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Electromags - Chapter 1

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Electronics Engineering (CR 061)

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LESSON CONTENT

According to Hayt & Buck, Vector analysis is a mathematical shorthand. It has some new symbols, some new rules, and a pitfall here and there like most new fields, and it demands concentration, attention, and practice.

 Why do we need to study Vector Analysis? - According to Yoonchang Jeong, a deficiency in vector analysis in the study of electromagnetics is similar to a deficiency in algebra and calculus in the study of physics. (ocw.snu.ac/sites/default/files/NOTE/02%20Vector%20Analysis.pdf) - Vector analysis is a mathematical tool used to explain and predict physical phenomena in the study of mechanics. A vector is a depiction or symbol showing movement or a force carried from point A to point B. A vector has properties of both magnitude and direction. A scalar only has the property of magnitude. (Strain, 2012) - It is easier for engineers to take a more rigorous and complete course in the mathematics department after they have been presented with a few physical pictures and applications. (Hayt & Buck, 2001).

WATCH: youtube/watch?v=MKKytaEmCGA (Introduction Vector Analysis) youtube/watch?v=rcbdjQb-yyE (Vector Analysis: Introduction to Vector Analysis)

Scalars and Vectors Many of the quantities that physicists use are easy to specify; measurements of mass, length, time, area, volume and temperature can all be expressed as simple numbers together with appropriate units of measurement. Such quantities are known as scalar quantities or scalar. However, some quantities are more tricky to deal with. For instance, if you want to travel from one place to another you will not only want to know how far apart the two places are, you will also need to know the direction that leads from one to other. The physical quantity that combines distance and direction is called displacement and is clearly more complicated than distance alone. Displacement is a simple example of a large class of physical quantities known collectively as vector quantities or vectors. (University, 2004)

SOURCE: sciencestruck

Scalars The term scalar refers to a quantity whose value may be represented by a single (positive or negative) real number. The x, y, and z we used in basic algebra are scalars and the quantities they represent are scalars. If we speak of a body falling a distance L in a time t, or the temperature T at any point in a bowl of soup whose coordinates are x, y, and z, then L, t, T, x, y, and z are all scalars. Scalars are printed in italic type, for example A. (Hayt & Buck, 2001)

Scalar Quantities: The physical quantities which are specified with the magnitude or size alone are scalar quantities.

SOURCE: grc.nasa

NOTE: Voltage is also a scalar quantity, although the complex representation of a sinusoidal voltage, an artificial procedure, produces a complex scalar, or phasor, which requires two real numbers for its representation, such as amplitude and phase angle, or real part and imaginary part. (Hayt & Buck, 2001)

Vectors The word “vector” comes from the Latin word vectus (or vehere – meaning to carry). A vector is a depiction or symbol showing movement or a force carried from point A to point B. Vectors play an important role in physics (specifically in kinematics) when discussing velocity and acceleration. A velocity vector contains a scalar (speed) and a given direction. Acceleration, also a vector, is the rate of change of velocity. (Strain, 2012)

A vector quantity has both a magnitude and a direction in space. We shall be concerned with two- and three-dimensional spaces only, but vectors may be defined in n-dimensional space in more advanced applications. Force, velocity, acceleration, and a straight line from the positive to the negative terminal of a storage battery are examples of vectors. In this book, as in most others using vector notation, vectors will be indicated by boldface type, for example, A. When writing longhand or using a typewriter, it is customary to draw a line or an arrow over a vector quantity to show its vector character. (Hayt & Buck, 2001)

Vector quantity is characterized by both a magnitude and a direction.

and start at the origin b.) Draw the 2nd vector and again, start at the origin. c.) Make a parallelogram e.) Draw diagonal from the origin

addition, for we may always express A-B as A+(-B) ; the sign and direction of the second vector are reversed, and this vector is then added to the first by the rule for vector addition. (Hayt & Buck, 2001).

Vector Addition - Parallelogram Method a.) Draw the 1st vector

SOURCE: Microsoft Word - PS_OV_Vector Tail to Tip Method

Subtraction of Vectors (Ibrahim, 2011)

WATCH: (4) youtube/watch?v=VgqsM-XdBD 0 (Adding and subtracting vectors) (5) youtube/watch?v=-maGrJzBW 0 M (Subtracting two vectors algebraically and graphically (6) youtube/watch?v=_mKUe0DJqzs (Parallelogram Law of Vector Addition | Don't Memorise) (7) youtube/watch?v=_LrDgGhCTJU (Physics - Mechanics: Vectors (6 of 21) Adding Vectors Graphically - Tip-To-Toe Method)

Vector Multiplication and Vector Division by a Scalar According to Hayt & Buck, vectors may be multiplied by scalars. The magnitude of the vector changes, but its direction does not when the scalar is positive, although it reverses direction when multiplied by a negative scalar. Multiplication of a vector by a scalar also obeys the associative and distributive laws of algebra, leading to

