Nathan Ida Third Edition - Engineering Electromagnetics

Even if no demonstrations are performed, it is recommended to read them as part of studying the chapter - the student will find some of the explanations useful. Amount of substance Themol(mol) is the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kg of carbon-12. The entities can be atoms, molecules, ions, electrons or any other particles.) The accepted number of entities (i.e. molecules) is known as Avogadro's number and is equal to approximately 6.02211023.

Table 1 The base SI units

Introduction

The discussion in this chapter starts with the definition of scalars and vectors in Cartesian coordinates.

Scalars and Vectors

Magnitude and Direction of Vectors: The Unit Vector

The magnitude of the vector isv, and this is written directly from the geometry in Figure 1.2bas. Note also that the magnitude of ^xog^y is one, since these are the unit vectors in the direction of the vector components of v, namely xandy.

Figure 1.1 The relations between vector A, the unit vector A ˆ , and the magnitude of the vector jAj

Vector Addition and Subtraction

The vector addition is also distributive, but we show this only in section 1.2.3. a) The sum of the two vectors. To calculate the differences, we add the vector components of the two vectors, observing the sign of each vector component: .

Figure 1.5 (a) Two vectors A and B and their x, y, and z components. (b) Addition of vectors A and B by adding their components

Vector Scaling

Products of Vectors

The Scalar Product

Example 1.4 Calculate the projection of a general vector onto another general vector, and the vector component in the direction of B. Figure 1.12 Definition of the dot product between vectors A and B. Solution: Using the dot product, the cosine of the angle between the vectors is evaluated by the draw.

Figure 1.12 Definition of the scalar product between vectors A and B. The smaller angle between the vectors is used

The Vector Product

In addition to the non-commutative properties of the vector product, the following properties are also observed: The formula in (b) is called the scalar equation of the plane while the formf(x,y,z)¼nC¼0 is referred to as the vector equation of the plane.

Figure 1.14 The vector product between vectors A and B

Multiple Vector and Scalar Products

Calculate the electric field intensity parallel to the surface at the center of the plate. A simpler method is to calculate the divergence of the electric field in the charge distribution.

Figure 1.18 Interpretation of the scalar triple product as a volume

Definition of Fields

Scalar Fields

An example of a time-dependent scalar field could be a weather map where temperatures vary with time. Similarly, a time-dependent electric potential distribution in a block of material is a time-dependent scalar field.

Vector Fields

Solution: For a vector field, we need to show both the magnitude of the vector and its direction. The arrow starts at the location (point) where the field appears and points in the direction of the field, and the length of the arrow indicates the size of the field.

Figure 1.21 Representation of scalar fields. (a) Values at given coordinates. (b) Contours of constant value

Systems of Coordinates

• The Cartesian Coordinate System
• The Cylindrical Coordinate System
• The Spherical Coordinate System
• Transformation from Cylindrical to Spherical Coordinates

For a point connecting vector (1) to point (2), the projection on each axis is the difference in coordinates corresponding to point (2) (head) and point (1) (tail) of the vector. (b) scalar component iA direction eBis projection iA. a). Note that the vector components in cylindrical coordinates are not constant (they depend on the angle ϕ except for the coordinates of points P1 and P2).

Figure 1.24 Direction of a surface. (a) The surface is always positive in the direction out of the volume.

Position Vectors

To re-enter the atmosphere, the speed is reduced by 1,000 km/h by firing a small rocket in the opposite direction of the satellite's motion:. What is the speed of the particle. a) Calculate the angle between the trajectories of the two objects.

Figure 1.32 Vector A and its relationship with position vectors R 1 and R 2 and end points P 1 and P 2

Introduction

There can be no language more universal and simpler, freer from error and obscurity, that is, more worthy of expressing the immutable relations of natural things [than mathematics].

Integration of Scalar and Vector Functions

Line Integrals

The closed contour integral of A is also called the circulation of A about path C. The circulation of a vector around a closed path can be zero or non-zero, depending on the vector. The size depends on the location of the field (i.e. the different vector lengths at different locations). b) FromP2toP3, the element of path isdl¼xdx^ þydy.

