closed surface S at time t is

I =       …………..(2)

Suppose is ρ the Volume Charge density of charges inside the surface S. Then the total charge inside the Volume V is

q =       …………..(3)

Since the current is flowing outward is it mean that charge within the enclosed surface is decreasing with time. The time rate of decrease of charge is

= -       …………..(4)

The negative sign shows that the charge inside the surface S is decreasing with time. The above equation can be written as

= -       …………..(5)

Since the charge is conserved, the current flowing outward through the closed surface must be equal to the rate of decrease of charge. Thus 

I =

Using equation (2) and (5), it becomes

= -       …………..(6)

According to divergence theorem

= dV

Using this result in equation (6) we obtain

dV = -   

This equation holds good for arbitrary volumes. Therefore the integrands must be equal. Hence  

=0 …………..(7)

This equation is called equation of continuity. It is the mathematical statement of law of conservation of charge. 

Special case

Consider a conductor AB through which a steady current I is flowing. The current I is said to be steady if charge is flowing per unit time through any two any sections C and D is the same. This means that the total amount of charge entering the volume V through C is equal to the charge leaving the volume V through D. Thus there is no change in the volume charge density in the volume V with time i.e.

=0

Figure 7: Current through conductor AB

We may define steady current as that current in which the charge density is independent of time.

Using equation (7) we obtain

This is known as the equation of continuity for steady currents.

Key Takeaways

• The amount of electric current traveling per unit cross-section area is called as current density and expressed in amperes per square meter.
• Current density is a vector quantity having both a direction and a scalar magnitude.
• The formula for Current Density is given as J = I / A
• The equation =0 is called equation of continuity. It is the mathematical statement of law of conservation of charge. 
• Steady current can be define as that current in which the charge density is independent of time .This is known as the equation of continuity for steady currents.

5.3 AMPERE’S AND FARADAY’S LAWS

AMPERE’S LAW

Ampere’s circuital law (ACL) relates current to the magnetic field associated with the current. The magnetic field in space around an electric current is proportional to the electric current which serves as its source, just as the electric field in space is proportional to the charge which serves as its source.

Figure 8: Ampere’s circuital law

Ampere's Law states that

The integral around a closed path of the component of the magnetic field tangent to the direction of the path equals μ0 times the current intercepted by the area within the path.

= μ0I     ………..(1)

Ampere's Law tells that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop.

In order to apply Ampère’s Law

All currents have to be steady (i.e.do not change with time)

Only currents crossing the area inside the path are taken into account and have some contribution to the magnetic field

The total magnetic circulation is zero only in the following cases:-

• The enclosed net current is zero
• The magnetic field is normal to the selected path at any point
• The magnetic field is zero

Finally, be aware that the form of Ampère’s Law addressed here applies to magnetostatics only. In the presence of a time-varying electric field, the right side of ACL includes an additional term known as the displacement current.

FARADAY’S LAWS OF EMI

Faraday’s law of electromagnetic induction (referred to as Faraday’s law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF). This phenomenon is known as electromagnetic induction.

Faraday’s law states that a current will be induced in a conductor which is exposed to a changing magnetic field. Lenz’s law of electromagnetic induction states that the direction of this induced current will be such that the magnetic field created by the induced current opposes the initial changing magnetic field which produced it. The direction of this current flow can be determined using Fleming’s right-hand rule.

Let us understand Faraday’s law by faraday‘s experiment.

FARADAY’S EXPERIMENT

In this experiment, Faraday takes a magnet and a coil and connects a galvanometer across the coil. At starting, the magnet is at rest, so there is no deflection in the galvanometer i.e. the needle of the galvanometer is at the center or zero position. When the magnet is moved towards the coil, the needle of the galvanometer deflects in one direction.

Figure 9: Faraday’s law of electromagnetic induction

When the magnet is held stationary at that position, the needle of galvanometer returns to zero position. Now when the magnet moves away from the coil, there is some deflection in the needle but opposite direction, and again when the magnet becomes stationary, at that point respect to the coil, the needle of the galvanometer returns to the zero position.

Similarly, if the magnet is held stationary and the coil moves away, and towards the magnet, the galvanometer similarly shows deflection. It is also seen that the faster the change in the magnetic field, the greater will be the induced EMF or voltage in the coil.

Let us take a look at the following table. This table is showing the observation of faraday’s experiment.

CONCLUSION

From this experiment, Faraday concluded that whenever there is relative motion between a conductor and a magnetic field, the flux linkage with a coil changes and this change in flux induces a voltage across a coil.

Michael Faraday formulated two laws on the basis of the above experiments. These laws are called Faraday’s laws of electromagnetic induction.

FARADAY’S FIRST LAW

Any change in the magnetic field of a coil of wire will cause an emf to be induced in the coil. This emf induced is called induced emf and if the conductor circuit is closed, the current will also circulate through the circuit and this current is called induced current.
Method to change the magnetic field:

1. By moving a magnet towards or away from the coil
2. By moving the coil into or out of the magnetic field
3. By changing the area of a coil placed in the magnetic field
4. By rotating the coil relative to the magnet

FARADAY’S SECOND LAW

It states that the magnitude of emf induced in the coil is equal to the rate of change of flux that linkages with the coil. The flux linkage of the coil is the product of the number of turns in the coil and flux associated with the coil.

According to Faraday’s law of electromagnetic induction, the rate of change of flux linkage is equal to induced emf.

        ………..(9)

Considering Lenz’s Law

Where:

• Flux Φ in Wb = B.A
• B = magnetic field strength
• A = area of the coil

FACTORS ON WHICH INDUCED EMF DEPEND

• By increasing the number of turns in the coil i.e. N, from the formulae derived above it is easily seen that if the number of turns in a coil is increased, the induced emf also gets increased.
• By increasing magnetic field strength i.e. B surrounding the coil- Mathematically, if magnetic field increases, flux increases and if flux increases emf induced will also get increased. Theoretically, if the coil is passed through a stronger magnetic field, there will be more lines of force for the coil to cut and hence there will be more emf induced.
• By increasing the speed of the relative motion between the coil and the magnet – If the relative speed between the coil and magnet is increased from its previous value, the coil will cut the lines of flux at a faster rate, so more induced emf would be produced.

Key Takeaways

• Ampere’s circuital law (ACL) relates current to the magnetic field associated with the current.
• Ampere's Law states that the integral around a closed path of the component of the magnetic field tangent to the direction of the path equals μ0 times the current intercepted by the area within the path. = μ0
• Any change in the magnetic field of a coil of wire will cause an emf to be induced in the coil. This emf induced is called induced emf .
• If the conductor circuit is closed, the current will also circulate through the circuit and this current is called induced current.
• Magnetic field can be change by moving a magnet towards or away from the coil, by moving the coil into or out of the magnetic field, by changing the area of a coil placed in the magnetic field, by rotating the coil relative to the magnet.
• Faraday’s law states that the magnitude of emf induced in the coil is equal to the rate of change of flux that linkages with the coil.
• The flux linkage of the coil is the product of the number of turns in the coil and flux associated with the coil.

