EEE 403 Lecture 7: Magnetostatics: Ampere’s Law Of Force; Magnetic Flux Density; Lorentz Force; Biot-savart Law; Applications Of Ampere’s Law In Integral.

EEE 403 Lecture 7: Magnetostatics: 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; Magnetic Flux 1

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. 2

Overview of Electromagnetics
Fundamental laws of classical electromagnetics Maxwell’s equations Special cases Electro-statics Magneto-statics Geometric Optics Electro-magnetic waves Statics: Transmission Line Theory Input from other disciplines Circuit Theory Kirchoff’s Laws 3

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 magnetostatics is Ampere’s law of force. Ampere’s law of force is analogous to Coulomb’s law in electrostatics. 4

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 5

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. 6

Ampere’s Law of Force (Cont’d)
Experimental facts: Two parallel wires carrying current in the same direction attract. Two parallel wires carrying current in the opposite directions repel.  I1 I2 F12 F21   I1 I2 F12 F21 7

Experimental facts: A short current-carrying wire oriented perpendicular to a long current-carrying wire experiences no force. F12 = 0  I2 I1 8

Experimental facts: The magnitude of the force is inversely proportional to the distance squared. The magnitude of the force is proportional to the product of the currents carried by the two wires. 9

The direction of the force established by the experimental facts can be mathematically represented by unit vector in direction of current I1 unit vector in direction of current I2 unit vector in direction of force on I2 due to I1 unit vector in direction of I2 from I1 10

The force acting on a current element I2 dl2 by a current element I1 dl1 is given by Permeability of free space m0 = 4p  10-7 F/m 11

The total force acting on a circuit C2 having a current I2 by a circuit C1 having current I1 is given by 12

The force on C1 due to C2 is equal in magnitude but opposite in direction to the force on C2 due to C1. 13

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. 14

Magnetic Flux Density (Cont’d)
The magnetic flux density can be introduced by writing 15

where the magnetic flux density at the location of dl2 due to the current I1 in C1 16

Suppose that an infinitesimal current element Idl is immersed in a region of magnetic flux density B. The current element experiences a force dF given by 17

The total force exerted on a circuit C carrying current I that is immersed in a magnetic flux density B is given by 18

Force on a Moving Charge
A moving point charge placed in a magnetic field experiences a force given by The force experienced by the point charge is in the direction into the paper. B v Q 19

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. 20

The Biot-Savart Law The Biot-Savart law gives us the B-field arising at a specified point P from a given current distribution. It is a fundamental law of magnetostatics. 21

The Biot-Savart Law (Cont’d)
The contribution to the B-field at a point P from a differential current element Idl’ is given by 22

23

The total magnetic flux at the point P due to the entire circuit C is given by 24

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. Volume current density (current per unit cross sectional area) - most general. 25

For a surface distribution of current, the B-S law becomes For a volume distribution of current, the B-S law becomes 26

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.” 27

Ampere’s Circuital Law in Integral Form (Cont’d)
By convention, dS is taken to be in the direction defined by the right-hand rule applied to dl. S dl dS Since volume current density is the most general, we can write Iencl in this way. 28

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. 29

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. known unknown 30

Applications of Ampere’s Law (Cont’d)
In general, solutions to integral equations must be obtained using numerical techniques. However, for certain symmetric current distributions closed form solutions to Ampere’s law can be obtained. 31

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 Amperian path is a closed contour to which the magnetic flux density is tangential and over which equal to a constant value. 32

Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law
Consider an infinite line current along the z-axis carrying current in the +z-direction: I 33

Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d)
(1) Assume from symmetry and the right-hand rule the form of the field (2) Construct a family of Amperian paths circles of radius r where 34

(3) Evaluate the total current passing through the surface bounded by the Amperian path 35

Amperian path 36

(4) For each Amperian path, evaluate the integral
Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d) (4) For each Amperian path, evaluate the integral length of Amperian path. magnitude of B on Amperian path. 37

(5) Solve for B on each Amperian path
Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d) (5) Solve for B on each Amperian path 38

Applying Stokes’s Theorem to Ampere’s Law
 Because the above must hold for any surface S, we must have Differential form of Ampere’s Law 39

Ampere’s Law in Differential Form
Ampere’s law in differential form implies that the B-field is conservative outside of regions where current is flowing. 40