Figure 2.4 The four segments of the contour used for integration in Example 2.2

Surface Integrals

So the surface integral of a vector is the flux of this vector through the surface. In this case, the direction of travel is counterclockwise along the edge of the surface.

Figure 2.5 Flow through a surface. (a) Flow normal to surface s 1 . (b) Flow at an angle θ to surface s 2 and the relation between the velocity vector and the normal to the surface

Volume Integrals

Calculate the flux of vectorA through a surface defined by 0

Figure 2.10 An element of volume and the corresponding projections on the axes and planes

Differentiation of Scalar and Vector Functions

The Gradient of a Scalar Function

If you were to sail in the direction of the gradient in air pressure, you would always have the wind in your face. The unit vector is found by dividing the gradient by the magnitude of the gradient.

Figure 2.13 The relation between the scalar function U and its gradient

The Divergence of a Vector Field

The total flux from the volume is given by the integral of the closed surface of the vector A [see Eq. This indicates that there is a net flow from the volume through the closed surface.

The Divergence Theorem

The expression in parentheses is the divergence of the vectorA. The expression for the total flow through the small box becomes [see Eq. Since the total flow through the volume or through the surface that encloses the volume must be the same, we can Eq.

Circulation of a Vector and the Curl

The direction of the electric field strength is in the direction rr0, as shown in Figure 3.13. The magnitude of the electric field strength due to the elementary charge is dE¼ρvr0dr0dϕ0dz0.

Figure 2.20 Circulation. (a) Maximum circulation. (b) Zero circulation

Stokes ’ Theorem

Conservative and Nonconservative Fields

Substituting them into the expression for the electric field strength of a dipole, we get Ed 1. Calculate the electric field strength in free space at a distance [m] above an infinite plane charged with a uniform surface charge density ρs [C/m2].

Figure 3.1 (a) Line charge distribution. (b) Surface charge distribution. (c) Volume charge distribution

Null Vector Identities and Classification of Vector Fields

The Helmholtz Theorem

Second-Order Operators

The out operator as well as the gradient, divergence, and curl are first-order operators; The result are first-order partial derivatives of scalar or vector functions. 2A¼x^∇2Axþy^∇2Ayþz^∇2Az ð2:137Þ and can be obtained by directly applying the scalar Laplacian operator to each of the scalar components of the vectorA. The scalar components of the Laplacian vector are

Other Vector Identities

Cr2dl, where r2¼x2+y2, from the origin to the point P(1,3) along the line connecting the origin to P(1,3).d is the differential vector in Cartesian coordinates. a) Magnitude and direction of the pressure gradient. What is the curl of A. 2.29 Check Stokes' theorem for A¼x^ð2xyÞ y^2yz2z^2zy2 on the upper half-plane of the spherex2+y2+z2¼4 above the xyplane.

Introduction

A charge density of 1 C/m means that one coulomb of charge is distributed per each meter length of the device. A surface charge density of 1 C/m2 means that one coulomb of charge is spread over every square meter of the surface.

Charge and Charge Density

A volumetric charge density of 1 C/m3 means that one coulomb of charge is distributed over a one meter cube:. A non-uniform charge density occurs when the charge in different sections of the charge distribution depends on location.

Coulomb ’ s Law

We will expand on this shortly, but for now it is sufficient to look at the magnitude of the force. The net force is balanced by the centrifugal force due to orbit of the electron.

Figure 3.2 Relations between direction of forces and polarity of charge. (a) Two positive charges

The Electric Field Intensity

Electric Fields of Point Charges

What are the points on the axis at which the intensity of the electric field is zero (other than at infinity). Solution: The electric field intensity is the superposition of the electric field intensity of the three charges.

Figure 3.7 Electric field intensity at a general point in space, due to two point charges

Electric Fields of Charge Distributions

The intensity of the electric field at a point in space due to this elementary point charge is. In the case discussed here, we need to find the electric field intensity at the center of the plate.