5.4 MAXWELL’S EQUATIONS

Maxwell equations are of fundamental importance since they describe the whole of classical electromagnetic phenomena.

From a classical perspective, light can be described as waves of electromagnetic radiation. As such, Maxwell equations are very useful to illustrate a number of the characteristics of light including polarization.

We are just to stating these equations without derivation. Since our goal is simply to apply them, the usual approach will be followed.

Maxwell’s four equations are given by

∇·E = ρ/ε0        (1)

∇×E = −∂B/∂t        (2)

∇×H = J + ∂D/∂t       (3)

∇·B = 0        (4)

These equations illustrate the unique coexistence in nature of the electric field and the magnetic field. The first two equations give the value of the given flux through a closed surface, and the second two equations give the value of a line integral around a loop. In this notation,

∇=(∂/∂x, ∂/∂y, ∂/∂z)

E is the electric vector

B is the magnetic induction

ρ is the electric charge density

j is the electric current density

ε0 is the permittivity of free space

c is the speed of light.

In addition to Maxwell equations, the following identities are useful:

J =  σE         (5)

D = εE         (6)

B = μH         (7)

Here,

D is the electric displacement

H is the magnetic vector

σ is the specific conductivity

ε is the dielectric constant (or permittivity)

μ is the magnetic permeability

In the Gaussian systems of units, Maxwell equations are given in the form of

∇·B=0          (8)

∇·E=4πρ         (9)

∇×H=(1/c)(∂D/∂t+4πj)       (10)

∇×E=−(1/c)(∂B/∂t)        (11)

Free space or non-conducting medium. We know that non conducting medium means no current so conductivity is zero i.e. σ=0

So current density J=σE will also become zero as σ=0 Also free space means no charges which leads to ρ=0. These points mentioned below.

(a)    No condition current i.e σ=0, thus J=0 ( J=σE)

(b)   No charges (i.e ρ=0)

For the case of no charges or currents, that is, j = 0 and ρ = 0, and a homogeneous medium. Using these the Maxwell equation can rewritten as

∇.D=0 or ∇.E=0      as  ρ=0         (12)

∇  x E= -dB/dt  or    ∇ x E= -μ dH/dt    because  B = μH       (13)

∇ x H=d D/dt  or  ∇ x H = ε dE/dt         (J=0) and D = εE  (14)

∇.B=0                                      (15)

Maxwell's Equations are a set of 4 complicated equations that describe the world of electromagnetics. These equations describe how electric and magnetic fields propagate, interact, and how they are influenced by objects.

Maxwell was one of the first to determine the speed of propagation of electromagnetic (EM) waves was the same as the speed of light - and hence to conclude that EM waves and visible light were really the same thing.

These equations are rules the universe uses to govern the behaviour of electric and magnetic fields. A flow of electric current will produce a magnetic field. If the current flow varies with time (as in any wave or periodic signal), the magnetic field will also give rise to an electric field. Maxwell's Equations shows that separated charge (positive and negative) gives rise to an electric field - and if this is varying in time as well will give rise to a propagating electric field, further giving rise to a propagating magnetic field.

From a classical perspective, light can be described as waves of electromagnetic radiation. As such, Maxwell equations are very useful to illustrate a number of the characteristics of light including polarization.

The First Maxwell’s equation (Gauss’s law for electrostatics)

Gauss' Law is the first of Maxwell's Equations which dictates how the Electric Field behaves around electric charges. Gauss' Law can be written in terms of the Electric Flux Density and the Electric Charge Density as:

∇·E = ρ/ε0 

0r

∇·D = ρ 

Hence, Gauss' law is a mathematical statement that the total Electric Flux exiting any volume is equal to the total charge inside. Hence, if the volume in question has no charge within it, the net flow of Electric Flux out of that region is zero. If there is positive charge within a volume, then there exists a positive amount of Electric Flux exiting any volume that surrounds the charge. If there is negative charge within a volume, then there exists a negative amount of Electric Flux exiting (i.e. the Electric Flux enters the volume).

The Gauss’s law states that flux passing through any closed surface is equal to 1/ε0 times the total charge enclosed by that surface.

Integral form of Maxwell’s 1st equation

It is the integral form of Maxwell’s 1st equation.

Maxwell’s first equation in differential form

It is called the differential form of Maxwell’s 1st equation.

The Second Maxwell’s equation (Gauss’s law for magnetism)

The Gauss’s law for magnetism states that net flux of the magnetic field through a closed surface is zero because monopoles of a magnet do not exist.

We know that Gauss' Law for Electric Fields states that the divergence of the Electric Flux Density D is equal to the volume electric charge density. But the second equation, Gauss' Magnetism law states that the divergence of the Magnetic Flux Density (B) is zero.

Why? Why isn't the divergence of B equal to the magnetic charge density?

Well - it is. But it just so happens that no one has ever found magnetic charge - not in a laboratory or on the street or on the subway. And therefore, until this hypothetical magnetic charge is found, we set the right side of Gauss' Law for Magnetic Fields to zero:

∇·B = 0 

Now, you may have played with magnets when you were little, and these magnetic objects attracted other magnets similar to how electric charges repel or attract like electric charges. However, there is something special about these magnets - they always have a positive and negative end. This means every magnetic object is a magnetic dipole, with a north and South Pole. No matter how many times you break the magnetic in half, it will just form more magnetic dipoles. Gauss' Law for Magnetism states that magnetic monopoles do not exist - or at least we haven't found them yet.

Maxwell’s Third equation (Faraday's Law)

Faraday figured out that a changing Magnetic Flux within a circuit (or closed loop of wire) produced an induced EMF, or voltage within the circuit. Maxwell gives his second equation from this.

Faraday's Law tells us that a magnetic field that is changing in time will give rise to a circulating E-field. This means we have two ways of generating E-fields - from Electric Charges (or flowing electric charge, current) or from a magnetic field that is changing.

∇×E = −∂B/∂t        

According to Faraday’s law of electromagnetic induction

It is the differential form of Maxwell’s third equation.

Maxwell’s fourth equation (Ampere's Law)

Ampere's Law tells us that a flowing electric current gives rise to a magnetic field that circles the wire. In addition to this, it also says that an Electric Field that is changing in time gives rise to a magnetic field that encircles the E-field - this is the Displacement Current term that Maxwell himself introduced.

This means there are 2 ways to generate a solenoidal (circulating) H-field - a flowing electric current or a changing Electric Field. Both give rise to the same phenomenon.

