Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) PDF Download

Magnetized Material

In dealing with electrostatics, we had seen that the electrical polarization of the medium itself influences the electrical behavior of substances. In the same way we have internal currents within a material because there are moving charges in atoms and molecules which can create their own magnetic field.

There are two types effects here, the first is due to the orbital motion of the electrons and the second is due to the intrinsic spin magnetic moment of the electrons. The second effect is purely quantum mechanical in origin and its discussion is beyond the scope of this course. Like in the electrostatic case, the magnetic moment could be intrinsic to an atom or molecule or it could be induced by an externally applied magnetic field.

We will see later that in terms of theor magnetic behavior, substances fall broadly into three classes, viz. diamagnets, paramagnets and ferromagnets. Diamagnetism arises because of electromagnetic induction effect where the atomic currents oppose any change of flux through the atoms by an external magnetic field. We will discuss this effect later in more detail. The total current inside a material consists of two parts, the first is the conduction current, which is due to bodily motion of the free charges in a conductor and the second is due to atomic currents. The latter are too tiny to be taken care of individually and we consider such effects in an average sense.

Let  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) be the magnetic moment of the i-th atom inside a matter. We define magnetization as the net magnetic moment per unit volume 

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The quantity is very similar to the polarization in dielectric material. We will calculate the effect due to the magnetization of the material by calculating the vector potential corresponding to the microscopic currents. We had seen earlier that the magnetic vector potential due to a magnetic moment is given by the expression

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

where  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is the position vector of the point of observation with respect to the position of the magnetic moment.

Using this we can write down the expression for the vector potential at a positionMagnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)due to magnetic moments I n a magnetized material having a magnetization  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where, as before,

we have used the primed quantities to indicate the variable to be integrated

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

As we did in the electrostatic case, we can convert this into two integrals, one over the volume and the other over the surface of the material, We can rewrite the vector potential as

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

We now use the vector identity for the curl of a product of a scalar with a vector,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

using which we can write,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The first term can be converted to a surface integral in a manner very similar to the way we converted volume integral of a divergence to a surface integral,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Thus  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) takes the role of a surface current. We now identify, as we did in the electrostatic case, a bound volume current and a bound surface current, defined by

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The origins of these currents can be understood by realizing that there are large number of internal currents within the material which may not cancel within the volume. The surface term is understood similar to the way the polarization charge was introduced as even when internal currents cancel out, the currents at the boundary do not have neighbouring loops to cancel. Let us calculate the magnetic field due to magnetization of the material.

Let us start with the expression for the vector potential. If the gradient is taken with respect to the point of observation, we can write,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

We use the identity,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

where A and B are arbitrary vector fields not to be confused with the vector potential and the field respectively.

Using this identity, we have,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Since the  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) operators are with respect to the unprimed index,  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is not differentiated with respect to it.

Using the fact that  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) we can perform the first integration and are left with,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The last term is expressed in terms of gradient of a vector which is to be understood, as explained earlier as gradient of three components of the vector on which it acts. Using the vector identity stated above, and, using the fact that the operatorMagnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) does not act on vectors or scalars which are functions of primed indices, we can write the second term of the above equation as

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

the last line follows because the curl in the previous expression is zero. Thus we get, for the magnetic field,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Since curl of a gradient is zero, we get,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

If we add the conduction current contribution to the above, we get Ampere’s law for the combined case to be

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The integral form of the Ampere’s law follows,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

where the magnetization current is defined in terms of the magnetization current density,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

We can rewrite Ampere’s law as

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

which now includes the contribution due to the magnetic medium. We now define, a new vector

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

which, incidentally, is also known as the magnetic field. There is a lot of confusion in literature because of this. The B field that we have been using so far, has been called the “magnetic field of induction” or the “magnetic flux density” in literature. However, whenever there is no confusion we will call each of these fields as the magnetic field and the context should make it clear which field we are referring to. When both are used in the same discussion, we will simply call them as the B field or the H field. The line integral of the H- field gives

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The situation is very similar to the use of E and D in electrostatics. The H field tries to artificially separate the contribution due to the applied magnetic field and is thought to arise from the conduction current alone. This is artificial because there is no physical way in which one can achieve this separation and what an experiment would measure is the B field which includes both the contributions. Note that there is a dimensional difference between H and B, the former is measured in units of A/m.