Figure 3.13 Electric field intensity due to a charged line element

The Electric Flux Density and Electric Flux

Since F¼qE, the use of electric flux density does not eliminate the need to consider dielectric permittivity. After calculating the electric flux density using Eq. 3.56), the electric field strength can be calculated from Eq. a).

Applications

In large scrubbers at coal-fired power plants, scrubbers are installed in flue structures in smokestacks, and cleaning is usually accomplished by "shaking" the particles off the electrodes and physically removing them from a collection pit at the bottom of the stacks. Applications: Electrostatic Deflection - Electron Gun and Deflection in Oscilloscopes, Cathode Tube The principle of electrostatic deflection is widely used in instruments such as oscilloscopes.

Figure 3.25 Electrostatic spraying

Experiments

If you touch the ball, some of the charge from the rod will be transferred to the ball and the ball will be repelled. Since both rod and ball have the same polarity, the ball will be repelled.

Summary

Calculate the electric field intensity at P1, P2 and P3 (at the axis of the cylinder) shown in Figure 3.46. Calculate the electric field intensity anywhere in space outside the electron (ie for R>R0).

Introduction

The Electrostatic Field: Postulates

More important is the fact that we have determined the divergence of the electric field strength as one of the conditions necessary to define a vector field. In other words, can we actually say that the curvature of a static electric field is always zero.

Figure 4.1 Two charge distributions that produce identical fields outside the sphere of radius R

Gauss ’ s Law

Applications of Gauss ’ s Law

However, under two special conditions, the evaluation of the intensity of the electric field is very simple. What is the magnitude of the electric field intensity at ground level, directly below the line.

Figure 4.4 (a) Two uniformly charged spherical shells. (b) A point charge surrounded by two uniformly charged spherical shells

The Electric Potential

Electric Potential Due to Point Charges

Thus, movement in the radial direction changes the potential (adds to the integral), while movement of the test charge perpendicular to the radial direction (along a circular path of constant radius) does not change the potential. The potential at a point P(R) in space is a scalar addition of the potentials of the individual charges.

Figure 4.10 Use of position vectors to calculate the potentials at points a and b

Electric Potential Due to Distributed Charges

To calculate the potential at a pointr1outside, we integrate as in the previous case from r¼btor¼r1: Vr1,b¼Vr1Vb¼. 160 4 Gauss's law and the electric potential. To find the potential on line B, we integrate in the direction of the electric field from x¼0 tox¼(d/2) –a.

Figure 4.14 Calculation of potential due to (a) line charge distribution, (b) surface charge distribution, (c) volume charge distribution

Calculation of Electric Field Intensity from Potential

From the general expression for the potential, we can calculate the strength of the electric field with a gradient: E¼ ∇V. 4πε0R2 ½ V To calculate the electric field strength, we use the gradient in spherical coordinates and write it down.

Figure 4.19 The electric dipole and calculation of potential at point P due to the dipole

Materials in the Electric Field

• Conductors
• Dielectric Materials
• Polarization and the Polarization Vector
• Electric Flux Density and Permittivity
• Dielectric Strength

We mentioned that the tangential component of the electric field intensity at the surface of a conductor must be zero simply by the definition of a conductor. The net effect is an induced surface charge density (polarization) due to the effect of the external electric field.

Figure 4.22 The process by which tangential components of the electric field intensity at the surface of a conductor vanish

Interface Conditions

Interface Conditions Between Two Dielectrics

Interface Conditions Between Dielectrics and Conductors

Capacitance

The Parallel Plate Capacitor

Capacitance of Infinite Structures

Connection of Capacitors

Energy in the Electrostatic Field: Point and Distributed Charges

Energy in the Electrostatic Field: Field Variables

Forces in the Electrostatic Field: The Principle

Applications

Experiments

Summary

Introduction

Poisson ’ s Equation for the Electrostatic Field

Laplace’s Equation for the Electrostatic Field

Solution Methods

Uniqueness of Solution

Solution by Direct Integration

The Method of Images

Separation of Variables: Solution to Laplace ’ s Equation