The modified form of ampere Law is given by

H.dL=(J+dD/dt)

Take integration on both sides we have

∫H.dL=∫(J+dD/dt).dS

Apply stoke’s theorem to L.H.S. Of above equation, we get

∫( ∇xH).dS=∫ H.dL

Now the above equation is written as

∫( ∇xH).dS =∫(J+dD/dt).dS

By cancelling the surface integral on both sides we have

∇xH =J+dD/dt

Which  is the differential form of Maxwell’s equation.

This can also be written in the form B

The Ampere-Maxwell Law

Begin with the Ampere-Maxwell law in integral form.

∮CB⋅dl=μ0∫SJ⋅dS+μ0ϵ0ddt∫SE⋅dS{\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} +\mu _{0}\epsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {S} }Invoke Stokes' theorem.

∫S(∇×B)⋅dS=μ0∫SJ⋅dS+μ0ϵ0ddt∫SE⋅dS{\displaystyle \int _{S}(\nabla \times \mathbf {B} )\cdot \mathrm {d} \mathbf {S} =\mu _{0}\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} +\mu _{0}\epsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {S} }

Set the equation to 0. ∫S(∇×B)⋅dS−μ0∫SJ⋅dS−μ0ϵ0ddt∫SE⋅dS=0∫S(∇×B−μ0J−μ0ϵ0∂E∂t)⋅dS=0{\displaystyle {\begin{aligned}\int _{S}(\nabla \times \mathbf {B} )\cdot \mathrm {d} \mathbf {S} -\mu _{0}\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} -\mu _{0}\epsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {S} &=0\\\int _{S}\left(\nabla \times \mathbf {B} -\mu _{0}\mathbf {J} -\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\cdot \mathrm {d} \mathbf {S} &=0\end{aligned}}}

Physical Significance Of Each Equation

Physical significance of Maxwell’s Ist equation

∇·E = ρ/ε0

According to this total electric flux through any closed surface is 1/0 times the total charge enclosed by the closed surfaces, representing Gauss's law of electrostatics, As this does not depend on time, it is a steady state equation. Here for positive , divergence of electric field is positive and for negative  divergence is negative. It indicates that  is scalar quantity.

Physical significance of Maxwell’s 2nd equation

∇·B = 0

It represents Gauss law of magnetostatic as ∇·B = 0 resulting that isolated magnetic poles or magnetic monopoles cannot exist as they appear only in pairs and there is no source or sink for magnetic lines of forces. It is also independent of time i.e. steady state equation.

Physical significance of Maxwell’s 3rd equation

∇×E = −∂B/∂t 

It shows that with time varying magnetic flux, electric field is produced in accordance with Faraday is law of electromagnetic induction. This is a time dependent equation.

Physical significance of Maxwell’s 4th equation

∇×H = J + ∂D/∂t 

This is a time dependent equation which represents the modified differential form of Ampere's circuital law according to which magnetic field is produced due to combined effect of conduction current density and displacement current density.

Key Takeaways

• The First Maxwell’s equation  is known as Gauss’s law for electrostatics. It is given by ∇·E = ρ/ε0              . According to this total electric flux through any closed surface is 1/0 times the total charge enclosed by the closed surfaces, representing Gauss's law of electrostatics,
• The Second Maxwell’s equation is known as Gauss’s law for magnetism. . It is given by ∇·B = 0.              The Gauss’s law for magnetism states that net flux of the magnetic field through a closed surface is zero because monopoles of a magnet do not exist.
• Maxwell’s Third equation is known as Faraday's Law. It is given by ∇×E = −∂B/∂t. Faraday figured out that a changing Magnetic Flux within a circuit (or closed loop of wire) produced an induced EMF, or voltage within the circuit.
• Maxwell’s fourth equation is known as Ampere's Law. It is given by ∇×H = J + ∂D/∂t              .Ampere's Law tells us that a flowing electric current gives rise to a magnetic field that circles the wire. In addition to this, it also says that an Electric Field that is changing in time gives rise to a magnetic field that encircles the E-field - this is the Displacement Current.

5.5 POLARISATION

In dielectric material molecules are tightly bounded to the nucleus and not able to move freely as in case of conductor. But when a dielectric material is placed in an electric field, electric charges do not flow through the material as they flow in case of an electrical conductor but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the direction opposite to the field. In dielectric molecules are tightly bound to the nucleus and not able to move freely as in case of conductors. The dielectric material is largely used in the manufacturing of capacitor.

Response of Dielectric to External Electric Field

In case of Polar Molecules -When the electric field is not present, it causes the electric dipole moment of these molecules in a random direction. This is why the average dipole moment is zero. If the external electric field is present, the molecules align themselves in the direction electric field and resulted in having dipole moment.

In case of Non-Polar Molecules – As we know nonpolar molecule has zero dipole moment. In spite of zero dipole, when a dielectric nonpolar material is placed in an electric field. The positive and the negative charges in a nonpolar molecule experience forces in opposite directions. This force causes the separation between the charges and hence nonpolar molecule experiences induced dipole moment.

Figure 10 (a) (left) Non Polar Molecule    Figure 10 (b) (right) Polar Molecule

In previous section we have seen about the dielectric material’s behaviour and we have seen that when the dielectric material is placed in an electric field, then the molecule of polar and non-polar dielectric gains the dipole moment. Then the dielectric is said to be polarised.

Definition:

The diploe moment per unit volume of dielectric material is called the electric polarization of dielectric. It is represented by vector P.

In S.I. System, unit of polarization is C/m2.

When an external electric field is applied to a dielectric material, its behaviour can be determined and is known as Dielectric Polarization that can be understood by the displacement of charges(positive and negative) when an electric field is applied

The main task of the dielectric polarization is to relate macroscopic properties to microscopic properties. Where macroscopic property can be dielectric constant to polarizability.

Polarization occurs through the action of an electric field or other external factors, such as mechanical stress in the case of piezoelectric crystals (piezoelectric crystals are those solid materials which accumulates electric charge within them). Dielectric Polarization can also arise spontaneously in pyroelectric crystals, particularly in ferroelectrics (Ferroelectricity is a property of certain materials that have a spontaneous electric polarization that can be reversed by the application of an external electric field).

Figure 11:  polarisation of dielectric molecules

Above figure explains the polarisation of dielectric molecules when the electric field is applied.

Dielectric polarization occurs when a dipole moment is formed in an insulating material because of an externally applied electric field. When a current interacts with a dielectric material, the dielectric material will respond with a shift in charge distribution with the positive charges aligning with the electric field and the negative charges aligning against it.

Type of dielectric polarization

There are four types of dielectric polarization

1. Electronic polarization

2. Ionic polarization

3. Orientational polarization

4. Space charge polarization

Key Takeaways

• A dielectric is non-conducting material which stores electrical charges.
• The molecules of a dielectric is of two types Polar molecules and Non-polar molecules
• The diploe moment per unit volume of dielectric material is called the electric polarization of dielectric. It is represented by vector P. In S.I. System, unit of polarization is C/m2.