(r+s)(A+B) = r(A+B) + s(A+B)= rA +rB + sA + sB

EXAMPLE: (Vector Addition, Vector Subtraction and Vector Multiplication by a scalar) 1) 3(Ā) Ā= 3(Ā)= or 3(Ā)=

-2(Ā)= or
Let a = (-2, 4), b = (1, 5), and c = (2, -5). Find the component form of the vector a) a+b =(-2 + 1)ax + (4+5)ay = -ax + 9ay or (-1,9) b) a-c =(-2-(2))ax + (4-(-5))ay = -4ax + 9ay or (-4,9)

c) 4a + 3c

a= 4[ − ]=(-8,16)

In the cartesian coordinate system we set up three coordinate axes mutually at right angles to each other, and call them the x, y, and z axes. It is customary to choose a, right-handed coordinate system, in which a rotation (through the smaller angle) of the x axis into the y axis would cause a righthanded screw to progress in the direction of the z axis. Using the right hand, the thumb, fore- finger, and middle finger may then be identified, respectively, as the x, y, and z axes. Figure 1 shows a right-handed cartesian coordinate system. A point is located by giving its x, y, and z coordinates. These are, respectively, the distances from the origin to the intersection of a perpendicular dropped from the point to the x, y, and z axes. An alternative method of interpreting coordinate values, and a method corresponding to that which must be used in all other coordinate systems, is to consider the point as being at the common intersection of three surfaces, the planes x = constant, y = constant, and z = constant, the constants being the coordinate values of the point.

Figure 1 shows the points P and Q whose coordinates are (1,2,3) and (2, -2,1), respectively. Point P is therefore located at the common point of intersection of the planes x = 1, y = 2, and z = 3, while point Q is located at the intersection of the planes x 2, y =-2, z = 1.

As we encounter other coordinate systems in this module, we should expect points to be located by the common intersection of three surfaces, not necessarily planes, but still mutually perpendicular at the point of intersection.

If we visualize three planes intersecting at the general point P, whose coordinates are x, y, and z, we may increase each coordinate value by a differential amount and obtain three slightly displaced planes intersecting at point P', whose coordinates are x + dx, y + dy, and z + dz. The six planes define a rectangular parallelepiped whose volume is dv = dxdydz; the surfaces have differential areas dS of dxdy, dydz, and dzdx. Finally, the distance dL from P to P' is the diagonal of the parallelepiped and has a length of √( dx)^2 + (dy)^2 + (dz)^2. The volume element is shown in Fig. 1; point P' is indicated, but point P invisible corner. All this is familiar from trigonometry or solid geometry and as yet involves only scalar quantities. We shall begin to describe vectors in terms of a coordinate system in the next section.

Given vectors Ā and B

Distance of two vectors: RAB = r B – rĀ For Magnitude: B = bxax + byay + bzaz

| |=B √(bx)^2 + (by)^2 + (bz)^ B For Direction: arB=|B|

Coordinate Systems (MIT-CoordinateSystems)

A coordinate system consists of four basic elements: (1) Choice of origin (2) Choice of axes (3) Choice of positive direction for each axis (4) Choice of unit vectors for each axis.

We illustrate these elements below using Cartesian coordinates. (1) Choice of Origin

Choose an origin O. If you are given an object, then your choice of origin may coincide with a special point in the body. For example, you may choose the mid-point of a straight piece of wire.

(2) Choice of Axis

Now we shall choose a set of axes. The simplest set of axes are known as the Cartesian axes, x-axis, y-axis, and the z-axis. Once again, we adapt our choices to the physical object. For example, we select the -axis so that the wire lies on the -axis, as shown in Figure B.1:

Then each point P in space our S can be assigned a triplet of values (xp, yp, zp), the Cartesian coordinates of the point P. The ranges of these values are: −∞ < xp < +∞, −∞ < yp < +∞, −∞ < zp < +∞.

The collection of points that have the same the coordinate yp is called a level surface. Suppose we ask what collection of points in our space S have the same value of P y = y. This is the set of points Syp={(x, y, z) ∈ S such that y=yp}. This set Syp is a plane, the x-z plane (Figure B.1), called a level set for constant yp. Thus, the y-coordinate of any point actually describes a plane of points perpendicular to the y -axis.

Using the given points in no, find the following: a) RBA RBA= rA-rB RBA= (6ax – 4az) – (2ax – 4ay – 2az) RBA= (6 – 2) ax + (0 - -4) ay + (-4 - -2) az RBA= 4ax + 4ay – 2ay

b) RAB RAB= rB-rA RAB= (2ax – 4ay – 2az) – (6ax – 4az) RAB= (2 – 6) ax + (-4 – 0) ay + (-2 - -4) az RAB= -4ax – 4ay + 2az

c) |rA| |rA| |rA|= 7 units

d) aAB

aAB=

aAB=−

= 6 units 6 aAB= -0 - 0 + 0

e) |rA-rB| rA-rB = 4ax + 4ay – 2ay

|r |rA-rB|= 6 units

Vector Components and Unit Vectors (Hayt & Buck, 2001) To describe a vector in the cartesian coordinate system, let us first consider a vector r extending outward from the origin. A logical way to identify this vector is by giving the three component vectors, lying along the three coordinate axes, whose vector sum must be the given vector. If the component vectors of the vector r are x, y, and z, then r=x + y + z. The component vectors are shown in Fig.

