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08.05.2020

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Lecture 7 Objectives

To begin our study of magnetostatics with Ampere’s law of force; magnetic flux density; Lorentz force; Biot-Savart law; applications of Ampere’s law in integral form; vector magnetic potential; magnetic dipole; and magnetic flux.

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Magnetostatics

** Magnetostatics** is the branch of electromagnetics dealing with the effects of electric charges in steady motion (i.e, steady current or DC).

The fundamental law of

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Magnetostatics (Cont’d)

In magnetostatics, the magnetic field is produced by steady currents. The magnetostatic field does not allow for

inductive coupling between circuits

coupling between electric and magnetic fields

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Ampere’s Law of Force

*Ampere’s law of force* is the “law of action” between current carrying circuits.*Ampere’s law of force* gives the magnetic force between two *current carrying circuits *in an otherwise empty universe.

Ampere’s law of force involves complete circuits since current must flow in closed loops.

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Ampere’s Law of Force (Cont’d)

The direction of the force established by the experimental facts can be mathematically represented by

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unit vector in direction of force on *I2* due to *I1*

unit vector in direction of *I2* from *I1*

unit vector in direction of current *I1*

unit vector in direction of current *I2*

Magnetic Flux Density

Ampere’s force law describes an “action at a distance” analogous to Coulomb’s law.

In Coulomb’s law, it was useful to introduce the concept of an *electric field* to describe the interaction between the charges.

In Ampere’s law, we can define an appropriate field that may be regarded as the means by which currents exert force on each other.

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Lorentz Force

If a point charge is moving in a region where both electric and magnetic fields exist, then it experiences a total force given by

The Lorentz force equation is useful for determining the equation of motion for electrons in electromagnetic deflection systems such as CRTs.

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Types of Current Distributions

*Line current density** *(current) - occurs for infinitesimally thin filamentary bodies (i.e., wires of negligible diameter).** Surface current density** (current per unit width) - occurs when body is perfectly conducting.

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Ampere’s Circuital Law in Integral Form

** Ampere’s Circuital Law** in integral form states that “the circulation of the magnetic flux density in free space is proportional to the total current through the surface bounding the path over which the circulation is computed.”

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Ampere’s Law and Gauss’s Law

Just as Gauss’s law follows from Coulomb’s law, so Ampere’s circuital law follows from Ampere’s force law.

Just as Gauss’s law can be used to derive the electrostatic field from symmetric charge distributions, so Ampere’s law can be used to derive the magnetostatic field from symmetric current distributions.

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Applications of Ampere’s Law

Ampere’s law in integral form is an *integral equation* for the unknown magnetic flux density resulting from a given current distribution.

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known

unknown

Applications of Ampere’s Law (Cont’d)

Closed form solution to Ampere’s law relies on our ability to construct a suitable family of ** Amperian paths**.

An

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Vector Magnetic Potential

Vector identity: “the divergence of the curl of any vector field is identically zero.”

Corollary: “If the divergence of a vector field is identically zero, then that vector field can be written as the curl of some vector potential field.”

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Vector Magnetic Potential (Cont’d)

In some cases, it is easier to evaluate the vector magnetic potential and then use *B** = A, *rather than to use the B-S law to directly find

In some ways, the vector magnetic potential

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Vector Magnetic Potential (Cont’d)

In classical physics, the vector magnetic potential is viewed as an auxiliary function with no physical meaning.

However, there are phenomena in quantum mechanics that suggest that the vector magnetic potential is a real (i.e., measurable) field.

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Magnetic Dipole

A ** magnetic dipole** comprises a small current carrying loop.

The point charge (

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Magnetic Dipole (Cont’d)

The magnetic dipole is analogous to the electric dipole.

Just as the electric dipole is useful in helping us to understand the behavior of dielectric materials, so the magnetic dipole is useful in helping us to understand the behavior of magnetic materials.

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Magnetic Dipole Moment

The magnetic dipole moment can be defined as

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Direction of the dipole moment is determined by the direction of current using the right-hand rule.

Magnitude of the dipole moment is the product of the current and the area of the loop.

Electromagnetic Force

The first term in the Lorentz Force Equation represents the electric force **F**e acting on a charge q within an electric field is given by.