Using Stoke’s theorem, we can write the curl of H as

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

This enables us to relate H to the scalar magnetic potential by

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

If  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) we can write  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) For small applied field the system has a linear response to the field, i.e.,  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) so that we can write  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) , the proportionality factor Xis known as the magnetic susceptibility of the material. As we stated earlier, there are three basic type of magnetic material. There are substances which are known as “paramagnetic” for which the magnetic susceptibility is positive while for “diamagnetic” material , it is negative. There is a third class of material known as “ferromagnets” which show hysteresis behavior. For substances which show linear response, we can write,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

where 1+Xis known as “relative permeability” .

Boundary conditions on B and H fields are obtained by using tricks similar to that used earlier. Gauss’s law  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) leads to the normal component of magnetic field of induction being continuous at an interface and the Ampere’s law for the H field,  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) leads to a discontinuity in the tangential component of the H field at a surface having a surface current,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Example : A Uniformly Magnetized Sphere

We will use the scalar potential concept to discuss this case. Recall that the scalar magnetic potential for a magnetic moment  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is given by

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

This is very similar to the corresponding electrostatic expression for the scalar potential due to an electric dipole moment,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Thus, following the way we obtained an expression for the scalar potential for a dielectric material and separated the contribution into two parts, a volume part and a surface part, we can write,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

(This is like defining a “magnetic charge density” with a volume component  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and a surface component  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) In reality, magnetic charges do not exist but it helps drawing parallel to the electric case.)

In this case, since the magnetization is uniform, the volume component vanishes and we are left with

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The normal to the surface being in the radial direction, if we take the direction of magnetization along the z direction,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

we have,  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) We will expand  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) in spherical harmonics

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Where the primes quantities refer to coordinates of  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and the unprimed quantities to coordinates of  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Note that  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The surface integral is essentially an integration over the solid angle which picks up, because of orthogonality condition on spherical harmonics  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) term only, leaving us with

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Inside the material,  r > = R, r<= r  which gives for the magnetic potential inside,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Outside the sphere, r>= r, r<=rR and we get,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Using these, we can get expressions for the magnetic field both inside and outside. Inside, since the scalar potential only depends on z,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

which gives Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Outside the sphere, we use spherical polar coordinates and since there is no azimuthal dependence of the potential, we have,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Which is a familiar expression for the magnetic field outside.

The field inside is uniform while the field outside must form closed loops.

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Tutorial Assignment

  1. A long cylindrical rod of radius R has a uniform magnetization  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) parallel to its axis. There are no free currents. Find the magnetic field both inside and outside the cylinder due to magnetization.
  2. A cylindrical wire, made of a linear magnetic material of permeability μ has a current | distributed uniformly over its cross section of radius R. Calculate all the bound current densities and determine the total current

Solutions to Tutorial Assignments

1. Since the magnetization is uniform, there are no bound volume currents. There will however be bound surface currents, given by

 Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

 Since the current is in the (azimuthal) direction of  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) the problem is similar to that of a solenoid. The field outside the cylinder is zero but the field inside is along the axis and is given by  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

2. As the free current is given, we can use Ampere’s law to determine the H field, which is given by 

 Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Since the material is linear we can write the corresponding B field by multiplying the field inside by μ and the outside field by μ0 i.e.

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Thus the magnetization vector  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is non zero only inside the material and is given by

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The volume bound current density is

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Since the magnitudes of the bound current densities are constant, the total bound current from volume is obtained by multiplying the current density with πR2 and the surface bound current is obtained by multiplying with the circumference 2πR. These two contributions being equal and opposite, the net bound current is zero.

Self Assessment Questions

1. A long cylindrical rod of radius R has a magnetization  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) in the azimuthal direction. There are no free currents. Find the magnetic field both inside and outside the cylinder due to magnetization.

2. In the preceding problem, take the magnetization to be given by  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where r is the distance from the cylindrical axis. Find the magnetic field both inside and outside the cylinder due to magnetization

3. Repeat Problem 2 by calculating the H field and adding the magnetization contribution.

4. A magnetized material is in the shape of a cube of side a as shown . The magnetization is directed along the z direction and varies linearly from its value zero on its surface on the y-

z plane to a value  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) on the plane x= α Calculate the bound currents both on the surface and inside the material.