5.6 PERMITTIVITY AND DIELECTRIC CONSTANT

Dielectric constant

Dielectric constant it is a quantity measuring the ability of a substance to store electrical energy in an electric field.

Or

The ratio of the permittivity of the substance to the permittivity of the free space

Dielectric constant is also called Relative Permittivity Dielectric Constant is expressed by Greek letter kappa ‘κ’. It is dimensionless quantity.

It is mathematically expressed as:

κ =

Where,

• κ is the dielectric constant
• 𝜺 is the permittivity of the substance
• 𝜺0 is the permittivity of the free space

Dielectric constant in terms of capacitance

Dielectric constant is equal to the ratio of the capacitance of a capacitor filled with the given material to the capacitance of an identical capacitor in a vacuum without the dielectric material.

If C is the value of the capacitance of a capacitor filled with a given dielectric and C0 is the capacitance of an identical capacitor in a vacuum, the dielectric constant κ, is simply expressed as

κ = 

Dielectric Strength

It is defined as the maximum electric field which the material can sustain without breaking down.

When a high electric field is applied to the dielectric, the outer electrons get detached from their parent atoms. The dielectric then behaves as a conductor. In other words every dielectric starts conducting when if an external field is applied to it. This value of electric field depends on the nature of material of dielectric and is called dielectric strength.

Table 1: List of value of dielctric constant and dielectric strength of some material

Relation  between dielectric constant and susceptibility

Electric displacement, also known as dielectric displacement and usually denoted by D, is a vector field in a non-conducting medium, a dielectric.

In a dielectric material, the presence of an electric field E causes the bound positive and negative charges in the material to slightly separate, inducing a local electric dipole moment. The electric displacement field "D" is defined as

D= 𝜺0E+P

D =D ≡ ε 0 E + P , {\displaystyle \mathbf {D} \equiv \varepsilon _{0}\mathbf {E} +\mathbf {P} ,} DD                           

Electric displacement 

E = External electric field in which the dielectric is placed

𝜺0 = the permittivity of the free space

P = Polarization Density

In S.I. System, Unit of Electric displacement is C/m2.

D = E+P

D=  E      and  P = eE

Substituting these values in eq (1), we get

E = E+eE

= +e

= 1 +

But

κ = = Dielectric constant

κ = 1 +

Clearly values of κ  for all dielectric is greater than one, and for empty space e=0 and κ =1.

5.7 INTERNAL FIELDS IN A SOLID, CLAUSIUS-MOSSOTTI EQUATION

In dielectric solids, the atoms or molecules experience not only the external applied electric field but also the electric field produced by the dipoles. The resultant electric field acting on the atoms or molecules of dielectric substance is called the local field or an internal field.

To find an expression for local electric field on a dielectric molecule or an atom, we consider a dielectric material in the electric field of intensity E, between the capacitor plates so that the material is uniformly polarized, as a result opposite type of charges are induced on the surface of the dielectric near the capacitor plates. The local field is calculated by using the method suggested by Lorentz.

According to this method,

Internal field or Local field in solids

Consider a dielectric material and is subjected to external field of intensity E1. The charges are induced on the dielectric plate and the induced electric field intensity is taken as E2. Let E3 be the field at the center of the material. E4 be the induced field due to the charges on the spherical cavity. The total internal field of the material is 

     ………..(1)

Figure 12: Local field in solids

Now consider the Electric field intensity applied E1

        ………..(2)

We know

       ………..(3)

Substituting the Electric flux density D in E1, we get

       ………..(4)

       ………..(5)

E2 is the Electric field intensity due to induced or polarized charges 

        ………..(6)

Here the charge is induced due to the induced field so the electric flux density D changes to the electric polarization P

        ………..(7)

Since we have considered that the specimen is non-polar dielectric material, at the center of the specimen the dipole moment is zero and hence the electric field intensity at the center is zero due to symmetric structure.

        ………..(8)

Now consider a circle from the center of the dielectric material. Calculation of the electric field intensity E4 on the surface of spherical cavity.

As we know the polarization P is the induced charge per unit area

       ………..(9)

Here the polarization changes to its component so we will take the component which is contributing.

       ………..(10)

Now this equation can be solved by finding out the values of the charge dq in the surface are dA. We know the Electric field intensity E

        ………..(11)

Multiplying with the cosine angle on both the sides we get

      ………..(12)

Now by applying all the present condition for the above equation we 

      ………..(13)

Now substituting the charge dq in the above equation we get

     ………..(14)

Figure 13: Sphere under applied field

If dA is the surface area of the sphere of radius r lying between θ and θ+dθ is the direction with reference to the direction of the applied force, then

DA= 2π(PQ)(QR)

But

Sinθ= PQ/r,

PQ = rsinθ         ………..(15)

And

dθ=QR/r,

QR=r dθ        ………..(16)

Hence dA= 2πrsinθrdθ= 2πr2sinθdθ    ………..(17)

Now substituting all the values in the electric field intensity on the spherical cavity E4 we get,

     ………..(18)

Substituting (17) in (18), we get

       ………..(19)

Integrating with in the limits 0 to π

      ………..(20)

On solving the integration we get,

         ………..(21)

So the total electric field 

      ………..(22)

Hence the Internal field obtained is

        ………..(23)

CLAUSSIUS -MOSOTTI EQUATION

It gives the relation between the dielectric constant and the ionic polarizability of atoms in dielectric material. If there are N number of atoms, the dipole moment per unit volume which is called Polarization is given by,

       ………..(24)

We know internal field

       ………..(25)

From above equations

      ………..(26)

We know polarization from the relation between polarization and dielectric constant

      ………..(27)

From the above two equations we get,

      ………..(28)

Key Takeaways

• In dielectric solids, the atoms or molecules experience not only the external applied electric field but also the electric field produced by the dipoles.
• The resultant electric field acting on the atoms or molecules of dielectric substance is called the local field or an internal field.
• The local field is calculated by using the method suggested by Lorentz.
• Internal field or Local field in solids is given by sum of external field of intensity, induced electric field intensities.

• Hence the Internal field obtained is given as

• Claussius - Mosotti equation gives the relation between the dielectric constant and the ionic polarizability of atoms in dielectric material and th relation is given as                                                                                   

5.8 FERROELECTRICS AND PIEZOELECTRIC

Ferroelectrics

Ferroelectricity was first discovered in 1921 in a complex compound called Rochelle salt (Sodium potassium tartrate tetrahydrate). At that time it was the only material possessing the extraordinary property of reversible polarization. Later around 1935 ferroelectricity was observed in potassium dihydrogen phosphate (KH2PO4). The most popular and classical ferroelectric material BaTiO3 was discovered during World War II in 1941. BaTiO3 originally looked as a potential material for capacitors because of its high dielectric constant.

The high dielectric constant originates from the ferroelectric nature of BaTiO3. The physical quantity corresponding to the stored electric energy per unit area is called electric displacement D; it is related to the electric field E by the following simple expression

D= ε0E +P = εε0E

Where ε and ε0 are the material’s relative permittivity and permittivity of the vacuum respectively and P is the dielectric polarization, respectively.