The electromagnetic force is given by *Lorentz Force Equation (*After Dutch physicist Hendrik Antoon Lorentz (1853 – 1928))

The electric force is in the direction of the electric field.

The Lorentz force equation is quite useful in determining the paths charged particles will take as they move through electric and magnetic fields. If we also know the particle mass, m, the force is related to acceleration by the equation

Since the magnetic force is at right angles to the magnetic field, the work done by the magnetic field is given by

Magnetic Force

The magnetic force is at right angles to the magnetic field.

The magnetic force requires that the charged particle be in motion.

It should be noted that since the magnetic force acts in a direction normal to the particle velocity, the acceleration is normal to the velocity and the magnitude of the velocity vector is unaffected.

The second term in the Lorentz Force Equation represents magnetic force **F**m(N) on a moving charge q(C) is given by

where the velocity of the charge is **u** (m/sec) within a field of magnetic flux density **B **(Wb/m2). The units are confirmed by using the equivalences Wb=(V)(sec) and J=(N)(m)=(C)(V).

Magnetic Force

D3.10: At a particular instant in time, in a region of space where **E** = 0 and **B** = 3**a**y Wb/m2, a 2 kg particle of charge 1 C moves with velocity 2**a**x m/sec. What is the particle’s acceleration due to the magnetic field?

To calculate the units:

P3.33: A 10. nC charge with velocity 100. m/sec in the z direction enters a region where the electric field intensity is 800. V/m **a**x and the magnetic flux density 12.0 Wb/m2 **a**y. Determine the force vector acting on the charge.

Given:

q= 10 nC, **u** = 100 **a**z (m/sec), E = 800 **a**x V/m, B = 12.0 **a**y Wb/m2.

Given:

q= 1 nC, m = 2 kg, **u** = 2 **a**x (m/sec), E = 0, B = 3 **a**y Wb/m2.

Newtons’ Second Law

Lorentz Force Equation

Equating

Magnetic Force on a current Element

Consider a line conducting current in the presence of a magnetic field. We wish to find the resulting force on the line. We can look at a small, differential segment dQ of charge moving with velocity **u**, and can calculate the differential force on this charge from

The velocity can also be written

Therefore

Now, since dQ/dt (in C/sec) corresponds to the current I in the line, we have

(often referred to as the *motor equation*)

We can use to find the force from a collection of current elements, using the integral

segment

velocity

Magnetic Force – An infinite current Element

Let’s consider a line of current *I* in the +**a**z direction on the z-axis. For current element Id**L**a, we have

The magnetic flux density **B**1 for an infinite length line of current is

We know this element produces magnetic field, but the field cannot exert magnetic force on the element producing it. As an analogy, consider that the electric field of a point charge can exert no electric force on itself.

*What about the field from a second current element Id Lb on this line?*

From Biot-Savart’s Law, we see that the cross product in this particular case will be zero, since Id

Magnetic Force – Two current Elements

By inspection of the figure we see that ρ = y and **a** = -**a**x. Inserting this in the above equation and considering that d**L**2 = dz**a**z, we have

Now let us consider a second line of current parallel to the first.

The force d**F**12 from the magnetic field of line 1 acting on a differential section of line 2 is

The magnetic flux density **B**1 for an infinite length line of current is recalled from equation to be

Magnetic Force on a current Element

To find the total force on a length L of line 2 from the field of line 1, we must integrate d**F**12 from +L to 0. We are integrating in this direction to account for the direction of the current.

This gives us a repulsive force.

Had we instead been seeking **F**21, the magnetic force acting on line 1 from the field of line 2, we would have found **F**21 = -**F**12.

Conclusion:

1) Two parallel lines with current in opposite directions experience a force of repulsion.

2) For a pair of parallel lines with current in the same direction, a force of attraction would result.

Magnetic Force on a current Element

In the more general case where the two lines are not parallel, or not straight, we could use the Law of Biot-Savart to find **B**1 and arrive at

This equation is known as *Ampere’s Law of Force* between a pair of current carrying circuits and is analogous to Coulomb’s law of force between a pair of charges.