Fundamental Postulates of Magnetostatics
Ampere’s law in differential form No isolated magnetic charges B is solenoidal 41

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.” 42

Vector Magnetic Potential (Cont’d)
Since the magnetic flux density is solenoidal, it can be written as the curl of a vector field called the vector magnetic potential. 43

The general form of the B-S law is Note that 44

Furthermore, note that the del operator operates only on the unprimed coordinates so that 45

Hence, we have 46

For a surface distribution of current, the vector magnetic potential is given by For a line current, the vector magnetic potential is given by 47

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 B. In some ways, the vector magnetic potential A is analogous to the scalar electric potential V. 48

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. 49

Magnetic Dipole A magnetic dipole comprises a small current carrying loop. The point charge (charge monopole) is the simplest source of electrostatic field. The magnetic dipole is the simplest source of magnetostatic field. There is no such thing as a magnetic monopole (at least as far as classical physics is concerned). 50

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. 51

Consider a small circular loop of radius b carrying a steady current I. Assume that the wire radius has a negligible cross-section. x y b 52

The vector magnetic potential is evaluated for R >> b as 53

The magnetic flux density is evaluated for R >> b as 54

Recall electric dipole The electric field due to the electric charge dipole and the magnetic field due to the magnetic dipole are dual quantities. 55

Magnetic Dipole Moment
The magnetic dipole moment can be defined as Magnitude of the dipole moment is the product of the current and the area of the loop. Direction of the dipole moment is determined by the direction of current using the right-hand rule. 56

Magnetic Dipole Moment (Cont’d)
We can write the vector magnetic potential in terms of the magnetic dipole moment as We can write the B field in terms of the magnetic dipole moment as 57

Divergence of B-Field The B-field is solenoidal, i.e. the divergence of the B-field is identically equal to zero: Physically, this means that magnetic charges (monopoles) do not exist. A magnetic charge can be viewed as an isolated magnetic pole. 58

Divergence of B-Field (Cont’d)
No matter how small the magnetic is divided, it always has a north pole and a south pole. The elementary source of magnetic field is a magnetic dipole. N S I N S 59

Magnetic Flux The magnetic flux crossing an open surface S is given by
Wb Wb/m2 60

Magnetic Flux (Cont’d)
From the divergence theorem, we have Hence, the net magnetic flux leaving any closed surface is zero. This is another manifestation of the fact that there are no magnetic charges. 61

Magnetic Flux and Vector Magnetic Potential
The magnetic flux across an open surface may be evaluated in terms of the vector magnetic potential using Stokes’s theorem: 62

Electromagnetic Force
The electromagnetic force is given by Lorentz Force Equation (After Dutch physicist Hendrik Antoon Lorentz (1853 – 1928)) 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 The first term in the Lorentz Force Equation represents the electric force Fe acting on a charge q within an electric field is given by. The electric force is in the direction of the electric field.

Magnetic Force The second term in the Lorentz Force Equation represents magnetic force Fm(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). 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. Since the magnetic force is at right angles to the magnetic field, the work done by the magnetic field is given by

Magnetic Force D3.10: At a particular instant in time, in a region of space where E = 0 and B = 3ay Wb/m2, a 2 kg particle of charge 1 C moves with velocity 2ax m/sec. What is the particle’s acceleration due to the magnetic field? Given: q= 1 nC, m = 2 kg, u = 2 ax (m/sec), E = 0, B = 3 ay Wb/m2. Newtons’ Second Law Lorentz Force Equation Equating 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 ax and the magnetic flux density 12.0 Wb/m2 ay. Determine the force vector acting on the charge. Given: q= 10 nC, u = 100 az (m/sec), E = 800 ax V/m, B = 12.0 ay Wb/m2.

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 velocity The velocity can also be written segment 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

Magnetic Force – An infinite current Element
Let’s consider a line of current I in the +az direction on the z-axis. For current element IdLa, we have The magnetic flux density B1 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 IdLb on this line? From Biot-Savart’s Law, we see that the cross product in this particular case will be zero, since IdL and aR will be in the same direction. So, we can say that a straight line of current exerts no magnetic force on itself.