1 Instead of one vector, we now have three, but this is a step forward, because the three vectors

are of a very simple nature, each is always directed along one of the coordinate axes.

In other words, the component vectors have a magnitude which depends on the given vector (such as r above), but they each have a known and constant direction. This suggests the use of unit vectors having unit magnitude, by definition, and directed along the coordinate axes in the direction of the increasing coordinate values. We shall reserve the symbol a for a unit vector and identify the direction of the unit vector by an appropriate subscript. Thus ax, ay, and az are the unit vectors in the cartesian coordinate system. They are directed along the x, y, and z axes, respectively, as shown in Fig. 1.

If the component vector y happens to be two units in magnitude and directed toward increasing values of y, we should then write y = 2ay. A vector rp pointing from the origin to point P(1,2,3) is written rp= ax + 2ay + 3az. The vector from P to Q may be obtained by applying the rule of vector addition. This rule shows that the vector from the origin to P plus the vector from P to Q is equal to the vector from the origin to Q. The desired vector from P(1,2,3) to Q(2,-2,1) is therefore

RAB ; |RAB |=

|RAB| 𝐀𝐀𝐀𝐀− 𝐀𝐀𝐀𝐀 +𝐀𝐀𝐀𝐀

√(−4) 2 + (−4) 2 + 22

Solution: We first construct the vector extending from the origin to point G, G= 2ax – 2ay – az

We continue by finding the magnitude of G,

|G|= = 3

and finally expressing the desired unit vector as the quotient, aG= 𝐀 = 𝐀ax - 𝐀ay - 𝐀az = 0 – 0 – 0 |𝐀 | 𝐀 𝐀 𝐀

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Electromags - Chapter 1

Course: Electronics Engineering (CR 061)

95 Documents

Students shared 95 documents in this course

University: Samar State University

Was this document helpful?

LESSON CONTENT

According to Hayt & Buck, Vector analysis is a mathematical shorthand. It has some new

symbols, some new rules, and a pitfall here and there like most new fields, and it demands

concentration, attention, and practice.

 Why do we need to study Vector Analysis?

-According to Yoonchang Jeong, a deficiency in vector analysis in the study of electromagnetics

is similar to a deficiency in algebra and calculus in the study of physics.

( http://ocw.snu.ac.kr/sites/default/files/NOTE/02%20Vector%20Analysis.p d f )

-Vector analysis is a mathematical tool used to explain and predict physical phenomena in the

study of mechanics. A vector is a depiction or symbol showing movement or a force carried from

point A to point B. A vector has properties of both magnitude and direction. A scalar only has the

property of magnitude. (Strain, 2012)

-It is easier for engineers to take a more rigorous and complete course in the mathematics

department after they have been presented with a few physical pictures and applications. (Hayt

& Buck, 2001).

WATCH:

https://www.youtube.com/watch?v=MKKytaEmC G A ( I ntroduction Vector Analysis)

https://www.youtube.com/watch?v=rcbdj Q b - y y E ( V ector Analysis: Introduction to Vector Analysis)

1. Scalars and Vectors

Many of the quantities that physicists use are easy to specify; measurements of mass, length,

time, area, volume and temperature can all be expressed as simple numbers together with

appropriate units of measurement. Such quantities are known as scalar quantities or scalar.

However, some quantities are more tricky to deal with. For instance, if you want to travel from one

place to another you will not only want to know how far apart the two places are, you will also need

to know the direction that leads from one to other. The physical quantity that combines distance and

direction is called displacement and is clearly more complicated than distance alone. Displacement

is a simple example of a large class of physical quantities known collectively as vector quantities or

vectors. (University, 2004)

SOURCE: sciencestruck.com

1.1. Scalars

The term scalar refers to a quantity whose value may be represented by a single (positive or

negative) real number. The x, y, and z we used in basic algebra are scalars and the quantities they

represent are scalars. If we speak of a body falling a distance L in a time t, or the temperature T at

any point in a bowl of soup whose coordinates are x, y, and z, then L, t, T, x, y, and z are all scalars.

Scalars are printed in italic type, for example A. (Hayt & Buck, 2001)

Scalar Quantities: The physical quantities which are specified with the magnitude or size alone

are scalar quantiti e s .

Electromags - Chapter 1

Electronics Engineering (CR 061)

Samar State University

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Preview text

LESSON CONTENT

1 Instead of one vector, we now have three, but this is a step forward, because the three vectors

RAB ; |RAB |=

√(−4) 2 + (−4) 2 + 22

|G|= = 3

Electromags - Chapter 1

Course: Electronics Engineering (CR 061)

University: Samar State University

Students also viewed

Related documents