Magnetic Force

D3.11: A pair of parallel infinite length lines each carry current I = 2A in the same direction. Determine the magnitude of the force per unit length between the two lines if their separation distance is (a) 10 cm, (b)100 cm. Is the force repulsive or attractive? (Ans: (a) 8 mN/m, (b) 0.8 mN/m, attractive)

Case (a) y = 10 cm

Magnetic force between two current elements when current flow is in the same direction

Magnetic force per unit length

Case (a) y = 10 cm

Magnetic Materials

Material | mr | |

Diamagnetic | bismuth | 0.99983 |

Paramagnetic | air | 1.0000004 |

Ferromagnetic | cobalt | 250 |

The degree to which a material can influence the magnetic field is given by its *relative permeability*,r, analogous to *relative permittivity* r for dielectrics.

In free space (a vacuum), r = 1 and there is no effect on the field.

We know that current through a coil of wire will produce a magnetic field akin to that of a bar magnet.

We also know that we can greatly enhance the field by wrapping the wire around an iron core. The iron is considered a *magnetic material* since it can influence, in this case amplify, the magnetic field.

Relative permeabilities for a variety of materials.

In the presence of an external magnetic field, a magnetic material gets magnetized (similar to an iron core). This property is referred to as magnetization **M** defined as

where is the material’s permeability, related to free space permittivity by the factor *r*, called the *relative permeability.*

Magnetic Flux Density

Where

where *m* (“chi”) is the material’s *magnetic susceptibility*.

The total magnetic flux density inside the magnetic material including the effect of magnetization **M** in the presence of an external magnetic field **H** can be written as

Substituting

Magnetostatic Boundary Conditions

Will use Ampere’s circuital law and Gauss’s law to derive normal and tangential boundary conditions for magnetostatics.

Ampere’s circuit law:

The current enclosed by the path is

We can break up the circulation of **H** into four integrals:

Path 1

Path 3

Path 2

Path 4

Path 1:

Path 2:

Now combining our results (i.e., Path 1 + Path 2 + Path 3 + Path 4), we obtain

A more general expression for the first magnetostatic boundary condition can be written as

where **a**21 is a unit vector normal going from media 2 to media 1.

Magnetostatic Boundary Conditions

Path 3:

Path 4:

Equating

Tangential BC:

ACL:

The tangential magnetic field intensity is continuous across the boundary when the surface current density is zero.

We know that

Important Note:

Special Case: If the surface current density K = 0, we get

Magnetostatic Boundary Conditions

If K = 0

Using the above relation, we obtain

The tangential component of the magnetic flux density B is not continuous across the boundary.

Therefore, we can say that

(or)

Magnetostatic Boundary Conditions

Gauss’s Law for Magnetostatic fields:

To find the second boundary condition, we center a Gaussian pillbox across the interface as shown in Figure.

We can shrink *h* such that the flux out of the side of the pillbox is negligible. Then we have

Normal BC:

Magnetostatic Boundary Conditions

Thus, we see that the normal component of the magnetic flux density must be continuous across the boundary.

We know that

Important Note:

Using the above relation, we obtain

The normal component of the magnetic field intensity is not continuous across the boundary (but the magnetic flux density is continuous).

Therefore, we can say that

Normal BC:

Magnetostatic Boundary Conditions

__Example 3.11__: The magnetic field intensity is given as **H**1 = 6**a**x + 2**a**y + 3**a**z (A/m) in a medium with r1 = 6000 that exists for z < 0. We want to find **H**2 in a medium with r2 = 3000 for z >0.

**Step (a) and (b):** The first step is to break **H**1 into its normal component (a) and its tangential component (b). **Step (c):** With no current at the interface, the tangential component is the same on both sides of the boundary. **Step (d):** Next, we find **B**N1 by multiplying **H**N1 by the permeability in medium 1. **Step (e):** This normal component **B** is the same on both sides of the boundary. **Step (f):** Then we can find **H**N2 by dividing **B**N2 by the permeability of medium 2. **Step (g):** The last step is to sum the fields .