When the centre of the positive and negative charges in the crystal structure do not coincide naturally (without application of any external field) such crystals are said to possess a spontaneous polarization. When, upon applying an electric field, such a spontaneous polarization can be reversed then the material is called ferroelectric.

Ferroelectrics are simply defined as the “class of materials which exhibits spontaneous electric polarization (in the absence of electric field) which reversible by applying an electric field”.

The term ferro in the ferroelectrics has nothing to do with the ferrous i.e., iron content in the materials, rather this term used to explain that this class of materials exhibit spontaneous electric polarization and electric hysteresis, analogous to the ferromagnetic properties of spontaneous magnetization and magnetic hysteresis.

The ferroelectric nature of a material depends crucially on the atomic structure. Other ferroelectric materials are Lead zirconate titanate (PZT), Lead lanthanum zirconate titanate (PLZT), Lithium niobate (LiNbO3), and relaxors like Lead magnesiumniobate (PMN).

Piezoelectric

All ferroelectric materials are piezoelectric (but the reverse is not true). In Piezoelectric “Piezo” derived from the Greek word “piezein” and the meaning of piezein is squeeze or press. So its name itself explaining the effect.

Direct Piezoelectric Effect

The piezoelectric effect transforms kinetic or mechanical energy into electrical energy, due to crystal deformation. When the material is compressed or squeezed, this Mechanical stress applied to the material generates electricity.

Microphones, speakers, buzzers, pressure sensors, hydrophones, and many other sensing types of devices use direct piezoelectric effect.

The above figure shows the direct piezoelectric effect

Figure 14: Direct Piezoelectric Effect

Inverse Piezoelectric Effect 

We can reverse the same effect. Inverse piezoelectric effect – when an electric field is applied to a crystal it aligns all dipoles present in the material in order. This realignment results in deformation of crystal i.e. converting electrical energy into kinetic or mechanical energy. The reversal of the piezoelectric effect is called the inverse piezoelectric effect.

Using the inverse piezoelectric effect we can develop devices that produce acoustic sound waves. Even medical ultrasound and sonar transducers use the inverse piezoelectric effect.

The above figure shows the inverse piezoelectric effect

Figure 15: The Inverse Piezoelectric Effect

The basic equations describing the direct and converse piezoelectric effects can be expressed in terms of electric and elastic properties as follows

D = dE + εσE

S = sEσ + dE 

Where D is the dielectric displacement, σ the stress, E the electric field, S the strain, d the piezoelectric coefficient, s the material compliance constant (inverse of the elastic constant) and ε is the material’s dielectric constant (permittivity). The superscript in the equations indicates that the quantity is held constant. In the case of εσ stress is held constant which implies that the piezoelectric element is mechanically unconstrained and in the case of sE, the electric field is held constant which implies electrodes on the elements are mutually shortened.

The direct and converse piezoelectric effects can also be conveniently expressed in terms of following tensor notations:

Pi = dijkσjk 

Xij = dkijEk

In case of the direct piezoelectric effect Pi is the polarization generated along the i-axis, which is proportional to the applied stress σjk through the piezoelectric coefficients dijk. For the converse piezoelectric effect Xij is the strain generated in a particular orientation of the crystal when an electric field Ek is applied along a certain k-axis.

Examples of Piezoelectric Materials

All piezoelectric materials are non-conductive for the piezoelectric effect to occur and work. Quartz is natural and first Piezoelectric Material whereas Lead Zirconate Titanate, Barium Titanate, and Lithium Niobate are man-made Piezoelectric Materials.

Key Takeaways

• Ferroelectricity was first discovered in 1921 in a complex compound called Rochelle salt (Sodium potassium tartrate tetrahydrate).
• Ferroelectrics are simply defined as the “class of materials which exhibits spontaneous electric polarization (in the absence of electric field) which reversible by applying an electric field”.
• The ferroelectric nature of a material depends crucially on the atomic structure.
• All ferroelectric materials are piezoelectric (but the reverse is not true).
• The piezoelectric effect transforms kinetic or mechanical energy into electrical energy, due to crystal deformation. When the material is compressed or squeezed, this Mechanical stress applied to the material generates electricity.
• Inverse piezoelectric effect – when an electric field is applied to a crystal it aligns all dipoles present in the material in order.
• Quartz is natural and first Piezoelectric Material whereas Lead Zirconate Titanate, Barium Titanate, and Lithium Niobate are man-made Piezoelectric Materials.

5.9 MAGNETISATION, PERMEABILITY AND SUSCEPTIBILITY

The magnetic effects in magnetic materials are due to atomic magnetic dipoles in the materials. These dipoles result from effective current loops of electrons in atomic orbits, from effects of electron spin & from the magnetic moments of atomic nuclei.

The electric currents in an atom are caused by orbital and spin motions of electrons and those of its nucleus. Since all these motions of charged particles form closed electric currents, they are equivalent to “magnetic dipoles”. When such dipoles are subjected to an external electric field, they experience a torque which tends to align their magnetic moments in the direction of the externally applied field.

Some Important Definitions

Magnetic dipole

Each tiny dimension of a magnetic material (or) atoms in magnetic materials is called magnetic dipole. This magnetic dipole produces magnetic moment depending on the alignment with respect to the applied magnetic field.

Magnetic flux (Ф)

It is defined as the amount of magnetic lines of forces passing perpendicularly through unit area of a given material. It is denoted by ‘Ф’

Ф=AB

Where A= Area of cross section of the material in m2

B = magnetic Induction in Wb/ m2

Units: Weber (Wb)

Intensity of Magnetization (M)

When a material is magnetized, it develops a net magnetic moment. The magnetic moment per unit volume is called Intensity of magnetization.

Magnetization (M) =

Units: Amp/m

Magnetic Induction (B)

Magnetic induction at a point is defined as the force experienced by a unit North Pole Placed at that point. It is denoted by ‘B’i.e.

B =

Unit is weber / m2

Magnetizing field strength (H)

When a medium is exposed to a magnetic field of intensity ‘H’, it causes an induction ‘B’ in the medium.i.e. B H

B = μH

Where μ = absolute permeability of the medium. If the medium is air or vacuum B=μ0H

μ0=permeability of free space i.e. air or vacuum

μ0=4𝜋×10-7 H/m

Units for H: Amp /m

Permeability (μ)

It indicates with which the material allows magnetic lines of force to pass through it.

Or

It is the ability of the medium to pass magnetic lines of forces through it.