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Solutions to Self Assessment Questions

1. The bound volume current is zero since the magnetization has constant magnitude. The surface current density is  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The magnetic field inside is zero as there is no volume current inside. The current outside is given by

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

2. Bound current densities are as follows  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

By symmetry the fields at The total current enclosed within a length L of the cylinder is  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus the magnetic field outside is zero.

Since the current is in azimuthal direction, in order to calculate the magnetic field, we need to take an Amperian loop which is in the form of a rectangle of height L whose one side is at a distance r from the axis and the other parallel side is outside. Since the field outside is zero, we have,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

which gives

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

3. Since there are no free currents, H=0. Thus  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Since the magnetization is zero outside, the field is zero. Inside, the field is equal to  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

4. Since the magnetization varies linearly with x, the magnetization is given by  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The bound current density in the volume is  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The surface bound currents are as follows. On the face x=0, there is no bound current because M=0 on that face. On the face  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) the normal direction is î so that the bound current density is  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) On the top face, the normal direction is  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and the magnetization is  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) so that the bound current density is zero. We have a contribution from the y=α face for which the normal is directed along J and the bound current density is  Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) the current on the opposite face is oppositely directed.

The document Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Electromagnetic Fields Theory (EMFT).
All you need of Electrical Engineering (EE) at this link: Electrical Engineering (EE)
11 videos|45 docs|73 tests

Up next

FAQs on Magnetized Material - Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

1. What is magnetized material in electronics and communication engineering?
Ans. Magnetized material refers to a material that has been subjected to a magnetic field, resulting in the alignment of its atomic or molecular magnetic moments in a preferred direction. In electronics and communication engineering, magnetized materials are utilized in various devices such as magnetic sensors, inductors, transformers, and magnetic storage devices like hard drives.
2. How is magnetized material used in electronics and communication engineering?
Ans. Magnetized materials play a crucial role in electronics and communication engineering. They are used in the construction of inductors and transformers, which are essential components in electronic circuits for energy storage and signal processing. Magnetized materials are also utilized in magnetic sensors that detect and measure magnetic fields, enabling the development of compasses, position sensors, and non-contact switches.
3. What are the advantages of using magnetized materials in electronics and communication engineering?
Ans. There are several advantages to using magnetized materials in electronics and communication engineering. Firstly, they provide a means of storing and transferring energy efficiently in devices such as inductors and transformers. Secondly, magnetized materials enable the development of sensitive magnetic sensors that can detect and measure magnetic fields accurately. Lastly, they are essential in the construction of magnetic storage devices like hard drives, allowing for high-capacity data storage.
4. Can magnetized materials be demagnetized?
Ans. Yes, magnetized materials can be demagnetized. The process of demagnetization involves subjecting the material to a demagnetizing field, which disrupts the alignment of the atomic or molecular magnetic moments. This can be achieved by applying an alternating magnetic field or gradually reducing the external magnetic field to zero. Demagnetization is commonly used in applications where it is necessary to remove or reset the magnetic properties of a material.
5. What are some common applications of magnetized materials in electronics and communication engineering?
Ans. Magnetized materials find numerous applications in electronics and communication engineering. Some common applications include magnetic sensors used in compasses, position sensors, and non-contact switches. They are also utilized in the construction of inductors and transformers, which are vital components in various electronic circuits. Furthermore, magnetized materials are crucial in magnetic storage devices like hard drives, enabling high-capacity data storage and retrieval.
11 videos|45 docs|73 tests
Download as PDF

Up next

Explore Courses for Electrical Engineering (EE) exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

MCQs

,

Important questions

,

Exam

,

Objective type Questions

,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

,

video lectures

,

Viva Questions

,

practice quizzes

,

study material

,

mock tests for examination

,

shortcuts and tricks

,

pdf

,

ppt

,

Semester Notes

,

Sample Paper

,

past year papers

,

Previous Year Questions with Solutions

,

Summary

,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

,

Extra Questions

,

Magnetized Material | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

,

Free

;