Magnetic Force – Two current Elements
Now let us consider a second line of current parallel to the first. The force dF12 from the magnetic field of line 1 acting on a differential section of line 2 is a = -ax ρ = y The magnetic flux density B1 for an infinite length line of current is recalled from equation to be By inspection of the figure we see that ρ = y and a = -ax. Inserting this in the above equation and considering that dL2 = dzaz, we have

To find the total force on a length L of line 2 from the field of line 1, we must integrate dF12 from +L to 0. We are integrating in this direction to account for the direction of the current. a = -ax ρ = y This gives us a repulsive force. Had we instead been seeking F21, the magnetic force acting on line 1 from the field of line 2, we would have found F21 = -F12. 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.

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 B1 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) Magnetic force between two current elements when current flow is in the same direction Magnetic force per unit length Case (a) y = 10 cm Case (a) y = 10 cm

Magnetic Materials and Boundary Conditions

Relative permeabilities for a variety of materials.
Magnetic Materials 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. Material mr Diamagnetic bismuth gold silver copper water Paramagnetic air aluminum platinum 1.0003 Ferromagnetic (nonlinear) cobalt nickel iron (99.8% pure) iron (99.96% pure) Mo/Ni superalloy 250 600 5000 280,000 1,000,000 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.

Magnetic Flux Density 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 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 Where where  is the material’s permeability, related to free space permittivity by the factor r, called the relative permeability.

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: Path 1 Path 4 Path 2 The current enclosed by the path is Path 3 We can break up the circulation of H into four integrals: Path 1: Path 2:

Path 3: Path 4: Now combining our results (i.e., Path 1 + Path 2 + Path 3 + Path 4), we obtain ACL: Equating Tangential BC: A more general expression for the first magnetostatic boundary condition can be written as where a21 is a unit vector normal going from media 2 to media 1.

Special Case: If the surface current density K = 0, we get If K = 0 The tangential magnetic field intensity is continuous across the boundary when the surface current density is zero. Important Note: We know that (or) Using the above relation, we obtain Therefore, we can say that The tangential component of the magnetic flux density B is not continuous across the boundary.

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:

Normal BC: Thus, we see that the normal component of the magnetic flux density must be continuous across the boundary. Important Note: We know that Using the above relation, we obtain Therefore, we can say that The normal component of the magnetic field intensity is not continuous across the boundary (but the magnetic flux density is continuous).

Example 3.11: The magnetic field intensity is given as H1 = 6ax + 2ay + 3az (A/m) in a medium with r1 = 6000 that exists for z < 0. We want to find H2 in a medium with r2 = 3000 for z >0. Step (a) and (b): The first step is to break H1 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 BN1 by multiplying HN1 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 HN2 by dividing BN2 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 B=mo(M+H) But M depends on H, Define the magnetic susceptibility m . Hence M= m H Hence, we get B=mo(m H +H) = B=moH (m +1) Finally define the relative permeability mr = (m +1) B=mo mr H= m H

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

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

Type of Magnetic Materials
We will study 6 types: Diamagnetic Paramagnetic Ferromagnetic Anti-ferromagnetic Ferrimagnetic Super paramagnetic

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

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 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

Diamagnetic Materials
Note that mr = 1 + susceptibility Material Susceptibility x 10-5 Bismuth -16.6 Carbon (diamond) -2.1 Carbon (graphite) -1.6 Copper -1.0 Superconductor -105

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

Paramagnetic Materials
Note that mr = 1 + susceptibility Material Susceptibility x 10-5 Iron oxide (FeO) 720 Iron amonium alum 66 Uranium 40 Platinum 26 Tungsten 6.8

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

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)

Curie Temperatures: Iron: 770oC Nickel: 354o C Cobalt: 1115o C Medium Relative Permeability μr Mu Metal 20,000 Permalloy 8000 Electrical Steel 4000 Ferrite (nickel zinc) 16-640 Ferrite (manganese zinc) >640 Steel 100 Nickel

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

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

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)

Super Paramagnetic Materials
Composed of Ferromagnetic particles inside non-ferromagnetic materials Domains occur but can not expand when exposed to external field Used in magnetic tapes for audio and video tape recording

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)

EEE 403 Lecture 7: Magnetostatics: Ampere’s Law Of Force; Magnetic Flux Density; Lorentz Force; Biot-savart Law; Applications Of Ampere’s Law In Integral.

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EEE 403 Lecture 7: Magnetostatics: Ampere’s Law Of Force; Magnetic Flux Density; Lorentz Force; Biot-savart Law; Applications Of Ampere’s Law In Integral.

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