There are three Permeabilities i.e. μ1,μ0, μr

μ = μ0μr

Where μ = Absolute permeability of the medium

μ0 = Permeability of free space i.e. air or vacuum

μr = Relative permeability of the medium

Magnetic moment

Magnetic moment μm = (current) × (area of circulating orbit)

μm = (I) × (𝜋r2)

Units: Amp-m2

When the magnetic dipoles (atoms consisting of charged particles like protons & neutrons) undergo orbital motion (or) spin motion produces a magnetic moment. Since motion of charged particles is considered as closed electric current loops which in turn produces a magnetic moment.

Magnetic susceptibility (χ)

If H is the applied magnetizing field intensity and M is the amount of magnetization of the material,

Then χ =

χ = 0 in vacuum

χ = +ve for paramagnetic and Ferro magnetic materials

χ = -ve for diamagnetic materials

Units: It has no units.

5.10 CLASSIFICATION OF MAGNETIC MATERIALS

Magnetic materials are classified as follows:

• Diamagnetic
• Paramagnetic
• Ferro magnetic
• Anti-Ferro magnetic

Diamagnetic

The orbital motion of electrons around the nucleus produces a magnetic field perpendicular to the plane of the orbit. Thus each electron orbit has finite orbital magnetic dipole moment. Since the orbital planes are oriented in random manner, the vector sum of magnetic moments is zero and there is no resultant magnetic moment for each atom.

In the presence of an external magnetic field, some electrons are speeded up and some are slowed down. The electrons whose moments were anti-parallel are speeded up according to Lenz’s law and this produces an induced magnetic moment in a direction opposite to the field. The induced moment disappears as soon as the external field is removed.

When placed in a non-uniform magnetic field, the interaction between induced magnetic moment and the external field creates a force which tends to move the material from stronger part to weaker part of the external field. It means that diamagnetic material is repelled by the field.

This action is called diamagnetic action and such materials are known as diamagnetic materials. Examples: Bismuth, Copper and Water etc.

The properties of diamagnetic materials are

• Magnetic susceptibility is negative.
• Relative permeability is slightly less than unity.
• The magnetic field lines are repelled or expelled by diamagnetic materials when placed in a magnetic field.
• Susceptibility is nearly temperature independent.
• Examples: Cu, Au, Zn, H20, Bi etc. organic materials

Paramagnetic materials

In some magnetic materials, each atom or molecule has net magnetic dipole moment which is the vector sum of orbital and spin magnetic moments of electrons. Due to the random orientation of these magnetic moments, the net magnetic moment of the materials is zero.

In the presence of an external magnetic field, the torque acting on the atomic dipoles will align them in the field direction. As a result, there is net magnetic dipole moment induced in the direction of the applied field. The induced dipole moment is present as long as the external field exists.

The properties of paramagnetic materials are:

• Magnetic susceptibility is positive and small.
• Relative permeability is greater than unity.
• The magnetic field lines are attracted into the paramagnetic materials when placed in a magnetic field.
• Susceptibility is inversely proportional to temperature.Ferromagnetic materials

Ferromagnetic Material

An atom or a molecule in a ferromagnetic material possesses net magnetic dipole moment as in a paramagnetic material. A ferromagnetic material is made up of smaller regions, called ferromagnetic domain. Within each domain, the magnetic moments are spontaneously aligned in a direction. This alignment is caused by strong interaction arising from electron spin which depends on the inter-atomic distance. Each domain has net magnetisation in a direction. However the direction of magnetisation varies from domain to domain and thus net magnetisation of the specimen is zero.

In the presence of external magnetic field, two processes take place

1.     the domains having magnetic moments parallel to the field grow in size

2.     the other domains (not parallel to field) are rotated so that they are aligned with the field.

As a result of these mechanisms, there is a strong net magnetisation of the material in the direction of the applied field

When placed in a non-uniform magnetic field, the ferromagnetic materials will have a strong tendency to move from weaker to stronger part of the field. Materials which exhibit strong magnetism in the direction of applied field are called ferromagnetic materials. Examples: Iron, Nickel and Cobalt.

The properties of ferromagnetic materials are:

• Magnetic susceptibility is positive and large.
• Relative permeability is large.
• The magnetic field lines are strongly attracted into the ferromagnetic materials when placed in a magnetic field.
• Susceptibility is inversely proportional to temperature.

Antiferromagnetism

In the periodic table the only element exhibiting antiferromagnetism at room temperature is chromium. Antiferromagnetic materials are very similar to ferromagnetic materials but the exchange interaction between neighbouring atoms leads to the anti-parallel alignment of the atomic magnetic moments. Therefore, the magnetic field cancels out and the material appears to behave in the same way as a paramagnetic material. Like ferromagnetic materials these materials become paramagnetic above a transition temperature, known as the Neel temperature, TN. (Cr: TN=37ºC).

The properties of antiferromagnetic materials are:

• They have permanent magnetic dipoles
• They do not possess permanent magnetic dipole moment. Since in the absence of field they have no spontaneous magnetization due to anti parallel spin
• The relative permeability μr>1
• Susceptibility χ is small but negative
• 𝜒depends on temperature

Key Takeaways

• Magnetic materials are classified as Diamagnetic, Paramagnetic, Ferromagnetic, Anti-Ferro magnetic.
• Diamagnetic material is repelled by the field and having negative Magnetic susceptibility.
• In paramagnetic materials the magnetic field lines are attracted to the paramagnetic materials when placed in a magnetic field. Magnetic susceptibility is positive and small.
• Materials that exhibit strong magnetism in the direction of the applied field are called ferromagnetic materials. Magnetic susceptibility is positive and large.
• Antiferromagnetic materials are very similar to ferromagnetic materials but the exchange interaction between neighbouring atoms leads to the anti-parallel alignment of the atomic magnetic moments. Susceptibility χ is small but negative

5.11 FERROMAGNETISM AND FERROMAGNETIC DOMAINS

A ferromagnetic material is made up of smaller regions, called ferromagnetic domain. Within each domain, the magnetic moments are spontaneously aligned in a direction. This alignment is caused by strong interaction arising from electron spin which depends on the inter-atomic distance. Each domain has net magnetisation in a direction. However the direction of magnetisation varies from domain to domain and thus net magnetisation of the specimen is zero.

In the presence of external magnetic field, two processes take place

1.     the domains having magnetic moments parallel to the field grow in size

2.     the other domains (not parallel to field) are rotated so that they are aligned with the field.

As a result of these mechanisms, there is a strong net magnetisation of the material in the direction of the applied field

This magnetic domain theory for ferromagnetic was first proposed by Pierre-Ernest Weiss in 1906. The microscopic ordering of electron spins characteristic of ferromagnetic materials leads to the formation of regions of magnetic alignment called domains.

The magnetic moments of atoms dictate the magnetic properties of a material. In ferromagnetic materials, long-range alignments of magnetic moments, called domains, contain magnetic moments that all point in the same direction.

However, if material were to have all of its magnetic moments pointed in the same direction, this would create a very large external magnetic field. This field is not energetically minimizing as it stores large amounts of magnetostatic energy in the field.