Magnetization and Permeability

**M** can be considered the magnetic field intensity due to the dipole moments when an external field **H** is applied

Hence, the total magnetic field intensity inside the material is **M**+**H**The magnetic field density inside the material is

But M depends on H, Define the magnetic susceptibility m . Hence

Finally define the relative permeability mr = (m +1)

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The Nature of Magnetic Materials

Materials have a different behavior in magnetic fields

Accurate description requires quantum theory

However, simple atomic model (central nucleus surrounded by electrons) is enough for us

We can also say that B tries to make the Magnetic Dipole Moment m **m** in the same direction of **B**

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Magnetic Dipole Moments in Atom

There are 3 magnetic dipole moments:

Moment due to rotation of the electrons

Moment due to the spin of the electrons

Moment due to the spin of the nucleus

The 3 rotations are 3 loop currents

The first two are much more effective

Electron spin is in pairs, in two opposite direction

Hence, a net moment due to electron spin occurs only when there is an un-filled shell (or orbit)

The combination of moment decide the magnetic characteristics of the material

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Diamagnetic Materials

Without an external magnetic field, diamagnetic materials have no net magnetic field

With an external magnetic field, they generate a small magnetic field in the *opposite* direction

The value of this opposite field depends on the external field and the diamagnetic material

Most materials are diamagnetic (with different parameters)

We will see that the relative permeability mr but 1

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Why Diamagnetic Materials, 1

Each atom has *zero* total Magnetic Dipole Moment

No torque due to external field and do not add any field__But__ if some electrons have their magnetic dipole moment with the external field

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External field will cause a small outward force on electrons, which adds to their centrifugal force

Electrons cannot leave shells to next shell (not enough energy)

Coulombs attraction force with nucleus is the same

To stay in same orbit, centrifugal force must go down. Hence, velocity reduces

The magnetic dipole moment of the atom decreases

Net magnetic dipole moment of the atom is created, ** opposite** to B

Why Diamagnetic Materials, 2

__Also__ if some electrons have their magnetic dipole moment opposite to the external field

External field will cause a small inward force on electrons, which reduces the centrifugal force

Electrons cannot leave shells to next shell (not enough energy)

Coulombs attraction force with nucleus is the same

To stay in same orbit, centrifugal force must increase

The magnetic dipole moment of the atom increases

Net magnetic dipole moment of the atom is created, ** opposite** to B

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Paramagnetic Materials

Atoms have a *small* net magnetic dipole moment

The random orientation of atoms make the average dipole moment in the material zero

Without an external field, there is no magnetic property

When an external magnetic field is applied there is a small torque on atoms and they become aligned with the field

Hence, inside the material, atoms add their own field to the external field

The diamagnetism due to orbiting electrons is also acting

If the net effect is an increase in the field B, the material is called paramagnetic

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Ferromagnetic Materials

Atoms have large dipole moment, they affect each other

Interaction among the atoms causes their magnetic dipole moments to align within regions, called *domains*Each domain have a strong magnetic dipole moment

A ferromagnetic material that was never magnetized before will have magnetic dipole moments in many directions

The average effect is cancellation. The net effect is zero

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Ferromagnetic Materials

When an external magnetic field B is applied the domains with magnetic dipole moment in the same direction of B increase their size at the expense of other domains

The internal magnetic field increases significantly

When the external magnetic field is removed a residual magnetic dipole moment stay, causing the permanent magnet

The only ferromagnetic material at room temperature are Iron, Nickel and Cobalt

They loose ferromagnetism at temperature > Curie temperature (which is 770o C for iron)

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Antiferromagnetic Materials

Atoms have a net dipole moment

However, the material is such that atoms dipole moments line-up in opposite direction

Net dipole moment is zero

No much difference when an external magnetic field is present

Phenomena occurs at temperature well below room temperature

No engineering importance at present time

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Ferrimagnetic Materials

Similar to antiferromagnetic materials, atoms dipole moments line-up in opposite direction

However, the dipole moments are not equal. Hence, there is a net dipole moment

Ferrimagnetic materials behave like ferrormagnetic materials, but the magnetic field increase is not as large

Effect disappear above Curie temperature

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Ferrimagnetic Materials

The main advantage is that they have high resistance. Hence can be used as the core of transformers, specially at high frequency

Also used in loop antennas in AM radios

In this case the losses Eddy current are much smaller than iron core

Example material: Iron Oxide (Fe3O4) and Nickel Ferrite (NiFe2O4)

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Magnetization and Permeability

Now let us discuss the magnetic effect of magnetic material in a quantitative manner

Let us call the current inside the material due to electron orbit, electron spin and atom spin by the bound current Ib

The material includes many dipole moments **m** (units A m2) that add-up

Define the **Magnetization M** as the magnetic dipole moment per unit volume**M** has a unit A/m (which is similar to the units of **H**)

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