Thus, for the system to minimize its internal energy, it must minimize the external field produced. To do this, the material creates different domains within itself to redirect the magnetic field. The regions in-between these domains are known as domain walls.

Interactions of a material's exchange interaction, magnetocrystalline anisotropy, and minimization of external magnetic field determine the domain structure of a material.

The magnetisation within the domain is saturated and will always lie in the easy direction of magnetisation when there is no externally applied field. The direction of the domain alignment across a large volume of material is more or less random and hence the magnetisation of a specimen can be zero.

Magnetic domains exist to reduce the energy of the system. A uniformly magnetized specimen as shown in figure 16(a) has large magnetostatic energy associated with it. This is the result of the presence of magnetic free poles at the surface of the specimen generating a demagnetizing field, Hd. From the convention adopted for the definition of the magnetic moment for a magnetic dipole the magnetisation within the specimen points from the south pole to the north pole, while the direction of the magnetic field points from north to south. Therefore, the demagnetizing field is in opposition to the magnetisation of the specimen. The magnitude of Hd is dependent on the geometry and magnetisation of the specimen.

In general, if the sample has a high length to diameter ratio (and is magnetized in the long axis) then the demagnetizing field and the magnetostatic energy will be low.

The break-up of the magnetisation into two domains as illustrated in figure 16(b) reduces the magnetostatic energy by half. In fact, if the magnet breaks down into N domains then the magnetostatic energy is reduced by a factor of 1/N, hence figure16(c) has a quarter of the magnetostatic energy of figure 16(a). Figure 16(d) shows a closure domain structure where the magnetostatic energy is zero, however, this is only possible for materials that do not have a strong uniaxial anisotropy, and the neighbouring domains do not have to be at 180º to each other.

Figure 16: Schematic illustration of the break-up of magnetisation into domains: (a) single domain; (b) two domains; (c) four domains; (d) closure domains.

The introduction of a domain raises the overall energy of the system, therefore the division into domains only continues while the reduction in magnetostatic energy is greater than the energy required to form the domain wall.

The energy associated with a domain wall is proportional to its area. The schematic representation of the domain wall, shown in figure 17, illustrates that the dipole moments of the atoms within the wall are not pointing in the easy direction of magnetisation and hence are in a higher energy state.

Also, the atomic dipoles within the wall are not at 180º to each other and so the exchange energy is also raised within the wall. Therefore, the domain wall energy is an intrinsic property of material depending on the degree of magnetocrystalline anisotropy and the strength of the exchange interaction between neighbouring atoms. The thickness of the wall will also vary in relation to these parameters, as a strong magnetocrystalline anisotropy will favour a narrow wall, whereas a strong exchange interaction will favour a wider wall.

Figure 17: Schematic representation of a 180o domain wall

Minimum energy can therefore be achieved with a specific number of domains within a specimen. This number of domains will depend on the size and shape of the sample (which will affect the magnetostatic energy) and the intrinsic magnetic properties of the material (which will affect the magnetostatic energy and the domain wall energy).

The main implication of the domains is that there is already a high degree of magnetization in ferromagnetic materials within individual domains, but that in the absence of external magnetic fields those domains are randomly oriented. A modest applied magnetic field can cause a larger degree of alignment of the magnetic moments with the external field, giving a large multiplication of the applied field.

These illustrations of domains are conceptual only and not meant to give an accurate scale of the size or shape of domains.

The microscopic evidence about magnetization indicates that the net magnetization of ferromagnetic materials in response to an external magnetic field may occur more by the growth of the domains parallel to the applied field at the expense of other domains rather than the reorientation of the domains themselves as implied in the sketch.

Figure 18: Domain Alignment

The internal magnetic fields which come from the long-range ordering of the electron spins are much stronger, sometimes hundreds of times stronger, than the external magnetic fields required to produce these changes in domain alignment.

Key Takeaways

• The microscopic ordering of electron spins characteristic of ferromagnetic materials leads to the formation of regions of magnetic alignment called domains.
• For the system to minimize its internal energy, it must minimize the external field produced. To do this, the material creates different domains within itself to redirect the magnetic field. The regions in-between these domains are known as domain walls.
• This number of domains will depend on the size and shape of the sample and the intrinsic magnetic properties of the material.

5.12 HYSTERESIS

Magnetic Hysteresis

The lag or delay of a magnetic material known commonly as Magnetic Hysteresis relates to the magnetization properties of a material by which it first becomes magnetized and then de-magnetized.

The set of magnetization curves, M above represents an example of the relationship between B and H for soft-iron and steel cores but every type of core material will have its own set of magnetic hysteresis curves. You may notice that the flux density increases in proportion to the field strength until it reaches a certain value where it cannot increase any more becoming almost level and constant as the field strength continues to increase.

This is because there is a limit to the amount of flux density that can be generated by the core as all the domains in the iron are perfectly aligned. Any further increase will not affect the value of M, and the point on the graph where the flux density reaches its limit is called Magnetic Saturation also known as Saturation of the Core and in our simple example above the saturation point of the steel, the curve begins at about 3000 ampere-turns per metre.

As the magnetic field strength, ( H ) increases these molecular magnets become more and more aligned until they reach perfect alignment producing maximum flux density, and an increase in the magnetic field strength due to an increase in the electrical current flowing through the coil will have little or no effect.

Retentivity

Let’s assume that we have an electromagnetic coil with a high field strength due to the current flowing through it and that the ferromagnetic core material has reached its saturation point, maximum flux density. If we now open a switch and remove the magnetizing current flowing through the coil we would expect the magnetic field around the coil to disappear as the magnetic flux reduced to zero.

However, the magnetic flux does not completely disappear as the electromagnetic core material still retains some of its magnetism even when the current has stopped flowing in the coil. This ability for a coil to retain some of its magnetism within the core after the magnetization process has stopped is called Retentivity or remanence, while the amount of flux density remaining in the core is called Residual Magnetism, BR.

The reason for this that some of the tiny molecular magnets do not return to a completely random pattern and still point in the direction of the original magnetizing field giving them a sort of “memory”. Some ferromagnetic materials have a high retentivity (magnetically hard) making them excellent for producing permanent magnets.

While other ferromagnetic materials have low retentivity (magnetically soft) making them ideal for use in electromagnets, solenoids, or relays. One way to reduce this residual flux density to zero is by reversing the direction of the current flowing through the coil, thereby making the value of H, the magnetic field strength negative. This effect is called a Coercive Force, HC.

If this reverse current is increased further the flux density will also increase in the reverse direction until the ferromagnetic core reaches saturation again but in the reverse direction from before. Reducing the magnetizing current, i once again to zero will produce a similar amount of residual magnetism but in the reverse direction.

Then by constantly changing the direction of the magnetizing current through the coil from a positive direction to a negative direction, as would be the case in an AC supply, a Magnetic Hysteresis loop of the ferromagnetic core can be produced.

Figure 19: B-H Curve

The B-H Curve or Magnetic Hysteresis loop above shows the behaviour of a ferromagnetic core graphically as the relationship between B and H is non-linear.

Starting with an unmagnetized core both B and H will be at zero, point 0 on the magnetization curve.

If the magnetization current, i is increased in a positive direction to some value the magnetic field strength H increases linearly with i and the flux density B will also increase as shown by the curve from point 0 to point a as it heads towards saturation.

Now if the magnetizing current in the coil is reduced to zero, the magnetic field circulating around the core also reduces to zero. However, the coils magnetic flux will not reach zero due to the residual magnetism present within the core and this is shown on the curve from point a to point b.

To reduce the flux density at point b to zero we need to reverse the current flowing through the coil. The magnetising force which must be applied to null the residual flux density is called a “Coercive Force”. This coercive force reverses the magnetic field re-arranging the molecular magnets until the core becomes unmagnetised at point c.

An increase in this reverse current causes the core to be magnetised in the opposite direction and increasing this magnetisation current further will cause the core to reach its saturation point but in the opposite direction, point d on the curve.

This point is symmetrical to point b. If the magnetising current is reduced again to zero the residual magnetism present in the core will be equal to the previous value but in reverse at point e.

Again reversing the magnetising current flowing through the coil this time into a positive direction will cause the magnetic flux to reach zero, point f on the curve and as before increasing the magnetisation current further in a positive direction will cause the core to reach saturation at point a.

Then the B-H curve follows the path of a-b-c-d-e-f-a as the magnetising current flowing through the coil alternates between a positive and negative value such as the cycle of an AC voltage. This path is called a B-H Curve or Magnetic Hysteresis Loop.

The effect of magnetic hysteresis shows that the magnetisation process of a ferromagnetic core and therefore the flux density depends on which part of the curve the ferromagnetic core is magnetised on as this depends upon the circuits past history giving the core a form of “memory”. Then ferromagnetic materials have memory because they remain magnetised after the external magnetic field has been removed.

However, soft ferromagnetic materials such as iron or silicon steel have very narrow magnetic hysteresis loops resulting in very small amounts of residual magnetism making them ideal for use in relays, solenoids, and transformers as they can be easily magnetised and demagnetised.

Since a coercive force must be applied to overcome this residual magnetism, work must be done in closing the hysteresis loop with the energy being used being dissipated as heat in the magnetic material. This heat is known as hysteresis loss, the amount of loss depends on the material’s value of coercive force.

By adding additives to the iron metal such as silicon, materials with a very small coercive force can be made that have a very narrow hysteresis loop. Materials with narrow hysteresis loops are easily magnetised and demagnetised and known as soft magnetic materials.

Magnetic Hysteresis Loops for Soft and Hard Materials

Figure 20: Magnetic Hysteresis Loops for Soft and Hard Materials

Magnetic Hysteresis results in the dissipation of wasted energy in the form of heat with the energy wasted being in proportion to the area of the magnetic hysteresis loop. Hysteresis losses will always be a problem in AC transformers where the current is constantly changing direction and thus the magnetic poles in the core will cause losses because they constantly reverse direction.

Rotating coils in DC machines will also incur hysteresis losses as they are alternately passing north the south magnetic poles. As said previously, the shape of the hysteresis loop depends upon the nature of the iron or steel used and in the case of iron which is subjected to massive reversals of magnetism, for example, transformer cores, the B-H hysteresis loop must be as small as possible.

Key Takeaways

• The lag or delay of a magnetic material known commonly as Magnetic Hysteresis relates to the magnetization properties of a material by which it first becomes magnetized and then de-magnetized.
• There is a limit to the amount of flux density that can be generated by the core as all the domains in the iron are perfectly aligned. Any further increase will not affect the value of M, and the point on the graph where the flux density reaches its limit is called Magnetic Saturation also known as Saturation of the Core.
• This ability for a coil to retain some of its magnetism within the core after the magnetization process has stopped is called Retentivity or remanence, while the amount of flux density remaining in the core is called Residual Magnetism, BR.
• One way to reduce this residual flux density to zero is by reversing the direction of the current flowing through the coil, thereby making the value of H, the magnetic field strength negative. This effect is called a Coercive Force, HC.
• Materials with narrow hysteresis loops are easily magnetised and demagnetised and known as soft magnetic materials.

5.13 APPLICATIONS OF MAGNETIC MATERIALS

It is difficult to imagine a world without magnetic materials and they are becoming more important in the development of modern society. The need for efficient generation and use of electricity is dependent on improved magnetic materials and designs.

Magnetic materials encompass a wide variety of materials, which are used in a diverse range of applications. Magnetic materials are utilised in the creation and distribution of electricity, and, in most cases, in the appliances that use that electricity.

They are used for the storage of data on audio and videotape as well as on computer disks.

In the world of medicine, they are used in body scanners as well as a range of applications where they are attached to or implanted into the body.

The home entertainment market relies on magnetic materials in applications such as PCs, CD players, televisions, games consoles, and loudspeakers.

Non-polluting electric vehicles will rely on efficient motors utilising advanced magnetic materials.

The telecommunications industry is always striving for faster data transmission and miniaturisation of devices, both of which require development of improved magnetic materials.

Magnetic materials are classified in terms of their magnetic properties and their uses. If a material is easily magnetised and demagnetised then it is referred to as a soft magnetic material, whereas if it is difficult to demagnetise then it is referred to as a hard (or permanent) magnetic material.

Materials in between hard and soft are almost exclusively used as recording media and have no other general term to describe them. Other classifications for types of magnetic materials are subsets of soft or hard materials, such as magnetostrictive and magnetoresistive materials.

Ferrites are ideally suited for making a device like an inductor core, circulators, and memory devices and also for various microwave applications.

The earliest ferrite was naturally occurring magnetite. The first application of ferrites was needles magnetized by the magnetite which functioned as compasses and allowed mariners to find North without the use of the stars. .However, magnetite was found to have poor magnetic properties and was not useful for magnetic applications.

The first large-scale applications for ferrites were in the television industry where large tonnages were used for the TV tube deflection yokes and the high voltage fly back transformers.

These are used in radar, satellite communications, memory, and computer applications, there has been a corresponding growth in consumer markets in radio, television, videotape recorders, and finally, the internet.

As the markets have changed, the requirements of ferrites have changed as well. From the old analog circuits to the newer digital ones, there arose the need for high-frequency switched-mode power supplies to power computers and other digital devices. Another strong market for ferrites is in the automotive industry and most recently in hybrid cars.

Reference

1. Introduction to Electrodynamics -- David J. Griffiths
2. A textbook of Engineering Physics, Dr. M. N. Avadhanulu, Dr. P.G. Kshirsagar - S.Chand
3. Engineering Mechanics, 2nd ed.- MK Harbola, Cengage Learning
4. Introduction to Solid State Physics by Charles Kittel (for Magnetic part)